dissertation_davina_fink.pdf - Universität Stuttgart

Model Reduction applied to

Finite-Element Techniques for the

Solution of Porous-Media Problems

Von der Fakultat Bau- und Umweltingenieurwissenschaften

der Universitat Stuttgart zur Erlangung der Wurde

eines Doktor-Ingenieurs (Dr.-Ing.)

genehmigte Abhandlung

vorgelegt von

Dipl.-Ing. Davina Fink

aus

Kassel

Hauptberichter: Prof. Dr.-Ing. Dr. h. c. Wolfgang Ehlers

Mitberichter: Prof. Dr. Bernard Haasdonk

Prof. Dr.-Ing. Stefanie Reese

Tag der mundlichen Prufung: 9. Juli 2019

Institut fur Mechanik (Bauwesen) der Universitat Stuttgart

Lehrstuhl fur Kontinuumsmechanik

Prof. Dr.-Ing. Dr. h. c. W. Ehlers

2019

Report No. II-37Institut fur Mechanik (Bauwesen)Lehrstuhl fur KontinuumsmechanikUniversitat Stuttgart, Germany, 2019

Editor:

Prof. Dr.-Ing. Dr. h. c. W. Ehlers

c© Davina FinkInstitut fur Mechanik (Bauwesen)Lehrstuhl fur KontinuumsmechanikUniversitat StuttgartPfaffenwaldring 770569 Stuttgart, Germany

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopy-ing, recording, scanning or otherwise, without the permission in writing of the author.

ISBN 978 – 3 – 937399 – 37 – 9(D 93 – Dissertation, Universitat Stuttgart)

Acknowledgements

The work presented in this doctoral thesis was developed during my profession as anassistant lecturer and research associate at the Institute of Applied Mechanics (CivilEngineering), Chair of Continuum Mechanics, at the University of Stuttgart. At thispoint, I want to take the opportunity to gratefully acknowledge numerous people whocontributed in various ways to complete this work.

First of all, I would like to sincerely thank my supervisor Professor Wolfgang Ehlersfor giving me the opportunity to prepare my thesis at his institute under his constantscientific support. I personally appreciate not only his structured way of working andhis broad knowledge but also his well-balanced character. Furthermore, I would like tothank Professor Bernard Haasdonk, not only for evaluating my thesis, but also for mostvaluable discussions concerning the mathematical aspects of this work and for his timeto extensively reading and discussing this thesis. I am also grateful to Professor StefanieReese for taking the third supervision in my dissertation procedure.

Moreover, I would like to thank all my former colleagues at the institute for the great andfriendly atmosphere, creating a pleasant and efficient working environment. Thanks for allthe warm conversations, for the constant willingness to share experiences and knowledgewith me and for the time we spend apart from work. In particular, I would like to thankArndt Wagner for his enthusiasm and his huge personal engagement as teaching assistant,with which he significantly inspired me to head into the field of mechanics. Furthermore, Iam very thankful for all the discussions we had on various topics and for the proofreadingof parts of this work. Equally, I would like to thank my proofreaders Sami Bidier, LukasEurich, Chenyi Luo and Alixa Sonntag, not only for the time to read parts of my thesis,but also for all the pleasurable moments. It was a fortune to have kind and reliableoffice-mates. When I wrote my diploma thesis at the institute, Nils Karajan supervisedand taught me many important routine matters at the institute - many thanks for this.Beyond that, I would like to thank David Koch, Maik Schenke and Patrick Schroder fortheir constant administrative support and for all the discussions about technical and manyother issues, and Christian Bleiler and Mylena Mordhorst for the nice conversations andthe cosy get-together at various conferences and workshops. This, of course, also appliesto my former colleagues Kai Haberle, Said Jamei and Joffrey Mabuma, and all the othersfrom the institute.

My special thanks go to my wonderful parents for always believing and trusting in me.Finally, I would like to thank my beloved husband for his limitless support and encour-agement.

Stuttgart, September 2019 Davina Fink

Contents

Abstract V

Deutschsprachige Zusammenfassung IX

Nomenclature XIII

Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIV

Selected acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX

1 Introduction and overview 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 State of the art, scope and aims . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Continuum-mechanical fundamentals for the modelling of porous media 7

2.1 The concept of the Theory of Porous Media . . . . . . . . . . . . . . . . . 7

2.1.1 Macroscopic modelling approach . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Volume fractions and density functions . . . . . . . . . . . . . . . . 8

2.2 Kinematical relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Motion of a porous material . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Deformation and strain measures . . . . . . . . . . . . . . . . . . . 11

2.3 Balance relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Basic stress measures . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 Master balance principle and specific mechanical balance equations 14

2.4 Modelling approach of specific materials . . . . . . . . . . . . . . . . . . . 16

2.4.1 Biphasic porous-soil model . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 Multiphasic brain-tissue model with application to drug-infusion

processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.3 Extended biphasic intervertebral-disc model . . . . . . . . . . . . . 23

3 Numerical treatment 25

3.1 Finite-element method in space . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Weak formulations and boundary conditions . . . . . . . . . . . . . 25

3.1.2 Spatial discretisation using mixed finite elements . . . . . . . . . . 26

I

II Contents

3.2 Temporal discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Solution procedure of coupled problems . . . . . . . . . . . . . . . . . . . . 29

3.3.1 Porous-soil model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.2 Drug-infusion model for brain tissue . . . . . . . . . . . . . . . . . . 35

3.3.3 Intervertebral-disc model . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Model-reduction methods 43

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Model reduction via proper orthogonal decomposition . . . . . . . . . . . . 47

4.2.1 Fundamentals of the POD method . . . . . . . . . . . . . . . . . . 47

4.2.2 Modified POD method . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.3 Error bounds and error estimation for efficient reduced-order models 51

4.2.4 Selection of snapshots . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Model reduction of nonlinear systems . . . . . . . . . . . . . . . . . . . . . 58

4.3.1 Discrete-empirical-interpolation method . . . . . . . . . . . . . . . 58

5 Numerical examples with application to selected porous materials 63

5.1 Application of the POD approach to porous-soil models with linear systems

of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1.1 Quasi-static 2-d porous-soil model . . . . . . . . . . . . . . . . . . . 64

5.1.2 Dynamic porous-soil model . . . . . . . . . . . . . . . . . . . . . . . 70

5.1.3 Quasi-static 3-d porous-soil model . . . . . . . . . . . . . . . . . . . 74

5.2 Application of the POD-DEIM to a biphasic model undergoing large de-

formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.1 Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2.2 Reduced-order system and numerical results . . . . . . . . . . . . . 82

5.3 Reduced simulations of drug-infusion processes within a brain-tissue model 88

5.3.1 Application of the PODmethod for the simplified brain-tissue model

with linear system of equations . . . . . . . . . . . . . . . . . . . . 88

5.3.2 Application of the POD-DEIM approach for the general brain-tissue

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4 Application of the POD-DEIM approach to an intervertebral-disc model . 99

5.4.1 Reduced-order system . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.2 Problem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.3 Complexity of the snapshot choice . . . . . . . . . . . . . . . . . . . 102

5.4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.5 Generalised approach for an application-driven model reduction . . . . . . 113

Contents III

5.5.1 From a full-order system towards time-efficient simulations on the

basis of a reduced model . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Summary and outlook 117

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

A Selected relations of tensor calculus 121

A.1 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

A.2 Tensor analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

B Specific derivation of the overall systems of equations in abstract for-

mulation 127

B.1 Overall system of a quasi-static biphasic model of a porous material . . . . 127

B.2 Overall system of the dynamic porous-soil model . . . . . . . . . . . . . . . 130

B.3 Overall system of the simplified drug-infusion model for brain tissue . . . . 136

C Specific derivation of the reduced systems in abstract formulation 141

C.1 Reduced formulation of the quasi-static porous-soil model with linear sys-

tem of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.2 Reduced system of the dynamic porous-soil model . . . . . . . . . . . . . . 143

C.3 Reduced system of a nonlinear biphasic model of a porous material . . . . 145

C.4 Reduced system of the different drug-infusion models for brain tissue . . . 146

C.4.1 Simplified brain-tissue model with linear equation system . . . . . . 146

C.4.2 General brain-tissue model with nonlinear equation system . . . . . 147

Bibliography 149

Curriculum Vitae 167

Abstract

Computational simulations have a tremendous impact on a growing number of scientificfields. In the course of time, increasingly precise predictions could be accomplished bymeans of simulation results. Furthermore, a wide variety of practical tests and exten-sive measurements could be substituted by simulations. In addition, simulations providenew possibilities in the modern medicine. In particular, they enable to gain a deeperunderstanding of the complex processes in the human body or even to contribute to thesuccessful planning of a surgical intervention by providing supplementary process infor-mation. The increasing importance of computational simulations requires for a high levelof trustworthiness of the simulation results. In order to meet these requirements, sim-ulations based on sophisticated models and with a sufficiently fine discretisation of thegeometry are indispensable for the numerical realisation. Particularly when taking intoaccount the high structural complexity of the underlying materials, a detailed knowledgeabout the inner structure and the composition of the materials of various componentsis essential for a sufficiently accurate modelling. A broad variety of materials cannot bedescribed with classical continuum-mechanical models restricted to singlephasic (homo-geneous) materials. This applies in particular to materials with a porous micro-structure.The group of porous media, consisting of a porous soil whose pore space is filled withfluids and/or gases, includes, amongst others, partially saturated soils and biological tis-sue aggregates. Hence, a multiphasic and multicomponent modelling approach on thebasis of the Theory of Porous Media (TPM) appears suitable in order to describe thesecomplex materials. Concerning the numerical treatment of porous materials, the finite-element method (FEM) has been proven to be a well-suited technique for the solutionof arbitrary initial-boundary-value problems. The resulting simulations are able to de-scribe the physical phenomena by repeatedly solving the descriptive set of coupled partialdifferential equations (PDE). However, the necessarily high accuracy of the approxima-tion results and the complexity of the underlying models result in many applications inan extremely high dimension of the resulting equation system. The consequences aretime-consuming simulations, which are either too slow to satisfy the time constraintsand to enable practical applications (such as an accompanying use in clinical practice)or which cannot be performed as often as needed due to the high computation time.In order to counteract these problems, to increase the solution speed and to reduce thecomputational expenses, model-reduction methods are increasingly important and aregaining a considerable scientific interest. While, on the one hand, the detailed theoreti-cal basis of the modelling approach needs to be maintained and, on the other hand, anefficient numerical computation should be provided, the available performance capacityof projection-based model-reduction techniques can be used to provide fast simulationswhenever they are actually needed. Supported by the steadily growing potential of com-puting power and storage capacities, time-consuming simulations based on models withall their complexity can already be performed and all necessary information and datacan be stored beforehand in more and more application areas. Following the concept ofoffline/online decomposition, a possibly time-consuming offline phase (including the sim-

V

VI Abstract

ulations performed in advance and the reduction of the underlying system of equations)can be separated from a time-efficient online phase with fast simulations (performed usingthe reduced system). A computationally intensive offline phase pays off by the possibilityto rapidly produce required simulation results in daily routines or by a sufficient num-ber of individual computationally inexpensive simulations with varying material and/orsimulation parameters.

In the present contribution, the developments in the modelling and the simulation ofporous materials in the framework of the well-founded TPM are combined with the cur-rent state of research in the field of model reduction using projection-based methods.In terms of the continuum-mechanical modelling, this work is focusing on problems de-veloped on the basis of detailed and thermodynamically consistent TPM models. Earlystudies concentrated on the multiphasic and multicomponent modelling of porous mediawith application of the TPM have shown the outstanding suitability of this modellingapproach, cf., e. g., de Boer [15, 16] or Ehlers [39, 42, 43]. The present work makes use ofa biphasic model for the simulation of a saturated porous soil (cf. Ehlers [40], Ehlers &Eipper [47], Eipper [54], Ellsiepen [55]), a multiphasic and multicomponent model for thesimulation of drug-infusion processes in brain tissue (cf. Ehlers & Wagner [51, 52], Wagner[127], Wagner & Ehlers [128]) and an extended biphasic model for the description of aninhomogeneous and anisotropic intervertebral disc (cf. Ehlers et al. [49, 50], Karajan [81]).The continuum-mechanical fundamentals of the TPM, required for the description of thesemodels, are outlined in Chapter 2. Therefore, the TPM is introduced, all necessary kine-matical relations are provided and the balance relations are presented. Furthermore, thegeneral continuum-mechanical fundamentals are specified for the models used in this work.A convenient technique for the solution of arbitrary initial-boundary-value problems is theFEM, cf. Lewis & Schrefler [91], which is used in this contribution for the numerical treat-ment of the TPM models. Starting from the weak forms of the governing equations, thespatial and temporal discretisation strategies are described in Chapter 3. In this re-gard, a reduction of the descriptive set of (strongly) coupled partial differential equationsprovides an enormous benefit to significantly reduce the dimension of these systems and,thus, the computation time and the numerical effort of the FE simulations. Particularlywith regard to nonlinear systems, the computational effort is usually immense as high-dimensional equation systems need to be solved repeatedly for the determination of thenonlinearities. Following this, a suitable reduction of these systems essentially improvesthe efficiency by solving only a subset of equations of the original model. Under considera-tion of these circumstances, efficient reduced models for the simulation of different porousmaterials are provided in the present work by an application-driven approach. Thereby,only model-reduction techniques applied to the monolithic solution of the strongly coupledequation systems are considered. The applied model-reduction techniques are explainedin detail in Chapter 4. In particular, projection-based model-reduction techniques areused to transform a high-dimensional system to a low-dimensional subspace. The advan-tage of such an approach is to maintain the detailed theoretical basis of the modellingprocess while an efficient numerical computation is provided. In this contribution, themethod of proper orthogonal decomposition (POD) is used as a starting point for themodel reduction. The development of the POD method, also known as Karhunen-Loeveexpansion, cf. Sirovich [115], traces back to fluid-dynamic applications including turbu-

Abstract VII

lence, cf. Berkooz et al. [10]. Beyond that, the POD method was successfully applied tovarious problems in fluid flow (cf. Kunisch & Volkwein [88], Rowley et al. [110]), optimalcontrol (cf. Kunisch & Volkwein [86]), aerodynamics (cf. Bui-Thanh et al. [24], Hall et al.[73]), biomechanics (cf. Radermacher & Reese [102]) and structural mechanics (cf. Herktet al. [76], Radermacher & Reese [103]). However, since the POD-Galerkin approximationdoes in fact significantly reduce the dimension of the equation system but not the effortto evaluate the nonlinear terms, the computational effort of nonlinear problems cannot be(sufficiently) reduced when exclusively using the POD method. This drawback motivatesthe application of additional methods for the reduction of the nonlinear terms. Withinthe scope of this work, the discrete-empirical-interpolation method (DEIM), which is thediscrete variant of the empirical-interpolation method (EIM, cf. Barrault et al. [7]) andwhich was introduced by Chaturantabut & Sorensen [32], is used in combination with thePOD method to reduce arising nonlinearities. In the works of Kellems et al. [85] for amodel of spiking neurons, in Chaturantabut & Sorensen [34] for a model with applicationto non-linear miscible viscous fingering, in Nguyen et al. [98] for reacting flow applications,in Negri et al. [97] for parametrised systems or in Bonomi et al. [19] for the application toparametrised problems in cardiac mechanics, amongst others, it could be shown that theDEIM is able to significantly reduce the numerical effort of complex nonlinear processes.The high complexity of the underlying multiphasic and multicomponent modelling of thetreated materials and the resultant strongly coupled equation systems require for indi-vidual adaptations and modifications of the used reduction methods to achieve satisfyingresults. Therefore, the scope of this monograph is the development of an application-driven approach for providing reduced models, which are capable of simulating specificporous materials in a time-efficient manner. The necessary modifications are discussedin detail in this work and are additionally illustrated with examples in Chapter 5. Inthis regard, an in-depth knowledge of the form and the characteristics of the underlyingequation system is essential and is therefore treated intensively. For example, it shouldbe ensured that the block structure of the coupled equation systems is preserved whileconsidering the different temporal (physical) behaviour of the primary variables. Sincethe outlined modifications might be of great interest for other applications, a generalisedapproach for an adaptation to other models is finally presented.

Deutschsprachige Zusammenfassung

Rechnerbasierte Simulationen sind heutzutage aus immer mehr Wissenschaftsfeldern nichtmehr wegzudenken. So konnten im Laufe der Zeit zunehmend prazisere Vorhersagenaus Simulationsergebnissen gewonnen und eine Vielzahl von praktisch durchgefuhrtenTests und aufwandigen Messungen durch numerische Simulationen ersetzt werden. Zudemeroffnen Simulationen auch in der modernen Medizin neue Moglichkeiten, ein erweitertesVerstandnis der komplexen Vorgange im menschlichen Korper zu erhalten oder gar me-dizinische Eingriffe in der Praxis durch Simulationen zu unterstutzen. Die zunehmendeBedeutung rechnerbasierter Simulationen verlangt eine hohe Vertrauenswurdigkeit in dieSimulationsergebnisse. Um diesen Anforderungen gerecht zu werden, sind komplexe Mo-delle als Basis der Simulationen sowie eine ausreichend feine Diskretisierung der Geometriebei der numerischen Umsetzung unabdingbar. Insbesondere bei Materialien mit einer ho-hen struktureller Komplexitat ist eine Kenntis uber die vorliegende innere Struktur unddie Zusammensetzung der Materialien aus verschiedenen Bestandteilen fur eine ausrei-chend genaue Modellbildung unverzichtbar. Es existieren eine Vielzahl von Materialien,die sich nicht mit den klassischen Methoden der Kontinuumsmechanik fur ein einzelneshomogenes Material beschreiben lassen. Dies gilt insbesondere fur Materialien mit einerporoser Mikrostruktur. Zu der Gruppe der porosen Medien, bestehend aus einem porosenFestkorper, dessen Porenraume mit Flussigkeiten und/oder Gasen gefullt sind, gehorengesattigte Boden ebenso wie biologisches Gewebe. Zur Modellbildung dieser komplexenMaterialien hat sich eine mehrphasige und mehrkomponentige Beschreibung auf Basis derTheorie Poroser Medien (TPM) bewahrt. Des Weiteren konnte im Hinblick auf die nume-rische Simulation solcher poroser Materialien in einer Vielzahl von Arbeiten die Eignungder Finite-Elemente-Methode (FEM) als numerische Approximationsmethode dargelegtwerden. Die sich daraus ergebenden Simulationen konnen physikalische Phanomene be-schreiben, indem wiederholt das beschreibenden Gleichungssystem aus gekoppelten par-tiellen Differentialgleichungen gelost wird. Die Komplexitat der zugrundeliegenden Mo-delle und die notwendig hohe Genauigkeit der Approximationsergebnisse fuhrt jedoch invielen Anwendungen zu einer extrem hohen Dimension des resultierenden diskreten Glei-chungssystems. Die Folge sind rechenintensive Simulationen, die in der Regel entwederzu hohe Rechenzeiten aufweisen um damit die zeitlichen Beschrankungen zu erfullen undpraktische Anwendungen (wie den unterstutzenden Einsatz bei medizinischen Eingriffenim Klinikalltag) zu ermoglichen oder aufgrund der hohen Rechenkosten nicht ausreichendhaufig durchgefuhrt werden konnen. Dieser Problematik kann man entgegenwirken, indemman durch den Einsatz geeigneter Reduktionsmethoden die sich ergebenden hohen Re-chenzeiten und Rechenkosten numerischer Simulationen deutlich verringert. Folglich ist esnicht verwunderlich, dass Methoden der Modellreduktion in den letzten Jahren eine starkzunehmende Bedeutung und ein bemerkenswertes wissenschaftliches Interesse gewonnenhaben. Dabei ermoglichen projektionsbasierte Reduktionsmethoden, dass einerseits diekomplexen theoretischen Grundlagen der Modellbildung beibehalten werden und ande-rerseits bei Bedarf zeiteffiziente numerische Berechnungen durchgefuhrt werden konnen.Gestutzt durch das standig wachsende Potential an Rechenleistungen und Speicherka-

IX

X Deutschsprachige Zusammenfassung

pazitaten konnen in immer mehr Anwendungsbereichen rechenaufwandige Simulationen,basierend auf Modellen mit all ihrer Komplexitat, bereits im Vorfeld durchgefuhrt undalle notwendigen Informationen und Datensatze abgelegt werden. Die Basis fur die Ef-fizienz des Reduktionsprozesses stellt eine Offline/Online-Zerlegung dar. Bei dieser kanneine meist rechenintensive Offline-Phase, bestehend aus vorab durchgefuhrte Simulatio-nen und der Reduktion des zugrundeliegenden Gleichungssystems, von einer zeiteffizientenOnline-Phase (Simulationen basierend auf dem reduzierten Gleichungssystem) entkoppeltwerden. Eine teure Offline-Phase zahlt sich dadurch aus, dass man die Moglichkeit hat,im Alltag schnell Simulationsergebnisse erzeugen zu konnen. Zudem ergibt sich bei einergenugend hohen Anzahl an Einzelsimulationen, die mit wechselnden Parametern durch-gefuhrt werden, eine deutliche Reduktion der Rechenzeiten und Rechenkosten.

Ziel dieser Arbeit ist es, die Fortschritte in der Modellbildung und Simulation poroser Ma-terialien auf Basis der TPM mit dem aktuellen Stand der Forschung zur Modellreduktionunter Verwendung projektionsbasierter Methoden zu kombinieren. Hinsichtlich der konti-nuumsmechanischen Modellbildung werden im Rahmen dieser Arbeit Probleme behandelt,die auf Grundlage detaillierter thermodynamisch konsistenter Modelle unter Anwendungder TPM entwickelt wurden. Fruhe Arbeiten zur mehrphasigen und mehrkomponentigenBeschreibung poroser Medien auf Basis der TPM zeigen die Eignung dieses Modellan-satzes (de Boer [15, 16], Ehlers [39, 42, 43]). Die vorliegende Arbeit macht Gebrauch voneinem Zweiphasenmodell zur Beschreibung gesattigter Boden (Ehlers [40], Ehlers & Eipper[47], Eipper [54]), einem mehrphasigen und mehrkomponentigen Modell zur Simulationvon Infusionsprozessen in Gehirngewebe (Ehlers & Wagner [51, 52], Wagner [127], Wagner& Ehlers [128]) und einem erweiterten Zweiphasenmodell zur Beschreibung inhomogenerund anisotroper Bandscheiben (Ehlers et al. [49, 50], Karajan [81]). Die zur Beschrei-bung der zugrundeliegenden Modelle notwendigen kontinuumsmechanischen Grundlagender TPM werden in Kapitel 2 erlautert. Dabei werden neben einer Einfuhrung der TPMdie notwendigen kinematischen Beziehungen eingefuhrt und die benotigten Bilanzglei-chungen zusammengestellt. Daruber hinaus werden die allgemeinen kontinuumsmechani-schen Grundlagen fur die in dieser Arbeit verwendeten Modelle spezifiziert. Eine effizienteMoglichkeit zur Behandlung beliebiger Anfangs-Randwertprobleme bietet die FEM (Lewis& Schrefler [91]), mit deren Hilfe die TPM-Modelle in dieser Arbeit numerisch diskretisiertwerden. Ausgehend von den schwachen Formen der Bilanzgleichungen folgt in Kapitel 3eine Orts- und Zeitdiskretisierung der Bestimmungsgleichungen. Eine Reduktion der be-schreibenden Gleichungssysteme aus (stark) gekoppelten partiellen Differentialgleichung-en ist von großem Nutzen um die Dimension der entsprechenden diskretisierten Systemeund damit einhergehend die Rechenzeiten und Rechenkosten der numerischen Simulatio-nen signifikant zu reduzieren. Insbesondere bei nichtlinearen Systemen ist der Rechenauf-wand in der Regel immens, da zur Bestimmung der Nichtlinearitaten wiederholt hoch-dimensionale Gleichungssysteme gelost werden mussen. Eine geeignete Reduktion dieserSysteme ermoglicht es, dass nur eine kleine Teilmenge von Gleichungen gelost werdenmuss. Unter Berucksichtigung der oben genannten Sachverhalte werden in der vorliegen-den Arbeit durch ein anwendungsorientiertes Vorgehen effiziente reduzierte Modelle derverschiedenen porosen Materialien erarbeitet. Dabei wird sich auf Reduktionsmethodenbeschrankt, die auf die monolithische Losung des stark gekoppelten Gleichungssystemsder TPM-Modelle angewandt werden. Diese werden in Kapitel 4 naher erlautert. Kon-

Deutschsprachige Zusammenfassung XI

kret werden projektions-basierte Reduktionsmethoden genutzt, um ein hochdimensionalesSystem auf einen niedrigdimensionalen Unterraum zu transformieren. Wahrend auf dieseWeise die detaillierten theoretischen Grundlagen des Systems erhalten bleiben, ist es invielen Anwendungen moglich, den Rechenaufwand der numerischen Simulationen deutlichzu reduzieren. Die Reduktion der hochdimensionalen Gleichungssysteme erfolgt in dieserArbeit mittels der POD-Methode (method of proper orthogonal decomposition). Erste Ar-beiten zur Entwicklung dieser Methode (auch bekannt als Karhunen-Loeve-Zerlegung,siehe Sirovich [115]) sind bei fluiddynamischen Anwendungen unter Einbeziehung vonTurbulenzen zu finden, siehe Berkooz et al. [10]. Des Weiteren wurde die POD-Methodeerfolgreich bei einer Vielzahl von Problemen im Bereich von Fluidstromungen (Kunisch& Volkwein [88], Rowley et al. [110]), optimaler Steuerung (Kunisch & Volkwein [86]),Aerodynamik (Bui-Thanh et al. [24], Hall et al. [73]), Biomechanik (Radermacher & Reese[102]) und Strukturmechanik (Herkt et al. [76], Radermacher & Reese [103]) angewandt.Da die POD-Galerkin-Approximation zwar die Dimension des zu losenden Gleichungs-systems deutlich reduziert, nicht jedoch den Aufwand zur Bestimmung der nichtlinearenAnteile, kann der Rechenaufwand bei nichtlinearen Gleichungssystemen bei alleiniger Ver-wendung der POD-Methode in der Regel (wenn uberhaupt) nicht ausreichend reduziertwerden. Dies motiviert bei nichtlinearen Problemen den Einsatz erganzender Methoden.Im Rahmen dieser Arbeit wird zur Reduktion solcher nichtlinearer Terme die DEIM(discrete-empirical-interpolation method, siehe Chaturantabut & Sorensen [32]), die ei-ne diskrete Variante der EIM (empirical-interpolation method, siehe Barrault et al. [7])darstellt, verwendet. So konnte unter anderem in den Arbeiten von Kellems et al. [85] beider Anwendung auf ein Neuronen-Modell, in Chaturantabut & Sorensen [34] bei der Be-schreibung spezieller Effekte (viscous fingering) bei der Modellierung nichtlinearer viskoserStromungen, in Nguyen et al. [98] bei der Anwendung auf ein Modell zur Beschreibungreaktiver Stromungen, in Negri et al. [97] bei der Reduktion von parametrisierten Syste-men oder in Bonomi et al. [19] bei der Anwendung auf parametrisierte Probleme bei derModellierung von Kontraktionen des Herzens gezeigt werden, dass die DEIM in der Lageist, den numerischen Aufwand komplexer nichtlinearer Prozesse deutlich zu reduzieren.Die hohe Komplexitat der zugrundeliegenden mehrphasigen und mehrkomponentigen Be-schreibung der betrachteten Materialien und der sich daraus ergebenden stark gekoppeltenGleichungssysteme erfordert individuelle Anpassungen und Modifikationen der verwende-ten Reduktionsmethoden, um zufriedenstellende Ergebnisse zu erhalten. Daraus resultiertein anwendungsorientiertes Vorgehen zur Bereitstellung reduzierter Modelle, welches einezeiteffiziente Simulation der spezifischen Anwendungsbeispiele ermoglicht. Die notwendi-gen Modifikationen werden in dieser Arbeit ausfuhrlich erlautert und in Kapitel 5 mitentsprechenden numerischen Beispielen unterlegt. Da fur ein solches Vorgehen die Formund die Eigenschaften der zugrundeliegenden Gleichungssysteme eine wesentliche Rollespielen, wird zudem umfassend auf diese eingegangen. So sollte zum einen die sich erge-bende Block-Struktur der gekoppelten Gleichungssysteme beibehalten werden, und zumanderen das physikalische (zeitlich stark differenzierte) Verhalten der Primarvariablenberucksichtigt werden um numerisch effiziente Simulationsergebnisse zu erhalten. Da diein dieser Monographie erarbeiteten Modifikationen auch fur andere Anwendungen von ho-hem Interesse sein konnen, wird außerdem auf ein generalisiertes Vorgehen zur Adaptionauf anderer Modelle eingegangen.

Nomenclature

As far as possible, the notation in this monograph follows the common conventions ofmodern tensor calculus, such as in de Boer [14] or Ehlers [41]. The particular symbolsused in the context of porous-media theories are chosen according to the establishednomenclature given by de Boer [16] and Ehlers [39, 43].

Conventions

General conventions

( · ) placeholder for arbitrary quantities

δ( · ) test functions of primary unknowns

d( · ) or ∂( · ) differential or partial derivative operator

a, b, . . . or φ, ψ, . . . scalars (zero-order tensors)

a, b, . . . or φ, ψ, . . . vectors (first-order tensors)

A, B, . . . or Φ,Ψ, . . . second-order tensorsn

A,n

B, . . . orn

Φ,n

Ψ, . . . nth- or higher-order tensors

a, b, . . . or A, B, . . . general column vectors (n× 1) and matrices (n×m)

Index and suffix conventions

i, j, k, l, . . . indices

( · )α subscripts indicate kinematical quantities of a constituent ϕα

within porous-media or mixture theories

( · )α superscripts indicate non-kinematical quantities of a con-stituent ϕα within porous-media or mixture theories

( · )(·)0α initial values of non-kinematical quantities with respect to

the referential configuration of a constituent ϕα

( · )k( · )k =

∑

k ( · )k( · )k Einstein’s summation convention yields a summation over

indices that appear twice unless stated otherwise·

( · ) = d( · )/dt total time derivatives with respect to the overall aggregate ϕ

( · )′α = dα( · )/dt material time derivatives following the motion of a con-stituent ϕα

XIII

XIV Nomenclature

Symbols

Greek letters

Symbol Unit Description

α constituent identifier in super- and subscript, e. g., α = S, FαB [ N/m2 ] material parameter of a blood constituent ϕB

β identifier for the pore liquids in super- and subscript

βB [ - ] material parameter of a blood constituent ϕB

γβR [ N/m3 ] effective weight of a liquid constituent ϕβ

γS1 [ - ] parameter of the anisotropic part of the solid strain energy

Γ , Γ(·) domain boundary and Dirichlet and Neumann boundaries

∆π [ N/m2 ] osmotic pressure difference

ε, εα [ J/kg ] mass-specific internal energy of ϕ and ϕα

εα [ J/m3 s ] volume-specific direct energy production of ϕα

εbound [ - ] (estimated) bound for the error

εNRMS [ - ] normalised root-mean-square error

εtol pre-defined tolerance

ζα [ J/Km3 s ] volume-specific direct entropy production of ϕα

η, ηα [ J/Kkg ] mass-specific entropy of ϕ and ϕα

η, ηα [ J/Km3 s ] volume-specific total entropy production of ϕ and ϕα

θ, θα [ K ] absolute Kelvin’s temperature of ϕ and ϕα

ϑs identifier for the primary variables

κ [ - ] exponent governing the deformation dependency of KS

λr eigenvalues of the Gramian matrix C

λS0 [ N/m2 ] first Lame constant of ϕS

µβ, µβR [ N s/m2 ] partial and effective dynamic viscosity of ϕβ

µS0 [ N/m2 ] second Lame constant of ϕS

µS1 [ N/m2 ] parameter of the anisotropic part of the solid strain energy

ρ [ kg/m3 ] density of the overall aggregate ϕ

ρα, ραR [ kg/m3 ] partial and effective (realistic) density of ϕα

ρα [ kg/m3 s ] volume-specific mass production term of ϕα

σ, σα scalar-valued supply terms of mechanical quantities

ση, σαη volume-specific external entropy supply of ϕ and ϕα

ςi identifier for a group of similar primary variables

ϕ, ϕα entire aggregate model and particular constituent

φS0 [ ] half of the fibre angle between two fibres

Ψ , Ψα [ ·/m3 ] volume-specific densities of scalar mechanical quantities

Nomenclature XV

Ψ , Ψα [ ·/m3 ] volume-specific productions of scalar mechanical quantities

Ω , ∂Ω spatial domain and boundary of the aggregate body BΩe, Ω

h finite element and discretised finite-element domain

ε vector-valued error between the full and the reduced system

µ parameter configuration

ξi local coordinate system of a referential finite element

σ, σα vector-valued supply terms of mechanical quantities

ϕr basis vectors of the subspace V l

φ, φα vector-valued efflux terms of mechanical quantities

φη, φαη [ J/Km2 s ] entropy efflux vector of ϕ and ϕα

χα, χ−1α motion and inverse motion functions of the constituents ϕα

ψr basis vectors of the subspace Vk

Ψ, Ψα [ ·/m3 ] volume-specific densities of vectorial mechanical quantities

Ψ, Ψα [ ·/m3 ] volume-specific productions of vectorial mechanical quantities

εS [ - ] linear solid strain tensor

ξ coefficient matrix used in the DEIM approach

τ , τα [ N/m2 ] Kirchhoff stress tensor of ϕ and ϕα

Υ subspace matrix used for a Petrov-Galerkin projection

Φ, Φα general tensor-valued mechanical quantities

Φu subspace matrix used in the POD approach

Ψw subspace matrix used in the DEIM approach

Latin letters


cDm [mol/m3 ] molar concentration of a therapeutic agent ϕD

da, daα [m2 ] actual area element of ϕ and ϕα

dmα [ kg ] local mass element of ϕα

dt [ s ] time increment

dv, dvα [m3 ] actual volume element of ϕ and ϕα

dVα [m3 ] reference volume element of ϕα

eα [ J/m3 s ] volume-specific total energy production of ϕα

D [mol/m2 s ] area-specific therapeutic efflux of ϕD over the boundary ΓD

Jα [ - ] Jacobian determinant of ϕα

k [ - ] number of DEIM (or magic) points

kF [m/s ] Darcy flow coefficient (hydraulic conductivity)

KS [m2 ] intrinsic (deformation-dependent) permeability

l [ - ] reduced number of degrees of freedom

XVI Nomenclature

m [ - ] number of snapshots of the state variables

n, ne [ - ] number of nodal points for Ωh and Ωe

nα [ - ] volume fractions of a constituent ϕα

N [ - ] number of degrees of freedom

N j(·) [ - ] global basis function of a degree of freedom

p, pαR [ N/m2 ] overall pore pressure and liquid pore pressures

pdif [ N/m2 ] differential pressure of the pore liquids

pr [ - ] interpolation indices

PE [ - ] projection error

q [m/s ] volume efflux of the fluid over the boundary Γq

Q [m3/s ] application rate during the CED application

r, rα [ J/kg s ] mass-specific external heat supply (radiation) of ϕ and ϕα

sβ [ - ] saturation function of the pore liquids ϕβ

t, tn−1, tn [ s ] actual time and temporally discretised time steps

T [ s ] specific (final) time

vβ [m3/m2 s ] area-specific volume efflux of ϕβ over the boundary Γvβ

V , Vα [m3 ] overall volume of B and partial volume of Bα

w [ - ] number of Newton steps

111 vector of all ones

a0S [ - ] unit vector pointing in the fibre direction

aS [ - ] fibre direction within the actual configuration

b, bα [m/s2 ] mass-specific body force vector

dα [m/s ] diffusion velocity vector of ϕα

da [m2 ] oriented actual area element

dAα [m2 ] oriented reference area element of ϕα

dx [m ] actual line element

dXα [m ] reference line element of the constituent ϕα

ei [ - ] (Cartesian) basis of orthonormal vectors

f [ N ] total volume force vector

fα [ N ] volume force vector acting on Pα from a distance

g [m/s2 ] constant gravitation vector

hα [ N/m2 ] volume-specific total angular momentum production of ϕα

kα, kαc , k

αv [ N ] total, contact and volume force element of ϕα

mα [ N/m2 ] volume-specific direct angular momentum production of ϕα

n [ - ] outward-oriented unit surface normal vector

pα [ N/m3 ] volume-specific direct momentum production of ϕα

q, qα [ J/m2 s ] heat influx vector of ϕ and ϕα

Nomenclature XVII

r [m ] level arm of the surface traction with regard to the COG

sα [ N/m3 ] volume-specific total momentum production of ϕα

tα [ N/m2 ] surface traction vector of ϕα

t [ N/m2 ] external load vector acting on the boundary Γt

tF [ N/m2 ] external load vector of the fluid acting on the boundary ΓtF

u vector containing the primary variables

uS [m ] solid displacement vector

vF , vS [m/s ] fluid and solid velocity vector

wβ [m/s ] seepage velocity vector of ϕβ

x [m ] actual position vector of ϕ

Xα = x0α [m ] reference position vector of Pα

x, x′α [m/s ] velocity vector of the aggregate ϕ and the constituent ϕα

x, x′′α [m/s2 ] acceleration vector of the aggregate ϕ and the constituent ϕα

Aα [ - ] Almansian strain tensor of ϕα

Bα [ - ] left Cauchy-Green deformation tensor of ϕα

Cα [ - ] right Cauchy-Green deformation tensor of ϕα

Dα [ ·/s ] symmetric deformation velocity tensor of ϕα

DD [m2/s ] diffusion tensor of a therapeutic agent ϕD

Eα [ - ] Green-Lagrangean strain tensor of ϕα

Fα [ - ] material deformation gradient of ϕα

I [ - ] identity tensor (second-order fundamental tensor)

I [ - ] matrix consisting of identity matrices

Kα [ - ] Karni-Reiner strain tensor of ϕα

Kβ [m/s ] Darcy permeability (or hydraulic conductivity) tensor of ϕβ

Kβspec [m4/N s ] specific permeability tensor of ϕβ

KSβ [m2 ] intrinsic permeability tensor of ϕβ

L, Lα [ ·/s ] spatial velocity gradient of ϕ and ϕα

M [ Nm ] moment vector (mechanical reaction to the surface traction)

NNNϑsabstract matrix representing the global basis functions of ϑs

P, Pα [ N/m2 ] first Piola-Kirchhoff (or nominal) stress tensor of ϕ and ϕα

R [ N ] force vector (mechanical reaction to the surface traction)

Rα [ - ] proper orthogonal rotation tensor of the polar decomp. of Fα

S, Sα [ N/m2 ] second Piola-Kirchhoff stress tensor of ϕ and ϕα

T, Tα [ N/m2 ] overall and partial Cauchy (true) stress tensor of ϕ and ϕα

TαE [ N/m2 ] partial Cauchy extra stress tensor of ϕα

TSE, aniso [ N/m2 ] anisotropic contribution to the solid’s extra stress TS

E

TSE, iso [ N/m2 ] isotropic contribution to the solid’s extra stress TS

E

XVIII Nomenclature

TSE,mech [ N/m2 ] purely mechanical part of the solid’s extra stress TS

E

TSosm [ N/m2 ] osmotic part of the solid’s extra stress TS

E

Uα, Vα [ - ] right and left stretch tensors of the polar decomposition of Fα

Wα [ ·/s ] skew-symmetric spin tensor of ϕα

4

De [ N/m2 ] fourth-order elasticity tensor

Calligraphic letters


B, Bα aggregate body and body of the constituent ϕα

O origin of a coordinate system

P [ N/m2 ] hydraulic pore pressure

P, Pα material points of ϕ and ϕα

S, Sα surface of the overall and the constituent body

Su trial spaces of the primary variables

Tu test spaces of the primary variables

V, V l Hilbert space and l-dimensional subspace

b vector containing the body-force terms

c coefficient vector used in the DEIM approach

∆ykn vector of stage increments at the current Newton step k

f , fext generalised force vector and generalised external force vector

f reduced force vector

F vector containing the global and local system of equations

GGG(·), GGGh(·) abstract function vectors containing the weak forms

k generalised stiffness vector

k reduced stiffness vector

LLLq, LLLhq vector-valued operators containing the evolution equations

q discrete vector containing the history variables

r residual vector

rev vector containing the non-differential terms of LLLhq

u discrete vector containing the nodal unknowns of each dof

u approximation of the vector of unknowns u

ured reduced vector of unknowns

vr eigenvectors of the Gramian matrix C

w vector containing the nonlinear terms of the equation system

w approximation of the nonlinear-terms vector w

y abstract vector containing the all variables of q and u

z current set of arguments used in the Newton solver

Nomenclature XIX

A matrix made up of unit matrices resulting from the terms in q

C Gramian matrix

D generalised system (damping) matrix

D reduced system (damping) matrix

G weighting matrix

Jkn global residual tangent at the current Newton step k

K generalised stiffness matrix

K reduced stiffness matrix

N ju

global basis functions at a nodal point Pj in a finite element

P matrix containing information on the DEIM (or magic) points

Rkn nonlinear functional at the current Newton step k

U snapshot matrix containing snapshots of the state variables

Uϑspart of the snapshot matrix U allocated to ϑs

W snapshot matrix containing snapshots of the nonlinear terms

Wϑspart of the snapshot matrix W allocated to ϑs

Selected acronyms

Symbol Description

2-d two-dimensional

3-d three-dimensional

dof degree of freedom

AF anulus fibrosus

CED convection-enhanced (drug) delivery

COG centre of gravity

DAE differential-algebraic equations

DEIM discrete-empirical-interpolation method

DIRK diagonally implicit Runge-Kutta

DOF degrees of freedom

DDS discrete deformation states

ECM extracellular matrix

EIM empirical-interpolation method

FE finite element

FEM finite-element method

GNAT Gauss-Newton with approximated tensors

HR hyper reduction

HTA hierarchical tensor approximation

XX Nomenclature

IVD intervertebral disc

LBB Ladyshenskaya-Babuska-Brezzi

LDEIM localised discrete-empirical-interpolation method

MBS multi-body system

MDEIM matrix-discrete-empirical-interpolation method

MPE missing-point estimation

NP nucleus pulposus

NRMS normalised root mean square

ODE ordinary differential equations

PDE partial differential equations

POD proper orthogonal decomposition

PVL Pade via Lanczos

REV representative elementary volume

SVD singular-value decomposition

TM Theory of Mixtures

TPM Theory of Porous Media

TPWL trajectory piecewise linear

UDEIM unassembled discrete-empirical-interpolation method

Chapter 1:Introduction and overview

1.1 Motivation

Due to progressive technological development, computational simulations have a tremen-dous impact on an increasing number of scientific fields. In the course of time, increasinglyprecise predictions could be accomplished by means of simulation results. Furthermore,a wide variety of practical tests and extensive measurements could be substituted bysimulations. Typical established applications can be found in the material modelling inengineering sciences, in crash simulations in the automotive industry, in weather forecastsin meteorology or in earthquake simulations in computer-aided seismology. But also in themodern medicine, simulations provide new possibilities to gain a deeper understanding ofthe complex processes in the human body or even to contribute to the successful planningof a surgical intervention by providing supplementary process information. The increasingimportance of computational simulations requires for a high level of trustworthiness of thesimulation results. In order to meet these requirements, simulations with a sufficientlyfine discretisation of the geometry (especially for complex and/or irregular geometries ofthe considered materials) that are based on sophisticated models are indispensable for thenumerical realisation.

Particularly when taking into account the high structural complexity of the underlyingmaterials, a detailed knowledge about the inner structure and the composition of thematerials of various components is essential for a sufficiently accurate modelling. A broadvariety of materials cannot be described with classical continuum-mechanical models re-stricted to singlephasic (homogeneous) materials. This applies in particular to materialswith a porous micro-structure, which, for example, can be found in the scientific fieldsof geotechnical engineering or biomechanics. The group of porous media, consisting of aporous solid whose pore space is filled with fluids and/or gases, includes, amongst oth-ers, partially saturated soils and biological tissue aggregates. Hence, a multiphasic andmulticomponent modelling approach on the basis of the Theory of Porous Media (TPM)appears suitable in order to describe these complex materials. Concerning the numeri-cal treatment of porous materials, the finite-element method (FEM) has been proven tobe a well-suited technique for the solution of arbitrary initial-boundary-value problems.The resulting simulations are able to describe the physical phenomena by repeatedlysolving the descriptive set of coupled partial differential equations (PDE). However, thenecessarily high accuracy of the approximation results and the complexity of the under-lying models result in many applications in an extremely high dimension (in terms ofthe system’s degrees of freedom) of the resulting equation system. The consequences aretime-consuming simulations, which are either to slow to satisfy the time constraints and toenable practical applications (such as an accompanying use in clinical practice) or whichcannot be performed as often as needed due to the high computation time.

1

2 1 Introduction and overview

In order to counteract these problems, to increase the solution speed and to reduce thecomputational expenses, model-reduction methods are increasingly important and aregaining a considerable scientific interest. While, on the one hand, the detailed theoreti-cal basis of the modelling approach needs to be maintained and, on the other hand, anefficient numerical computation should be provided, the available performance capacityof projection-based model-reduction techniques can be used to provide fast simulationswhenever they are actually needed. Supported by the steadily growing potential of com-puting power and storage capacities, time-consuming simulations based on models with alltheir complexity can already be performed and all necessary information and data can bestored beforehand in more and more application areas. Following the concept of offline/on-line decomposition, a potentially time-consuming offline phase (including the simulationsperformed in advance and the reduction of the underlying system of equations) can beseparated from a time-efficient online phase with fast simulations (performed using thereduced system). A computationally intensive offline phase pays off by the possibility torapidly produce required simulation results in daily routines or by a sufficient number ofindividual computationally inexpensive simulations with varying material and/or simula-tion parameters.

1.2 State of the art, scope and aims

In the present contribution, the developments in the modelling and the simulation ofporous materials in the framework of the well-founded TPM, cf. de Boer [16], Bowen [21]or Ehlers [39, 42, 43], with use of the FEM are combined with the current state of researchin the field of model reduction using projection-based methods. The high complexity ofthe underlying multiphasic and multicomponent modelling of the treated materials andthe resultant strongly coupled equation systems require for individual adaptations andmodifications of well-known reduction methods to achieve satisfying results. Therefore,the scope of this monograph is the development of an application-driven approach forproviding reduced models, which are capable of simulating specific porous materials ina time-efficient manner (with sufficient accuracy of the considered quantities). In thisregard, the computation time and the crucial simulation results of simulations using theoriginal unreduced system (hereinafter referred to as “full” or “full-order” systems) or thereduced system are compared to each other in order to demonstrate the efficiency of theconsidered reduction methods. Moreover, possibilities for an adaptation of the evolvedmodifications to other models are presented.

Currently, there are several models available, which are used to describe the mechanicalbehaviour of porous materials. For a proper description, the developed models shouldbe, on the one hand, as simple as possible and, on the other hand, complex enough tocapture the relevant properties of the materials. Thereby, the requirements of the in-tended application have an important part in the appropriate description of the model.In terms of the continuum-mechanical modelling, this work is focusing on problems de-veloped on the basis of detailed and thermodynamically consistent TPM models. Earlystudies concentrated on the multiphasic and multicomponent modelling of porous mediawith application of the TPM have shown the outstanding suitability of this modelling

1.2 State of the art, scope and aims 3

approach, cf., e. g., de Boer [15, 16] or Ehlers [39, 42, 43]. In recent years, a wide field ofgeomechanical and biomechanical applications of the TPM have been successfully derived,cf., e. g., Graf [61] or Ehlers [44] for flow and transport processes in unsaturated soils orAcarturk [1], Ehlers [44], Ehlers et al. [49], Ehlers & Wagner [52] or Ricken & Bluhm[108] for biomechanical problems. The present work makes use of a biphasic model forthe simulation of a saturated porous soil (cf. Ehlers [40], Ehlers & Eipper [47], Eipper[54], Ellsiepen [55]), a multiphasic and multicomponent model for the simulation of drug-infusion processes in brain tissue (cf. Ehlers & Wagner [51, 52], Wagner [127], Wagner &Ehlers [128]) and an extended biphasic model for the description of an inhomogeneousand anisotropic intervertebral disc (cf. Ehlers et al. [49, 50], Karajan [81]).

Concerning the numerical treatment of the developed TPM models, the FEM is a con-venient technique for the solution of arbitrary initial-boundary-value problems, cf. Lewis& Schrefler [91]. Thereby, the spatial semi-discretisation of an initial-boundary-valueproblem leads to a system of differential-algebraic equations (DAE) in the time domain.Furthermore, suitable time-integration schemes, such as diagonally implicit Runge-Kutta(DIRK) methods (cf. Diebels et al. [37], Ellsiepen [55]), can be used for the numericalintegration of the semi-discrete systems. In this regard, a reduction of the descriptiveset of (strongly) coupled partial differential equations provides an enormous benefit tosignificantly reduce the dimension of these systems and, thus, the computation time andthe numerical effort of the FE simulations. Particularly with regard to nonlinear systems,the computational effort is usually immense as high-dimensional equation systems need tobe solved repeatedly for the determination of the nonlinearities. Following this, a suitablereduction of these systems essentially improves the efficiency by solving only a subset ofequations of the original model. The numerical implementation is realised with the finite-element solver PANDAS1, which is going back to Ehlers & Ellsiepen [48] and Ellsiepen[55] and is continuously maintained and further developed at the Institute of AppliedMechanics (Continuum Mechanics) at the University of Stuttgart. All computations wereperformed on a single core of an Intel i5-4590 with 32 GB of memory running at clockspeed of 3.30 GHz.

Under consideration of these circumstances, efficient reduced models for the simulationof different porous materials are provided in the present work by an application-drivenapproach. Herein, only model-reduction techniques applied to the monolithic solutionof the strongly coupled TPM model are considered. In particular, projection-basedmodel-reduction techniques are used to transform a high-dimensional system to a low-dimensional subspace. The advantage of these approaches is to maintain the detailedtheoretical basis of the modelling approach while an efficient numerical computation isprovided. Besides model-reduction techniques, parallelised solution methods or decou-pled solution strategies may also be considered as alternatives to increase the solutionspeed and, thus, reducing the computational expenses. On the one hand, parallelisedsolution methods can be applied to the full system (which is also used to perform the pre-computations required for the determination of the reduced basis) as well as to the reducedone. However, since it is out of the scope of this work, the combination of parallelised

1Porous media Adaptive Nonlinear finite element solver based on Differential Algebraic Systems, cf.http://www.get-pandas.com


solution methods with model-reduction techniques is not intended here. On the otherhand, decoupled solution strategies can break the problem down to smaller subproblems,which can be subsequently integrated in a staggered manner, cf. Zinatbakhsh [132] orMarkert et al. [93]. Nevertheless, this process may generate non-dissipative subproblemsand thus render the problem just conditionally stable, cf. Ehlers et al. [53].

Restricting to projection-based model-order-reduction methods, Antoulas & Sorensen [5]provide an overview of different techniques. In this regard, the so-called Krylov-basedmethods and the singular-value-decomposition(SVD)-based methods can be identified ascentral approaches. The Krylov-based methods are approximation methods, which arebased on the matching of the so-called moments (coefficients of series expansions) of thefull and the reduced model, cf. Grimme [64] or Freund [59]. Herein, the response func-tion of the model is interpolated by comparing the coefficients of the Taylor expansion,whereby the first moments of the transfer function of the reduced model need to matchwith the ones from the full system. Widely used Krylov-based methods are the Pade viaLanczos (PVL), the Arnoldi procedures and the multipoint rational interpolation. UsingSVD-based methods, a matrix is broken down into the product of three matrices by meansof a singular-value decomposition for the benefit that the singular values can be read offdirectly. Typical SVD-based methods are the balanced truncation, cf. Moore [96], andthe method of proper orthogonal decomposition (POD), which is used as a starting pointfor the model reduction in this contribution. The development of the POD method, alsoknown as Karhunen-Loeve expansion, cf. Sirovich [115], traces back to fluid-dynamic ap-plications including turbulence, cf. Berkooz et al. [10]. Beyond that, the POD method wassuccessfully applied to various problems in fluid flow (cf. Kunisch & Volkwein [88], Rowleyet al. [110]), optimal control (cf. Kunisch & Volkwein [86]), aerodynamics (cf. Bui-Thanhet al. [24], Hall et al. [73]), biomechanics (cf. Radermacher & Reese [102]) and structuralmechanics (cf. Herkt et al. [76], Radermacher & Reese [103]). The error bounds for POD-Galerkin approximations of linear and nonlinear parabolic equations have been provenby Kunisch and Volkwein [87, 88]. One great advantage of the POD method is that thismethod is independent from the type of the model and can be used for nonlinear systemsas well as for systems of second order. Its flexibility in application is based on analysinga given data set to determine a reduced basis. In order to generate this “training” data,pre-computations are performed in a time-consuming offline phase using the full modeland the state solutions (so-called snapshots) of the system are collected. Specifically, aset of snapshots consists of discrete samples of state variables at certain time instancesand associated with specific material parameters and particular initial and boundary con-ditions. After a reduced basis is determined, time-efficient simulations can be performedin the online phase using the reduced model. However, since the POD-Galerkin approxi-mation does in fact significantly reduce the dimension of the equation system but not theeffort to determine the nonlinear terms, the computational effort of nonlinear problemscannot be (sufficiently) reduced when exclusively using the POD method. This draw-back motivates the application of additional methods for the reduction of the nonlinearterms. In combination with the POD method, adaptive sub-structuring as presented inRadermacher & Reese [103], the trajectory-piecewise-linear (TPWL) method, analysed inRewienski & White [107], the lookup-table approach, cf. Herkt et al. [76] or Herkt [75], theGauss-Newton-with-approximated-tensors (GNAT) method, cf. Carlberg et al. [29], Carl-

1.3 Outline of the thesis 5

berg [30], the approach of hyper-reduction (HR), cf. Ryckelynck [111] and Ryckelynck &Missoum-Benziane [112], and the discrete-empirical-interpolation method (DEIM), whichis the discrete variant of the empirical-interpolation method (EIM, cf. Barrault et al. [7])and which was introduced by Chaturantabut & Sorensen [32], are commonly used tech-niques. At this point, it should be mentioned that the DEIM is similar to the empiricaloperator interpolation, which was presented in Haasdonk et al. [71] and extended to non-linear problems in Haasdonk & Ohlberger [68]. In the works of Kellems et al. [85] for amodel of spiking neurons, in Chaturantabut & Sorensen [34] for a model with applicationto non-linear miscible viscous fingering, in Nguyen et al. [98] for reacting flow applications,in Negri et al. [97] for parametrised systems or in Bonomi et al. [19] for the application toparametrised problems in cardiac mechanics, amongst others, it could be shown that theDEIM is able to significantly reduce the numerical effort of complex nonlinear processes.Hence, the POD method is used in this work in combination with the DEIM as additionalmethod for the reduction of nonlinear terms. In this regard, problem-dependent modifi-cations of the classical model-reduction approaches are realised exemplarily for the threeabove-mentioned applications.

To sum it up, the desired goal of this monograph is neither the development of a finite-element model for a specific multiphasic and multicomponent material on the basis of asophisticated continuum-mechanical modelling approach nor the derivation and formu-lation of an entirely new model-reduction technique. Instead, existing model-reductionapproaches are taken as a basis to enable fast simulations of already developed continuum-mechanical models of porous materials. However, it can be shown within this work thatthe used model-reduction methods in their original form cannot be applied straightfor-ward to the utilised complex material models. To provide time-efficient but also accuratesimulations in daily routines, specific, problem-dependent modifications of the existingmodel-reduction approaches are necessary and are therefore performed within this work.In this regard, an in-depth knowledge of the form and the characteristics of the under-lying equation systems is essential and is therefore treated intensively. For example, itshould be ensured that the block structure of the (strongly) coupled equation systemsis preserved while considering the different temporal (physical) behaviour of the primaryvariables. Moreover, the characteristics of the treated DAE systems make it impossibleto use already established error estimators with the consequence that alternative errorbounds need to be formulated. Ultimately, the derived modifications can be transferredto various other problems where physical phenomena occur on different time-scales. Fol-lowing this, the present work delivers a generalised approach for an adaptation of theevolved modifications to other models.

1.3 Outline of the thesis

Starting with Chapter 2, the continuum-mechanical fundamentals, required for the de-scription of porous media, are summarised on the basis of a general biphasic solid-fluidmixture. Therefore, the basic concept of the TPM is introduced, all necessary kinematicalrelations, such as the motion of an overall aggregate body, are provided and the balancerelations for porous media are presented for the particular constituents and for the entire


aggregate. Afterwards, the general continuum-mechanical fundamentals are specified forthree models of porous materials, namely a biphasic standard problem of a saturatedporous soil, a multiphasic and multicomponent description of human brain tissue withapplication to drug-infusion processes and an extended biphasic model for the descriptionof an inhomogeneous and anisotropic intervertebral disc.

The numerical treatment of the derived governing equations, in the framework of themixed finite-element method in space and the finite-difference method in time, is shown inChapter 3. In this regard, the weak formulations of the governing balance equations arepresented for the multi-phasic modelling approaches of the previously described examples.This is done in a detailed way for the biphasic standard problem of the porous soil,distinguishing between quasi-static and dynamic initial-boundary-value problems, andbriefly for the remaining examples.

Chapter 4 is concerned with the theoretical basis of the considered projection-basedmodel-reduction techniques. After an overview of the different methods is given, all nec-essary mathematical fundamentals of the reduction processes, applied for both linear andnonlinear problems, are shown. In this context, the main focus lies on an applicationof the POD method, either individually or, for the treatment of nonlinearities, in com-bination with the DEIM. In this regard, application-driven modifications of the classicalreduction processes are pointed out.

The application of the model-reduction methods for the three specified TPM models iscarried out in Chapter 5. Therein, the utilisation of the POD method for linear andfor nonlinear porous-media problems is discussed, as well as the application of the POD-DEIM for nonlinear problems. Starting with the relatively simple biphasic porous-soilmodel under quasi-static and linear-elastic material behaviour, more complex phenomena,such as dynamic or Neo-Hookean material behaviour, are investigated. Subsequently, ageneralised approach for an adaptation of the evolved modifications to other models ispresented.

Finally, a summary and an outlook are given in Chapter 6, discussing the presentedwork and showing possible further potentials of model-reduction implementations.

For a better understanding of the discussed topics, additional information regarding themathematical fundamentals is given in Appendix A and Appendix B. Furthermore,Appendix C provides a detailed presentation of the reduced systems of equations usedin this monograph.

Chapter 2:Continuum-mechanical fundamentals for themodelling of porous media

The purpose of this chapter is to review the theoretical fundamentals for the continuum-mechanical description of multiphasic and multicomponent materials using the frameworkof the well-founded Theory of Porous Media (TPM). Therefore, the concept of volumefractions is presented, followed by a brief overview of the kinematical relations of su-perimposed constituents, providing the introduction of relevant deformation and strainmeasures. In addition, general balance relations for the overall aggregate as well as forthe particular constituents are derived. Finally, this chapter is closed by a specificationof the continuum-mechanical fundamentals for the modelling of three particular porousmaterials.

2.1 The concept of the Theory of Porous Media

With regard to a continuum-mechanical description of multiphasic porous materials, theearliest approaches trace back to the work of Biot [12, 13], describing the consolidationproblem of biphasic geomaterials. The two major milestones for a further extension tothe current understanding of the TPM (cf. de Boer & Ehlers [17], Ehlers [39, 40, 42, 43])have been the development of the Theory of Mixtures (TM), cf. Truesdell & Toupin [124]and Bowen [20], and the enhancement by the concept of volume fractions (cf. Mills [94]and Bowen [21, 22]). An excellent overview of the historical evolution of the TPM is givenin de Boer [15], de Boer & Ehlers [18] and Ehlers [39, 45].

2.1.1 Macroscopic modelling approach

When regarding the complex inner structure of porous media by means of the TPM, ahomogenisation process over a locally defined representative elementary volume (REV),whose constituents are assumed to be in a state of ideal disarrangement, needs to beperformed and yields a model on the macroscale. Due to this, the real microstructureof porous materials can remain unknown and the information about the underlying mi-crostructure is assured by the concept of volume fractions. In a multiphasic and mul-ticomponent approach of an entire aggregate model ϕ, multiple constituents ϕα can beidentified, yielding ϕ =

⋃

α

ϕα. In this regard, the union symbol is used to characterise

that the aggregate consists of all the identified consituents (and is therefore not used in astrictly mathematical notation). In order to avoid any confusion, the terms component,phase and constituent are clarified here. Regarding the real composition of a material,several interacting or independent components are recognised and together form the in-tegrated whole of the described material. The term phase is used in this work to describe

7

8 2 Continuum-mechanical fundamentals for the modelling of porous media

the chemical state of aggregation, such as solid, liquid or gaseous. In general, each com-ponent can exist in different phase states. Moreover, the term constituent is used in thecontext of the modelling process. In this regard, all constituents add up to the entiretheoretical model of the specified material.

2.1.2 Volume fractions and density functions

In order to account for the local composition of the aggregate, local structure parametersare introduced. The volume V of the overall aggregate body B is divided into the sum ofits partial volumes V α of the constituent bodies Bα. This yields

V =

∫

Bdv =

∑

αV α . (2.1)

Therein, the partial volume V α can be described by the volume fraction nα, which allocatesfor each spatial point of the actual configuration the local part of the volume V α of theconstituent ϕα on the volume V of the overall aggregate ϕ, yielding

V α =

∫

Bα

dv =

∫

Bdvα =

∫

Bnα dv , with nα :=

dvα

dv. (2.2)

Proceeding from equations (2.1) and (2.2), the so-called saturation constraint for a mul-tiphasic approach can be introduced as

∑

αnα = 1 , (2.3)

assuming fully saturated conditions. Due to the applied homogenisation, two differentdensities can be defined for each constituent ϕα, the partial density ρα and the materialor realistic density ραR. Following the concept of volume fractions yields the relations

ρα :=dmα

dvand ραR :=

dmα

dvα, (2.4)

with the local mass element dmα of the constituent ϕα. The densities are related to eachother by

ρα = nα ραR . (2.5)

Following this, the partial density ρα changes with a change of the volume fraction nα orwith a change of the realistic density ραR. Consequently, the property of material incom-pressibility (ραR =const.) will not necessarily cause bulk incompressibility (ρα =const.).Finally, the overall density ρ of the aggregate body B can be obtained as the sum of allpartial densities

ρ =∑

αρα =

∑

αnα ραR . (2.6)

2.2 Kinematical relations 9

2.2 Kinematical relations

The following section offers a brief overview of the kinematical relations and providesthe introduction of relevant deformation and strain measures. Note that all quantitiesrelated to the motion of a constituent are indicated in the subscript ( · )α, whereas all non-kinematical quantities are indicated in the superscript ( · )α. For a detailed description ofthe kinematical quantities, the reader is referred to Ehlers [39].

2.2.1 Motion of a porous material

In order to describe nonlinear deformation processes of porous materials, the kinematicalrelations of an overall aggregate body B have to be considered. The body B is defined asthe connected manifold of the material points Pα, which can be described over the timet with respect to a fixed origin O.

At any time t, each spatial point x of the current configuration is occupied by materialpoints Pα of each constituent ϕα, stemming from different reference positions Xα at timet0, see Figure 2.1 exemplary for a classical biphasic approach. Thus, the continuum-me-chanical relations of singlephasic materials can be applied to each single constituent ϕα.

xXSXF

χF (XF , t)

χS(XS , t)

BB0

(t)(t0)

PS

PF PS, PF

O

Figure 2.1: Motion of a biphasic material with a solid constituent ϕS and a fluid constituentϕF , as discussed in Subsection 2.4.1.

Moreover, each constituent ϕα follows its own individual Lagrangean (material) place-ment function χα (Xα, t), such that the placement functions χα carry the points Pα of thereference configurations Xα to their current configuration x. Following this, the currentconfiguration x can be given as

x = χα (Xα, t) . (2.7)

The unique inverse motion functions χ−1α can only be computed, if non-singular functional

determinants (Jacobians) Jα exist, viz.:

Xα = χ−1α (x, t) , if Jα := det

∂χα

∂Xα6= 0 . (2.8)

Using the material time derivatives, the corresponding velocity and acceleration fields for


each constituent can be described in a Lagrangean setting as

x′α =

∂χα (Xα, t)

∂t= x′

α(Xα, t) and x′′α =

∂2χα (Xα, t)

∂t2= x′′

α(Xα, t) . (2.9)

Moreover, the Eulerian (spatial) representation of the velocity and acceleration fields canbe obtained by inserting the inverse motion function (2.8), yielding

x′α = x′

α

(

χ−1α (x, t), t

)

= x′α(x, t) and x′′

α = x′′α

(

χ−1α (x, t), t

)

= x′′α(x, t) . (2.10)

Furthermore, the so-called local barycentric velocity x (mixture velocity of the overallaggregate) is given by the density-weighted mean of the velocities x′

α, reading

x =1

ρ

∑

αρα x′

α . (2.11)

Based on this, the diffusion velocity

dα := x′α − x with

∑

αρα dα = 0 (2.12)

is introduced as the difference of the velocity of a constituent ϕα and the barycentricvelocity x. Note that all time derivatives ( · )′α and ˙( · ) used in the equations above arematerial (total) time derivatives. The Lagrangean formulation in (2.9) implies that thematerial time derivative dχα (Xα, t) of the motion function is equivalent to the partialtime derivative ∂χα (Xα, t), as the reference position Xα is fixed in time by an initialcondition at time t0. But using an Eulerian description, the spatial variable x implicitlydepends on the time t. Following this, the inner (implicit) derivative needs to be includedin the material time derivative. In this regard, the total time derivatives of an arbi-trary, steady and sufficiently steady differentiable field function (scalar-valued Υ(x(t), t)or vector-valued Υ(x(t), t)) with respect to the constituents ϕα and the overall aggregateϕ are given as

Υ′α =

dαΥ

dt=

∂Υ

∂t+ gradΥ · x′

α , Υ =dΥ

dt=

∂Υ

∂t+ gradΥ · x ,

Υ′α =

dαΥ

dt=

∂Υ

∂t+ (gradΥ)x′

α , Υ =dΥ

dt=

∂Υ

∂t+ (gradΥ) x .

(2.13)

Therein, the gradient operator grad( · ) = ∂( · )/∂x denotes the partial derivative withrespect to the actual position x.

In terms of porous-media theories including large solid deformations, it is suitable to usea Lagrangean description of the solid skeleton via the solid displacement field

uS := x − XS (2.14)

with respect to a basically given reference configuration. In contrast, the pore-liquid flowof the immiscible constituents is better expressed in a modified Eulerian setting via theseepage velocities

wβ := x′β − x′

S , (2.15)

describing the velocities of the pore liquids (constituents ϕβ) in relation to the velocity ofthe deforming solid skeleton.

2.2 Kinematical relations 11

2.2.2 Deformation and strain measures

The most important measure for the deformation in continuum mechanics is provided bythe material deformation gradient. The deformation gradients Fα of the constituents ϕα

and their inverses F−1α can be formally introduced reading

Fα =∂χα(Xα, t)

∂Xα

=: Gradα x and F−1α =

∂χ−1α (x, t)

∂x= gradXα , (2.16)

respectively. Therein, the gradient operator Gradα( · ) = ∂( · )/∂Xα denotes the partialderivative with respect to the reference position Xα of ϕα. The deformation gradients“transport” the line elements dXα of the reference configuration to the line element dxof the actual configuration (push-forward transformation):

dx = Fα dXα . (2.17)

The material deformation gradient FS of the solid plays a key role as the basic kinematicalquantity. The solid deformation gradient FS and its inverse F−1

S may be computed usingthe solid displacement field (2.14), yielding

FS =∂(XS + uS)

∂XS= I + GradS uS ,

F−1S =

∂(x − uS)

∂x= I − graduS ,

(2.18)

where I is the second-order identity tensor. According to equation (2.8), the existenceof uniquely invertible motions requires non-zero Jacobians Jα. Starting the deformationprocess at time t0 from an undeformed (natural) state, the associated initial conditionFα(t0) = I restricts the domain of detFα to positive values. Thus,

detFα = Jα > 0 with detFα(t0) = 1 . (2.19)

Proceeding from the fundamental theorem of polar decomposition, the solid deformationgradient FS can be multiplicatively split into an orthogonal rotation tensor RS with theproperties RT

S = R−1S and detRS = 1, and a symmetric, positive definite right or left

stretch tensor US or VS, respectively, yielding

FS = RS US = VS RS . (2.20)

Based on the fundamental transport theorem (2.17) for line elements, further transporttheorems for area and volume elements are obtained via

da = (cof Fα) dAα and dv = (detFα) dVα . (2.21)

Therein, da and dv denote the area and volume element in the actual configuration,whereas dAα and dVα represent the corresponding elements of ϕα in the referential con-figuration. As a next step, the right and the left Cauchy-Green deformation tensors


CS and BS, respectively, are introduced via the square of the line elements in the ac-tual and referential configurations using the push-forward transformation (2.17) and thecorresponding pull-back transformation dXα = F−1

α dx:

dx · dx = (FS dXS) · (FS dXS) = dXS · (FTS FS) dXS =: dXS ·CS dXS ,

dXS · dXS = (F−1S dx) · (F−1

S dx) = dx · (FT−1S F−1

S ) dx =: dx ·B−1S dx .

(2.22)

It can be directly deduced that

CS = FTS FS = US US and BS = FS F

TS = VS VS . (2.23)

Furthermore, strain measures are introduced via the difference of the squares of the lineelements given in (2.22) expressed with respect to the reference and the actual configura-tion, yielding

dx · dx − dXS · dXS = dXS · (CS − I) dXS =: dXS · 2ES dXS ,

dx · dx − dXS · dXS = dx · (I − B−1S ) dx =: dx · 2AS dx .

(2.24)

Therein, ES denotes the Green-Lagrangean strain tensor and AS is known as the Al-mansian strain tensor, which can be expressed as

ES = 12(CS − I) and AS = 1

2(I − B−1

S ) , (2.25)

respectively. Note in passing that there are several other possibilities to define and ex-press deformation and strain measures, cf., e. g., Truesdell & Noll [123]. However, theseexpressions have no relevance in this work and, therefore, need no further considerationhere.

In order to describe rate-dependent material behaviour, time derivatives of the introduceddeformation and strain measures are required. Proceeding from the material solid velocitygradient

(FS)′S =

dS

dt

(∂x

∂XS

)

=∂x′

S

∂XS

= GradS x′S =

∂x′S

∂x

∂x

∂XS

=: LS FS , (2.26)

the spatial solid velocity gradient is introduced via

LS =∂x′

S

∂x= gradx′

S = (FS)′S F

−1S . (2.27)

Moreover, the spatial solid velocity gradient LS can be decomposed into a symmetric partDS and a skew-symmetric part WS (known as spin tensor):

LS = DS + WS with

DS = 1

2(LS + LT

S ) = DTS ,

WS = 12(LS − LT

S ) = −WTS .

(2.28)

Applying the material time derivative to the right Cauchy-Green deformation tensor CS,the rate of the deformation tensor in the referential frame results in

(CS)′S = (FT

S FS)′S = (FT

S )′S FS + FT

S (FS)′S

= FTS LT

S FS + FTS LS FS = 2FT

S DS FS .(2.29)

2.3 Balance relations 13

2.3 Balance relations

The following section gives a brief summary of the fundamental balance principles ofcontinuum mechanics applied to the TPM. Therefore, the basic stress measures are intro-duced before the specific balances are derived on the basis of the master balance principle.The basis for the formulation of the balance relations for multiphasic materials is given inTruesdell’s metaphysical principles, cf. Truesdell [122]. For a more detailed description ofthe balance relations for porous media, the reader should refer to the work of Ehlers [43].A more comprehensive description can be found in de Boer & Ehlers [17] or Diebels [36].

2.3.1 Basic stress measures

In general, an external loading of a deformable body results in (finite) deformations and anassociated (inner) state of stress. According to the free-body principle, each constituentϕα of the overall aggregate body B can be subjected to an external mass-specific volumeforce bα(x, t), which is acting on all material points Pα, and an external contact (surface)force tα(x, t,n) acting on parts of the boundary surface S, whose orientation is indicatedby the outward-oriented unit surface normal vector n. Hereby, the volume force bα, whichis commonly induced as the gravitation, is denoted per mass unit, whereas the contactforce tα is denoted per surface area. Following this, the overall body is loaded with atotal volume force f and a total contact force t, reading

f = ρb =∑

α

ρα bα =∑

α

fα and t =∑

α

tα , (2.30)

respectively. In conclusion, the summation of the contact and volume forces acting on theoverall aggregate ϕ (kc and kv, respectively) and on the single constituents ϕα (kα

c andkαv , respectively) yields the corresponding total forces k and kα, viz.:

k =

∫

B

ρb dv

︸︷︷︸

kv

+

∫

S

t da

︸︷︷︸

kc

, kα =

∫

B

ρα bα dv

︸︷︷︸

kαv

+

∫

S

tα da

︸︷︷︸

kαc

. (2.31)

Moreover, the stress-tensor concept introduced by Cauchy for singlephasic materials candirectly be transferred to theories for multiphasic materials. Applying the Cauchy’stheorem, the Cauchy stress tensor T of the entire aggregate ϕ and the partial Cauchystress tensors Tα of the constituents ϕα can be introduced via

t(x, t,n) = T(x, t)n and tα(x, t,n) = Tα(x, t)n , (2.32)

respectively, representing the current (true) stress state at material points by relating theactual contact force element to the oriented area element da of the actual configuration:

dkαc = tα da = Tα n da = Tα da . (2.33)


Using the transport theorem (2.21)1 of the area elements, additional stress tensors can beformulated:

Tα da = Tα (cof Fα) dAα =

τα

︷︸︸︷

(detFα)Tα FT−1

α︸︷︷︸

Pα

dAα . (2.34)

Therein, the Kirchhoff stress (weighted Cauchy stress) τα relates the actual contactforce element to a weighted area element (detFα)

−1da of the actual configuration, whereasPα represents the (non-symmetric) 1st Piola-Kirchhoff stress (nominal stress) relatingthe actual contact force element to the area element dAα of the referential configuration.Furthermore, it is useful to introduce the symmetric 2nd Piola-Kirchhoff stress

Sα = F−1α Pα = F−1

α ταFT−1α , (2.35)

which operates on the reference configuration.

2.3.2 Master balance principle and specific mechanical balance

equations

Proceeding from the results of classical continuum mechanics, general balances for volume-specific mechanical quantities (scalar-valued Ψ or vector-valuedΨ) of the overall aggregatebody B can be formulated using the master-balance principle

d

dt

∫

BΨdv =

∫

S(φ · n) da +

∫

Bσ dv +

∫

BΨ dv (2.36)

in scalar-valued form or

d

dt

∫

BΨ dv =

∫

S(Φn) da +

∫

Bσ dv +

∫

BΨ dv (2.37)

in vector-valued form. Therein, φ ·n and Φn are the effluxes of the respective mechanicalquantity, which directly enter the aggregate body B over its surface S, σ and σ are theexternal mechanical quantities causing a volume-specific supply from distance, and Ψ andΨ are the production terms allowing for interactions of the body with its surrounding.

Applying the Gaußian integral theorem and reformulating these equations, the local bal-ance relations of the entire aggregate ϕ are given by

Ψ + Ψdiv x = divφ + σ + Ψ ,

Ψ + Ψ div x = divΦ + σ + Ψ .(2.38)

According to Truesdell’s metaphysical principles, the balance relations for porous mediacan be formulated in a summarised form for the entire aggregate ϕ or individually for eachconstituent ϕα. In order to account for interactions between the different constituents,

2.3 Balance relations 15

appropriate production terms have to be included in the balance equations of the con-stituents. Following this, the corresponding local balance equations for the constituentsϕα are determined in an analogous manner as the balances (2.38), yielding

(Ψα)′α + Ψα divx′α = divφα + σα + Ψα ,

(Ψα)′α + Ψα divx′α = divΦα + σα + Ψα .

(2.39)

Herein, the partial quantities ( · )α have the same physical meaning as the overall quantities( · ) in (2.38). However, the production terms ( · )α now characterise interactions betweenthe constituents.

As is usual in continuum thermodynamics, the specific balance relations for the overallaggregate ϕ and the constituents ϕα are introduced via postulated axioms. Comparingthese axioms with the global master balances (2.36) and (2.37), the specific balance rela-tions of mass, momentum, moment of momentum (m. o.m.), energy and entropy can beformulated. For a detailed formulation see, e. g., de Boer & Ehlers [17], Diebels [36], Ehlers[43] or Ellsiepen [55]. In a summarised form, the local balance equations for the overallaggregate ϕ, known from continuum mechanics of single-phase materials, can be found as

mass : ρ + ρ div x = 0 ,

momentum : ρ x = divT + ρb ,

m. o.m. : 0 = I×T −→ T = TT ,

energy : ρ ε = T · L − divq + ρ r ,

entropy : ρ η ≥ divφη + ση ,

(2.40)

where the overall density ρ is used as volume-specific mechanical quantity for the balanceof mass. In addition, the momentum ρ x of the entire body and the corresponding momentof momentum x× (ρ x) identify the volume-specific mechanical quantities in the balancesof momentum and moment of momentum. Concerning the energy and entropy balances,the volume-specific mechanical quantities are the overall energy (ρ ε+ 1

2x · (ρ x)) and the

entropy ρ η of the entire aggregate, respectively. Furthermore, ε is the internal energy, q isthe heat influx vector, r is the external heat supply, η is the mass-specific entropy, φη is theefflux of entropy and ση is the external entropy supply. Moreover, the entropy productionη ≥ 0 can never be negative in order to fulfil the second law of thermodynamics.

Accounting for the occurring interaction mechanisms by additional production terms, thebalance equations for the constituents ϕα are derived in direct analogy to the balance


equations of single-phasic materials yielding

mass : (ρα)′α + ρα divx′α = ρα ,

momentum : ρα x′′α = divTα + ρα bα + pα ,

m. o.m. : 0 = I×Tα + mα ,

energy : ρα (εα)′α = Tα · Lα − div qα + ρα rα + εα ,

entropy : ρα (ηα)′α = − div

(qα

θα

)

+ρα rα

θα+ ζα .

(2.41)

Therein, the partial entropy effluxes and the external entropy supplies of the constituentsare a priori assumed as

φαη = − 1

θαqα and σα

η =ρα rα

θα, (2.42)

respectively, where the absolute Kelvin’s temperatures θα of the constituents allow foran individual temperature field for each constituent. Moreover, the production terms pα,mα, εα and ζα correspond directly to the production of mass ρα, the total production ofmomentum sα, the total production of moment of momentum hα, the total production ofenergy eα and the total production of entropy ηα yielding

∑

α

ρα = 0 ,

sα = pα + ρα x′α , where

∑

α

sα = 0 ,

hα = mα + x× (pα + ρα x′α) , where

∑

α

hα = 0 ,

eα = εα + pα · x′α + ρα (εα + 1

2x′α · x′

α) , where∑

α

eα = 0 ,

ηα = ζα + ρα ηα , where∑

α

ηα ≥ 0 .

(2.43)

2.4 Modelling approach of specific materials

On the basis of the presented continuum-mechanical fundamentals of the preceding sec-tion, arbitrary models for different porous materials and various application areas can besimulated and a variety of models have already been successfully derived in recent yearsin the framework of the TPM. Within the scope of this work, three specific TPM modelsare addressed, namely a biphasic model for the simulation of saturated porous soils, amultiphasic and multicomponent model for the simulation of infusion processes in braintissue, and finally an extended biphasic model for the description of an inhomogeneousand anisotropic intervertebral disc. For this purpose, the continuum-mechanical funda-mentals are specified in this section for these models. Note that the order of the presented

2.4 Modelling approach of specific materials 17

TPM models is chosen with regard to their application in Chapter 5. Thus, the specificmodels are arranged according to their complexity concerning the performed reductionprocess.

2.4.1 Biphasic porous-soil model

Geomaterials, such as porous soils, rocks or sandstones, consist of a porous solid skeletonwhose pores are saturated with one or multiple fluids. In this work, a classical biphasicapproach on the basis of the TPM is used in order to simulate a saturated porous soil.Thereby, the purpose of this subsection is to give a brief introduction into the particularmodel, whereas more detailed information can be found in Ehlers [40], Ehlers & Eipper[47], Eipper [54], Ellsiepen [55], Heider [74] and references therein. With regard to Section2.1, the two immiscible parts of the model, namely the solid constituent ϕS and the fluidconstituent ϕF , lead to an entire aggregate model

ϕ =⋃

α

ϕα = ϕS ∪ ϕF , where α = S, F , (2.44)

see Figure 2.2. According to relation (2.3), the saturation constraint for a biphasic ap-proach can be formulated as

∑

αnα = nS + nF = 1 , (2.45)

assuming fully saturated conditions. The underlying fundamental principles of an isother-mal biphasic model enable the development of various customised models. In the follow-ing, a model with several preliminary assumptions, namely materially incompressibleconstituents (ραR =const.) and negligible extra stresses of the fluid (TF

E ≈ 0), is used.Moreover, both quasi-static processes (under small deformations) and dynamic processesare discussed.

microscale

macroscale

REV

concept of volume fractions

homogenised model dvS

dvF

ϕ = ϕS ∪ ϕF

Figure 2.2: Representative elementary volume (REV) of a soil with porous microstructure andhomogenisation to a biphasic TPM macroscale model.


Quasi-static porous-soil model

Restricting the validity of the model to slow processes, and thus assuming quasi-staticconditions for the overall aggregate as well as for the particular constituents, the overallacceleration and the constituent’s acceleration terms are neglected yielding

x ≈ 0 and x′′α ≈ 0 . (2.46)

Under consideration of the above mentioned constitutive assumptions, the governing bal-ance relations for quasi-static processes are finally given by the momentum balance (2.47)1and the volume balance (2.47)2 of the overall aggregate, viz.:

0 = divT + ρb = div (TSE − p I) + (nSρSR + nFρFR)b ,

0 = div (uS)′S + div (nF wF ) , with nF wF = −KF

spec(grad p − ρFR b) .(2.47)

Therein, the overall Cauchy stress tensor T = TS +TF is derived by a summation of itspartial stresses

TS = TSE − nSp I and TF = −nFp I , (2.48)

where TSE represents the extra stress of the solid, while I is the identity tensor and p

denotes the effective pore-fluid pressure. Moreover, the external volume force b is com-monly induced as the uniform and constant gravitational (body) force b = g, while KF

spec

denotes the specific permeability tensor, which depends on the fluid properties. Assumingisotropic permeability in the undeformed reference state, the specific permeability tensorcan be written as KF

spec = (kF/γFR) I, with the effective fluid weight γFR and the con-ventional hydraulic conductivity (Darcy permeability) kF > 0. Using the volume balanceof the solid skeleton, the volume fraction nS = nS

0S (detFS)−1 of the solid of the actual

configuration can be obtained from the (initial) volume fraction nS0S of the solid of the

reference configuration. With regard to the numerical solution, the coupled system ofequations (2.47) is governed by the solid displacement field uS and the effective pore-fluidpressure p as primary variables. For more details, the interested reader is referred toEhlers [40], Ehlers & Eipper [47], Eipper [54] or Ellsiepen [55] among others.

Dynamic porous-soil model

Considering the more general case of dynamic processes, the acceleration terms cannotbe neglected. Thus, the momentum balance of the fluid needs to be added as governingequation to be able to solve the overall system of equations via a numerical solutionstrategy. Following this, the governing balance relations are given viz.:

0 = nSρSR (uS)′′S + nFρFR

[(

(uS)′S + wF

)′S + grad

(

(uS)′S + wF

)

wF

]

−

− div (TSE − p I) − (nSρSR + nFρFR)b ,

0 = ρFR[(

(uS)′S + wF

)′S + grad

(

(uS)′S + wF

)

wF

]

+nFγFR

kFwF +

+grad p − ρFR b ,

0 = div (uS)′S + div (nFwF ) ,

(2.49)


namely the momentum balance (2.49)1 of the overall aggregate, the momentum balance(2.49)2 of the fluid and the volume balance (2.49)3 of the overall aggregate. Therein, thepore-liquid flow is expressed in a modified Eulerian setting via the fluid-seepage velocityvector

wF = x′F − x′

S = x′F − (uS)

′S , (2.50)

according to relation (2.15). Consequently, it is possible to solve the coupled system ofequations (2.49) by considering the solid displacement field uS, the fluid-seepage velocityvector wF and the pore-fluid pressure p as the primary variables of the system. For amore detailed derivation of the governing balance relations, the reader is referred to, e. g.,Ellsiepen [55].

2.4.2 Multiphasic brain-tissue model with application to drug-infusion processes

For the purpose of this work, the basis of the considered thermodynamically consistentdrug-infusion model is briefly described in this subsection. In this regard, a distinction ismade between a general brain-tissue model and a simplification thereof. For a detailedintroduction of the customised multi-component and multi-phasic model, the interestedreader is referred to the underlying works of Ehlers and Wagner [51, 52, 127, 128] andcitations therein. The simplified drug-infusion model is introduced in Fink et al. [58].

General multi-component model for brain tissue

The used brain-tissue model for the simulation of drug-infusion processes includes fourconstituents ϕα. In particular, three immiscible constituents are given by the solid skeletonϕS, the blood plasma ϕB and the overall interstitial fluid ϕI , which contains a miscibledissolved therapeutic solute ϕD, see Figure 2.3.

microscale

REV

homogenised model

dv

dvS

dvI

dvB

tissue cells

vascular system

interstitial fluid

therapeutic solutes

Figure 2.3: REV with exemplarily displayed micro-structure of brain tissue and macroscopicmultiphasic and multicomponental modelling approach (courtesy of Wagner [127]).

Moreover, preliminary assumptions are included, i. e., isothermal conditions, materiallyincompressible constituents and negligence of acceleration terms. Thus, the governingbalance relations with the primary variables solid displacement uS, effective pore-liquid


pressures pBR and pIR and molar concentration cDm of the therapeutic agent are given bythe momentum balance (2.51)1 of the overall aggregate, the volume balance (2.51)2 of ϕ

B,the volume balance (2.51)3 of ϕI and the concentration balance (2.51)4 of ϕD, viz.:

0 = divT + ρb

= div(

TSE − (sBpBR + sIpIR) I

)

+ (nSρSR + nBρBR + nIρIR)b ,

0 = (nB)′S + nBdiv (uS)′S + div (nBwB) ,

0 = (nI)′S + nIdiv (uS)′S + div (nIwI) ,

0 = (cDm)′S n

I + (nI)′S cDm + nIcDm div (uS)

′S + div (nIcDmwD) .

(2.51)

All remaining unknown quantities in (2.51) need to be described by a combination ofspecific balance relations with appropriate constitutive assumptions. In particular, theseepage velocities wB, wI and wD are described via

nB wB = − KSB

µBR(grad pBR +

pdifsB

grad sB − ρBR b) ,

nI wI = − KSI

µIR(grad pIR − ρIR b) ,

nIcDmwD = −DD grad cDm + nIcDmwI ,

(2.52)

where the differential pressure pdif = pBR − pIR indicates the pressure difference of thepore liquids, the saturation function sB = 1− sI = nB/(1−nS) specifies the volumetricalamount of blood (and, thus, of the interstitial fluid) in relation to the pore space (porosity)and the quantity nS

0S denotes the initial volume fraction of the solid. Moreover, µBR andµIR are the effective shear viscosities of the liquids. Furthermore, the local anisotropiesof the brain tissue are included via the anisotropic intrinsic permeability tensors KSB

and KSI , which can be evaluated from (patient-specific) medical imaging data. The sameapplies for the diffusivity DD of the therapeutic agent. Next to the intrinsic permeabilitytensor KSβ, which represents a purely geometric quantity, other kinds of permeabilitymeasures are widely used. In particular, these are the Darcy permeability (hydraulicconductivity) tensor Kβ = γβR KSβ/µβR, where γβR represents the effective weights ofthe liquids, and the specific permeability tensor Kβ

spec = KSβ/µβR, both depending onthe fluid properties. In this regard, the use of a specific permeability tensor Kβ

spec offersthe advantage of a single prefactor in the respective pore-liquid-flow equations. For amore detailed discussion on these different permeability measures, the reader is referredto, e. g., Wagner [127]. For the volume fractions and their temporal changes, the following


relations for fully saturated conditions are provided, viz.:

nS = nS0S (detFS)

−1 and (nS)′S = −nS div (uS)′S ,

nB = sB (1− nS) and nI = 1 − nS − nB ,

sB(pdif) =1

2( pdifαB

− βB)

( pdifαB

− βB − 2)

+

√

4 +(pdifαB

− βB)2

,

(nB)′S = (sB)′S (1 − nS) − sB(nS)′S , where (sB)′S =∂sB

∂pdif(pdif)

′S ,

(nI)′S = −(nS)′S − (nB)′S .

(2.53)

Therein, αB and βB denote material parameters, which allow for the adaption of thepore-pressure difference and the initial blood saturation to typical values as they exist inthe human brain. Furthermore, the overall Cauchy stress

T = TSE − (sB pBR + sI pIR) I , where TS

E = TSE,iso + TS

E,aniso

with

TSE, iso = 2

µS0

JSKS + λS0 (1 − nS

0S)2

(1

1 − nS0S

− 1

JS − nS0S

)

I ,

TSE, aniso =

µS1

JS (aS · aS)

(

(aS · aS )γS1 /2 − 1

)

(aS ⊗ aS) ,

(2.54)

consists of an isotropic and an anisotropic (transversely isotropic) part. This constitu-tive description enables the description of finite and anisotropic deformation processes.Therein, λS0 and µS

0 denote the first and second Lame constants and µS1 and γS1 are

material parameters governing the fibre stiffness of the solid skeleton. In addition,

KS = 12

(

GradSuS + (GradSuS)T + GradSuS (GradSuS)

T)

(2.55)

is the so-called Karni-Reiner strain tensor. Finally, aS = FS a0S indicates the fibredirection in the actual configuration. In conclusion, the model is closed and can be solvedfor the primary variables solid displacement uS, effective pore-liquid pressures pBR and pIR

and molar concentration cDm of the therapeutic agent. With regard to the correspondingnumerical solution of the coupled system of partial differential equations (PDE), the(Bubnov-)Galerkin mixed FEM is applied. This process will be described in Chapter 3.

Simplified drug-infusion model for brain tissue

Simulating the convection-enhanced drug-delivery (CED) procedure within brain tissue,the temporal and spatial spreading of the applied therapeutic agent is of particular in-terest. In order to represent this procedure in a brain-tissue model that is as simpleas possible, some simplifications of the previously described general model are comple-mented. Therefore, further preliminary assumptions are initially included, i. e., negligible


gravitational forces, geometrically linear material behaviour (small deformations) and aconstant blood volume fraction nB = nB

0S. Thus, the following relations for fully saturatedconditions are provided for the volume fractions and their temporal changes:

nS = nS0S (detFS)

−1 and (nS)′S = −nS div (uS)′S ,

nB = nB0S and (nB)′S = 0 ,

nI = 1 − nS − nB0S and (nI)′S = − (nS)′S ,

sB = 1 − sI =nB0S

1 − nS, grad sB ≈ 0 .

(2.56)

Following this, the governing balance relations, namely the momentum balance (2.57)1 ofthe overall aggregate, the volume balance (2.57)2 of ϕ

B, the volume balance (2.57)3 of ϕI

and the concentration balance (2.57)4 of ϕD, are simplified, yielding

0 = divT = div(

TSE − 1

1 − nS(nB

0S pBR + nI pIR) I

)

,

0 = nB0S div (uS)

′S + div (nB

0S wB) ,

0 = (1 − nB0S) div (uS)

′S + div (nIwI) ,

0 = nI (cDm)′S + (1 − nB

0S) cDm div (uS)

′S + div (nIcDmwD) .

(2.57)

There, the seepage velocities wB, wI and wD are described via

nB0S wB = − KSB

µBRgrad pBR ,

nI wI = − KSI

µIRgrad pIR ,

nIcDmwD = −DD grad cDm + nIcDmwI .

(2.58)

Furthermore, the overall Cauchy stress results in

T = TSE − 1

1 − nS(nB

0S pBR + nI pIR) I , (2.59)

with TSE from (2.54). As in the general model, the simplified model is closed and can be

solved for the primary variables solid displacement uS, effective pore-liquid pressures pBR

and pIR and molar concentration cDm of the therapeutic agent.

However, it should be noted that the general model provides a larger scope than the sim-plified model. Basically, the application of the TPM to the complex brain-tissue aggregateis motivated to describe the flow and transport processes within the extracellular matrix(ECM). Thereby, the fully coupled consideration of the blood constituent is investigatedto model in-vivo brain tissue based on all physical constituents. The derived simplifica-tions result in a model that does not make use of the full complexity of the modellingapproach. Nevertheless, the simplified model can be used for investigations where only thetemporal and spatial spreading of the applied therapeutic agent is of particular interest.


2.4.3 Extended biphasic intervertebral-disc model

In this subsection, the continuum-mechanical fundamentals are specified for an extendedbiphasic description of an inhomogeneous and anisotropic intervertebral disc (IVD). Whilehereinafter only a comprehensive description of the considered TPM model is presented,a detailed introduction of the customised model can be found in the underlying works ofEhlers et al. [49, 50], Karajan [81, 82] and citations therein.

Regarding an intervertebral disc allows for a rough distinction of two regions, see Figure2.4. Herein, a gelatinous core named nucleus pulposus (NP) can be outlined, which issurrounded by a lamellar structure named anulus fibrosus (AF), where the lamellae of theAF consist of a fibrous structure.

nucleus pulposus(NP)

anulus fibrosus(AF)

lamella of the AF

Figure 2.4: (a) Axial cut through an IVD (courtesy of H.-J. Wilke) and (b) schematic illus-tration of the IVD, cf. Karajan [81].

The underlying continuum-mechanical model is based on the TPM, which offers a greatpossibility to describe the porous micro structure of the IVD consisting of a fibre-reinforcedECM and an ionised pore fluid, cf. Ehlers [44], Ehlers et al. [49] and Karajan [81, 82].Following this, two immiscible constituents are given by the solid skeleton ϕS, which isextended by incorporating volume-free fixed negative charges ϕfc, and the fluid ϕF , whichis a (miscible) mixture of water ϕL and dissolved mobile positive and negative ions ϕ+

and ϕ−, see Figure 2.5.

microscale

macroscale

REV

concept of volume fractions

homogenised model

multicomponent

macro modeldvF

dvS

ϕ = (ϕS ∪ ϕfc) ∪ ϕF

ϕF = ϕL ∪ ϕ+ ∪ ϕ−

Figure 2.5: REV of the qualitative micro-structure of charged hydrated biological tissues andmulticomponent TPM macro model, by courtesy of Karajan [81].

Moreover, preliminary assumptions are included, i. e., materially incompressible con-stituents, isothermal conditions, uniform body force for all constituents (bα = g), negli-gence of acceleration terms (quasi-static conditions) and negligible extra stresses of the


fluid (TFE ≈ 0). Furthermore, the specific permeability tensor

KFspec =

KS(JS)

µFRI with KS(JS) = KS

0S

(JS − nS

0S

1 − nS0S

)κ

(2.60)

basically allows for a deformation-dependent intrinsic permeability KS(JS), where µFR

represents the dynamic fluid viscosity and KS0S denotes the initial intrinsic permeability of

the undeformed reference configuration. In this regard, the intrinsic permeability KS(JS)(expressed in [m2]) represents a purely geometric measure, which is independent of thefluid properties. However, the influence of a deformation-dependent permeability shouldbe negligible regarding the almost impermeable character of the ECM. Therefore, thenonlinearity of the deformation dependence is switched off (κ = 0) in the presentedexamples in Chapter 5, leading to a constant intrinsic permeability KS = KS

0S . Besidethe above mentioned intrinsic permeability KS, the Darcy flow coefficient (hydraulicconductivity) kF = γFRKS/µFR (expressed in [m/s]) is a commonly used form of apermeability measure. Moreover, the overall Cauchy stress

T = TS + TF = −nS P I + TSE − nF P I

= −P I + TSE = −P I − ∆π I + TS

E,mech = −p I + TSE,mech

(2.61)

can be formulated using either the unspecified hydraulic pore pressure P or the overallpressure p = P +∆π in the tissue, where ∆π represents the osmotic pressure difference.Therein, the overall Cauchy stress tensor T is derived by a summation of its partialstresses TS = −nSP I+TS

E and TF = −nFP I. Furthermore, the extra stress TSE of the

solid consists of an osmotic part TSosm = −∆π I and a purely mechanical extra stress

TSE,mech. Following this, either the pair P, uS or the pair p, uS can be used as primary

variables. According to Ehlers & Acarturk [46], an approach using the overall pressurep as unknown field quantity would lead to unstable numerical solutions. Subsequently,following Karajan [81], the hydraulic pressure P and the solid displacement field uS areused as primary variables in this work.

Finally, the governing balance relations, obtained by an addition of the respective con-stituent balances, are given by the overall aggregate balances, namely the momentumbalance (2.62)1 and the volume balance (2.62)2, viz.:

0 = divT + ρg = div (TSE − P I) + (nSρSR + nFρFR) g ,

0 = div (uS)′S + div (nF wF ) ,

with nF wF = −KFspec (gradP − ρFR g) = − KS

µFR(gradP − ρFR g) .

(2.62)

Therein, constitutive assumptions are made to describe the solid’s extra stress TSE , which

includes the intrinsic viscoelastic effects of the ECM. Thus, the model is closed and thecoupled system (2.62) of PDE can be solved numerically, as it is described in the nextchapter.

Chapter 3:Numerical treatment

This chapter is concerned with the numerical treatment of the specific models of porousmaterials discussed in the previous chapter. For this purpose, the finite-element method(FEM) has been proven to be a well-suited numerical tool to approximate the overallsolution of the descriptive set of coupled partial differential equations and to deal withmany degrees of freedom (DOF). Thus, the fundamentals of the FEM are briefly reviewed,followed by appropriate techniques for the temporal discretisation of the arising mixed for-mulation. Afterwards, the required weak formulations of the governing balance relationsas well as reliable solution schemes to solve for the unknown quantities of the discretisedsystems are presented for the specific material models described before.

3.1 Finite-element method in space

Basically, the FEM is established as a commonly used approximation technique in all fieldsof engineering and, in particular, in material modelling. In this regard, a broad varietyof references on the FEM can be found, e. g., Bathe [8], Schwarz [114] or Zienkiewicz &Taylor [131] among others, whereas Ehlers & Ellsiepen [48], Eipper [54], Ellsiepen [55] orAmmann [2] particularly consider the approximation of porous-media models using mixedfinite elements.

3.1.1 Weak formulations and boundary conditions

Within the framework of the FEM, the governing equations need to be brought into a form,which is suitable for a numerical treatment. For this reason, the corresponding primaryvariables ϑs have to be chosen and the local balance relations need to be converted froma local (strong) to an integral (weak) form. For the sake of a compact representation, theprimary unknown field variables ϑs are collected in vector of unknowns u = [ϑ1 ϑ2 ... ]T ,and the weak formulations are summarised in a function vector GGGu. Note that the weakformulations are presented in Section 3.3 for the respective material models described inSection 2.4. They have been derived from the strong forms by integrating over the spatialdomain Ω , which is occupied by the overall aggregate body B at time t. In particular,the variational formulations are obtained by weighting the respective terms of the strongformulations with independent test functions δϑs (collected in a vector of test functionsδu = [ δϑ1 δϑ2 ... ]T ) corresponding to the primary variables ϑs and including so-calledNeumann (natural) orDirichlet (essential) boundary conditions for each governing equa-tion. Therefore, the overall boundary (surface of the body) Γ = ∂Ω of the domain Ω ismathematically split for each of the governing equations into

Γ = ΓD ∪ ΓN with ΓD ∩ ΓN = ∅ . (3.1)

25

26 3 Numerical treatment

While the Dirichlet boundaries ΓD directly correspond to the primary variables ϑs, theNeumann boundaries ΓN contain the natural boundary conditions. Note in passing thatoverlapping boundary conditions between the primary variables are possible, whereasDirichlet and Neumann boundary conditions cannot be defined simultaneously for onespecific primary variable. Moreover, an appropriate ansatz for the unknown field quanti-ties (primary variables) as well as for the test functions needs to be formulated.

3.1.2 Spatial discretisation using mixed finite elements

The numerical implementation of the FEM requires the transfer of the continuous vari-ational problem into a (semi-)discrete formulation. Therefore, the continuous domain Ωneeds to be subdivided into E non-overlapping subdomains Ωe (finite elements), whichin sum yield the approximated domain Ωh, as exemplarily shown in Figure 3.1. Further-more, each finite element (FE) Ωe is defined by ne nodal points Pj

ej=1,...,ne, leading to a

total of n nodes of the FE mesh.

nodal point Pje

continuous domain Ω approximated domain Ωh

subdomain Ωe

Ω ≈ Ωh=

E⋃

e=1

Ωe

Figure 3.1: Exemplary spatial discretisation of a continuous domain Ω .

On the basis of the spatial discretisation, the infinite-dimensional spaces

Su(t) = u ∈ H1(Ω)d : u(x, t) = u(x, t) on ΓD ,Tu = δu ∈ H1(Ω)d : δu(x) = 0 on ΓD

(3.2)

of the continuously defined ansatz and test functions with their corresponding Sobolevspaces H1(Ω), where the superscript d ∈ 1, 2, 3 denotes the dimension of the problem,can be transformed into discrete n-dimensional spaces Sh

u(t) and T h

u. Thereby, the ansatz

functions Su(t) are affine spaces. Applying the well-known Bubnov-Galerkin method (orsimply Galerkin method), the same basis functions Nu are used for both, the ansatz andthe test functions of the respective primary variable. This yields

u(x, t) ≈ uh(x, t) = uh(x, t) +

n∑

j=1

N ju(x)uj(t) ∈ Sh

u(t) ,

δu(x) ≈ δuh(x) =

n∑

j=1

N ju(x) δuj ∈ T h

u,

(3.3)

where uh are the Dirichlet boundary conditions, N juare the global basis functions of the

ansatz or test functions and uj are the unknown nodal quantities (degrees of freedom) at

3.1 Finite-element method in space 27

the nodal point Pj of the FE mesh. Consequently, the ansatz functions naturally van-ish at Dirichlet boundaries, where discrete values are prescribed. Moreover, an efficientimplementation due to sparse matrices results from the consideration of normalised basisfunctions N j

u, which support only those finite elements which are attached to the respec-

tive nodal point Pj (and otherwise are equal to zero). Finally, the spatially discretisedvariational problem can be formulated via:

Find uh ∈ Shu(t) such that GGGh

u(δuh, uh; q) = 0 ∀ δuh ∈ T h

u(3.4)

at any time t ∈ [ t0, T ] for a given set of Neumann boundary conditions. Dependingon the underlying material model, internal variables q, such as, for example, inelasticdeformation tensors, may occur indirectly via the extra-stress tensor of the solid skeleton.Within this spatially discretised problem (3.4), the approximation of all (scalar-valued)unknown nodal quantities is investigated simultaneously in order to solve the problemstraightforward in a monolithic manner. One of the difficulties in the mixed formulationis to find proper ansatz functions. Since the mechanical extra-stress tensor of the solidskeleton implicitly depends on the gradient of the displacement field, its approximationneeds to be one order higher compared with the approximation used for the pore-fluidpressure. Furthermore, the pore-fluid pressure requires at least a linear discretisation asthe gradient of the pressure needs to be computed in the volume balance of the overallaggregate. Thus, the choice of linear as well as quadratic shape functions yields theusage of an element type, which is commonly denoted as extended Taylor-Hood element,cf. Figure (3.2). The used Taylor-Hood elements fulfil the so-called inf-sup condition(also referred to as Ladyshenskaya-Babuska-Brezzi (LBB) condition) and are, as far asthe author is aware, the best choice with respect to the numerical stability and accuracy.Concerning the individual ansatz functions for the specific material models (and thus thespecific primary variables), described in Section 2.4, the used shape functions are furtherdiscussed in Section 3.3.

nodal degrees of freedom

quadratic basis functions

linear basis functions

Figure 3.2: Extended tetrahedral and hexahedral three-dimensional Taylor-Hood elements.

Geometry mapping approaches provide a simple strategy to efficiently perform the nu-merical integration of the weak formulations over finite elements with arbitrary geometry.Therefore, a standard reference element with orthogonal local coordinates ξi is chosen inthe parameter space. Furthermore, the integrations and derivatives are evaluated in theresulting parameter space and a geometrical mapping between the parameter space andthe real space is performed. Thus, the global basis functions are obtained from a geometrytransformation to the global (physical) position x. Using an isoparametric concept, thegeometry and the displacements are expressed by the same set of basis functions.


3.2 Temporal discretisation

In order to choose a suitable time-discretisation method, the spatially discretised but (still)time-continuous equations need to be formulated in an abstract manner allowing for an

adequate characterisation. Therefore, all unknown nodal quantities uj =[ϑj1 ϑj2 ...

]Tof

the primary variables ϑs are collected in a generalised vector of unknownsu = [ϑ11 ... ϑ

n1 ϑ12 ... ϑ

n2 ... ]

T. Moreover, all nodal values in u and all history variables

in q are combined in the general vector of unknowns y := [u q]T and the only appear-

ing material time derivative ( · )′S is expressed for simplicity via ˙( · ). The occurrence ofinternal variables q makes it necessary to introduce a local residuum vector LLLq contain-ing the evolution equations (and its semi-discrete counterpart LLLh

q containing the set ofspace-discrete evolution equations of all integration points of the FE mesh), which areneeded to determine the internal variables. Thus, the abstract formulation of the implicitinitial-value problem can be given using the space-discrete vector F , yielding

F (t, y, y) =

[

GGGhu(t, u, u, q)

LLLhq(t, q, q, u)

]

!= 0 . (3.5)

In a next step, this system needs to be further discretised in the time domain. While here-inafter only a concise introduction of this issue is presented, a detailed discussion of tem-poral discretisation methods for coupled systems can be found in the works of Ammann[2], Diebels et al. [37], Ehlers & Ellsiepen [48], Ellsiepen [55] or Rempler [106]. Restrict-ing to implicit integration schemes and preferring a method from the class of single-stepmethods, the diagonally implicit Runge-Kutta (DIRK) method has been proven to pro-vide a suitable numerical integration scheme for the treated problems. For the purposeof this monograph, only the implicit (backward) Euler time-integration scheme, which isnaturally included in the DIRK method, is used for the temporal discretisation. Withinthis method, the temporal change yn of the primary and the history variables in the cur-rent (actual) time tn is approximated via a difference quotient using the solution vectorof the previous time tn−1 by

yn = y(tn) =yn − yn−1

tn − tn−1

=∆yn∆tn

with yn = yn−1 + ∆yn , (3.6)

where ∆tn denotes the (actual) time increment. Applying this implicit and uncondition-ally stable time-integration scheme to the system of differential-algebraic equations (3.5),the stage increment ∆yn of the current time-step n can be used as unknown quantityinstead of the stage solution yn in order to reduce round-off errors during the solution ofthe time-discrete system

Fn(tn, yn−1 +∆yn,1

∆tn∆yn) =: Rn(∆yn)

!= 0 , (3.7)

introducing a nonlinear functional Rn(∆yn). After solving system (3.7) with a Newton-iteration scheme for the unknown stage increment ∆yk

n at the current Newton iterationstep k, the actual stage increment ∆yk

n can be computed for giving values ykn−1 of the

3.3 Solution procedure of coupled problems 29

previous time-step. In order to apply the Newton-iteration scheme, the required residualtangent (Jacobian matrix)

Jkn =

dRkn

d∆ykn

=1

∆tkn

∂F kn

∂ykn

∣∣∣∣z

+∂F k

n

∂ykn

∣∣∣∣z

(3.8)

is computed based on a central difference quotient. Therein, z = (tkn, ykn, y

kn) represents

the current set of arguments. For the case that internal variables q occur (and sucha local residuum vector LLLh

q is introduced), a generalisation of the Block Gauß-Seidel-Newton method (also known as multilevel Newton procedure) is applied and makes useof the block-structured nature of the resulting residual tangent. Otherwise, the residualtangent can be formulated as

Jkn =

1

∆tkn

∂F kn

∂ukn

∣∣∣∣z

+∂F k

n

∂ukn

∣∣∣∣z

(3.9)

with z = (tkn, ukn, u

kn) and represents a usual sparse FE matrix consisting of a generalised

system matrix Dkn = ∂F k

n /∂ukn and a generalised stiffness matrix Kk

n = ∂F kn /∂u

kn, which

result after analytical linearisation. For more details, the reader is referred to the worksof Diebels et al. [37], Ehlers & Ellsiepen [48] or Ellsiepen [55]. Finally, after the solutionvector yk+1

n = ykn +∆yk

n is updated, the procedure is repeated via the next Newton stepuntil the norm of the residuum is less than a certain pre-defined tolerance εtol.

3.3 Solution procedure of coupled problems

After a generalised approach for the numerical treatment of a descriptive set of coupledpartial differential equations within the FEM has been discussed in the previous twosections, the underlying assumptions concerning the specific TPM models are treatedbelow. Therefore, the necessary weak formulations of the governing balance relationsfor the different material models, described in Section 2.4, are presented and transferredinto the respective abstract formulation of the implicit initial-value problems. Additionalinformation regarding the systematic derivation of the equilibrium equations is given inAppendix B.

3.3.1 Porous-soil model

Regarding the biphasic modelling of a porous soil as discussed in Subsection 2.4.1, the setof primary variables and also the strong formulations of the balance relations differ forthe assumption of either quasi-static or dynamic processes. Following this, the numericalsolution procedure, including the formulation of the weak formulations of the balancerelations and the determination of the global systems of equations, is hereinafter presentedseparately for theses two cases.


Slow (quasi-static) processes

In the strong formulation (2.47) of the quasi-static initial-boundary-value problem of abiphasic model, the primary variables ϑs are the solid displacement field uS and the pore-fluid pressure p. In order to obtain the corresponding weak formulations, the respectivebalance equations are weighted by independent test functions δuS and δp, respectively, andare integrated over the domain Ω . Furthermore, the boundary conditions are consideredand the product rule and the Gaußian integral theorem are applied. In order to allow forthe application of Neumann boundary conditions, the overall surface Γ of the domain Ωis mathematically split into Γ = ΓuS

∪Γt and Γ = Γp∪Γq, where Γt and Γq represent theNeumann (natural) parts of the surface, while ΓuS

and Γp are the Dirichlet boundaries.Thus, a family of functions uS(t), p(t) is called weak solution if for all times t ∈ [ t0, T ]both coupled equations

GuS(uS, p, δuS) ≡

∫

Ω

(TSE − p I) · grad δuS dv−

−∫

Ω

(nSρSR + nFρFR)b · δuS dv −∫

Γt

t · δuS da = 0 ,

Gp(uS, p, δp) ≡∫

Ω

ρFR div (uS)′S δp dv+

+

∫

Ω

ρFR KFspec (grad p − ρFR b) · grad δp dv +

∫

Γq

q δp da = 0 ,

(3.10)and for t = t0 the initial conditions

∫

Ω

(

uS(t0) − uS 0

)

· δuS dv = 0 and

∫

Ω

(

p(t0) − p 0

)

δp dv = 0 (3.11)

for arbitrary δuS and δp are fulfilled. Therein, t = (TSE −pI)n is the external load vector

acting on the boundary Γt, while q = nFwF ·n denotes the volume efflux of the fluid overthe boundary Γq. Moreover, uS = uS on ΓuS

and p = p on Γp exactly fulfil the Dirichletboundary conditions, whereas the test functions δuS and δp naturally vanish on Dirichletboundaries. Using an operator equation, the system of weak formulations (3.10) can bewritten as

Gu(u, u, δu) =

∫

Ω

(

D(u, u, δu) + K(u, δu))

dv −∫

Γ

F(δu) da = 0 , (3.12)


including the vector of unknowns u := [uS p ]T containing the primary variables, thevector of test functions δu := [ δuS δp ]T and the operators

D(u, u, δu) = ρFR div (uS)′S δp

︸︷︷︸

Dp

,

K(u, δu) = (TSE − p I) · grad δuS − (nS ρSR + nF ρFR)b · δuS

︸︷︷︸

KuS

+

+(

ρFR KFspec(grad p − ρFR b)

)

· grad δp︸︷︷︸

Kp

,

F(δu) = t · δuS

︸︷︷︸

FuS

− q δp︸︷︷︸

Fp

.

(3.13)

Herein, the extra stress TSE of the solid and the volume fractions nS and nF of the solid

and the pore fluid, respectively, are functions of the solid displacement field uS. Note inpassing that the terms in the operators (3.13) need to be brought to the same dimension(such that the operators D and K have the dimension 1/m3 and the operator F hasthe dimension 1/m2) before adding them. Following this, the result of the integration in(3.12) is dimensionless. In the next step, the resulting system of differential equationsis discretised using mixed Taylor-Hood elements with a quadratic approximation of thesolid displacement field and a linear approximation of the pore-fluid pressure. To enablea compact (abstract) formulation, the nodal unknowns of each primary variable (degreesof freedom) are collected in a generalised vector of unknowns u ∈ R

N , where N denotesthe total number of degrees of freedom, and analogously the time derivatives and thetest functions of the primary variables are summarised in abstract vectors u ∈ R

N andδu ∈ R

N , respectively, viz.:

u := [uϑ1 uϑ2 ]T = [u1

S ... unS p1 ... pn ]

T=: [uuuS ppp ]T ,

u := [ uϑ1 uϑ2 ]T = [ (u1

S)′S ... (u

nS)

′S (p1)′S ... (p

n)′S ]T

=: [ uuuS ppp ]T ,

δu := [ δuϑ1 δuϑ2 ]T = [ δu1

S ... δunS δp1 ... δpn ]

T=: [ δuuuS δppp ]T .

(3.14)Therein, abstract vectors uuuS ∈ R

3n (for the general three-dimensional FE model) andppp ∈ R

np are introduced to represent the nodal unknowns of uS and p, respectively. Sincelinear ansatz functions are used to approximate the pore-fluid pressure p (whereas the soliddisplacement field uS is approximated using quadratic ansatz functions), the number np

of nodal unknowns of the pore-fluid pressure is not equal to the number n of nodes (nodalpoints of the FE mesh). Within the FE tool, the degrees of freedom of the pore-fluidpressure are included only at the corner nodes of the elements and the nodal unknownsare written either node-wise or element-wise (depending on the chosen representation) intothe generalised vector of unknowns. However, for the sake of a compact representation,


the numbering of the nodal unknowns pj (and analogous δpj) follows here the globalnumbering of the nodes and the nodal unknowns of one primary variable are written afteranother. Within the spatial discretisation, the (discrete) ansatz and test functions aredefined in the form of

uS(x, t) ≈ uhS(x, t) = uh

S(x, t) +

n∑

j=1

N juS(x)uj

S(t) ,

p(x, t) ≈ ph(x, t) = ph(x, t) +

n∑

j=1

N jp (x) p

j(t) ,

δuS(x) ≈ δuhS(x) =

n∑

j=1

N juS(x) δuj

S ,

δp(x) ≈ δph(x) =n∑

j=1

N jp (x) δp

j ,

(3.15)

where N juS

= diag [N juS1, N j

uS2, N j

uS3] (for the three dimensions in space) and N j

p are theglobal basis functions of the ansatz or test functions and uh

S and ph are the Dirichletboundary conditions. In an abstract discrete formulation, the nodal unknowns of the FEmesh and the related nodal test functions are collected in

uh =[uuuhS ppph

]Twith uuuhS = NNNuS

uuuS and ppph = NNNp ppp ,

δuh =[δuuuhS δppph

]Twith δuuuhS = NNNuS

δuuuS and δppph = NNNp δppp(3.16)

using abstract matrices NNNuS∈ R

3n×3n and NNNp ∈ Rnp×np to represent the global basis func-

tions. In such a formulation, the Dirichlet boundary conditions uh do not need to beconsidered, as they are explicitly fulfilled during the assembling of the FE system. Basedon the above considerations and having in mind that the extra stress TS

E of the solid in(3.10) may indirectly depend on internal variables combined in a vector q, the overallsystem of the initial-value problem is described by the abstract matrix equation

F (t, y, y) =

[

GGGhu(t, u, u, q)

LLLhq(t, q, q, u)

]

=

[

D u + k(u, q) − f (t)

A q − rev(q, u)

]

!= 0 , (3.17)

where y(t0) = y0 accounts for initial conditions at an initial time t0. Herein, the functionvector GGGh

u represents a system of N linearly independent equations, where the ith entryGhu i is obtained by setting the discrete test function δui (representing the ith entry ofδu) to 1, while setting the remaining ones to δuk = 0 for k = 1, ..., i − 1, i + 1, ..., N .Following the classical notations of the FEM for linear-elastic problems, D ∈ R

N×N isdenoted as the generalised damping matrix (system matrix), k ∈ R

N as the generalisedstiffness vector and f ∈ R

N as the generalised force vector. If internal variables needto be solved, a local residuum vector LLLq is introduced (where LLLh

q corresponds to itssemi-discrete counterpart) containing the respective evolution equations. Classifying thesystem LLLh

q to be a system of ordinary differential equations (ODE), A can be identified


as a regular identity matrix while rev contains the non-differential terms of the evolutionequations. Due to the nonlinearities in k, the generalised stiffness matrix K ∈ R

N×N

results only after analytical linearisation. Therein, the particular blocks in the blockmatrix K are determined by Kts :=Kϑtϑs

= ∂kϑt/∂uϑs

=: ∂kt/∂uϑs, cf. Ellsiepen [55].

Thus, the equilibrium equation in global form, cf. Appendix B.1, yields

[

0 0

D21 0

]

︸︷︷︸

D

[

uuuS

ppp

]

︸︷︷︸

u(t)

+

[

k1(u, q)

k2(u, q)

]

︸︷︷︸

k(u(t), q)

=

[

f1(t)

f2(t)

]

︸︷︷︸

f (t)

.(3.18)

Therein, for the sake of a compact representation, the nodal values of the force vector f ,which are allocated to the primary variable ϑs, are combined in the vector fϑs

=: fs andthe values of the system matrixD, which are allocated to the primary variable ϑ2 = p andto be multiplied by the time derivatives of the primary variable ϑ1 = uS, are indicatedby Dϑ2ϑ1 =: D21. It becomes obvious that in the case of a quasi-static description,the generalised damping matrix does not possess the full rank and is consequently asingular matrix. Thus, (3.17) becomes a system of differential-algebraic equations (DAE),which suggests the use of an implicit time-integration scheme. When taking a closer lookat the formulation (3.18), the differential index of this DAE system can be found tobe 1, cf. Ellsiepen [55]. Proceeding from the special case of geometrically linear (smalldeformations) as well as materially linear behaviour, the stiffness matrixK is assumed tobe (approximately) time-invariant, so that the equilibrium equation (3.18) can be writtenin form of a linear system, cf. Appendix B.1, viz.:

[

0 0

D21 0

]

︸︷︷︸

D

[

uuuS

ppp

]

︸︷︷︸

u(t)

+

[

K11 K12

0 K22

]

︸︷︷︸

K

[

uuuS

ppp

]

︸︷︷︸

u(t)

=

[

f1 ext(t)

f2 ext(t)

]

︸︷︷︸

fext(t)

,(3.19)

where fext is the generalised external force vector, including the body-force terms b and thespace-discrete Neumann boundary terms f . A systematic derivation of the equilibriumequations (3.18) and (3.19), including a determination of the individual quantities, isprovided in Appendix B.1. For further explanations regarding the representation of thespace-discrete coupled equations in a matrix form using global mass and stiffness matrices,the interested reader is referred to the work of Heider [74] related to saturated porous-media dynamics in a geometrically linear biphasic model.

Dynamic processes

Regarding the more general case of dynamic initial-boundary-value problems, the fluidseepage velocity vector wF is also considered as primary variable in addition to the soliddisplacement vector uS and the pore-fluid pressure p. Following this, the strong for-mulations (2.49) are weighted by independent test functions δuS, δwF and δp and are


integrated over the domain Ω in order to obtain the corresponding weak formulations

GuS≡

∫

Ω

(TSE − p I) · grad δuS dv +

∫

Ω

nSρSR (uS)′′S · δuS dv+

+

∫

Ω

nFρFR[(

(uS)′S + wF

)′S + grad

(

(uS)′S + wF

)

wF

]

· δuS dv−

−∫

Ω

(nSρSR + nFρFR)b · δuS dv −∫

Γt

t · δuS da = 0 ,

GwF≡

∫

Ω

ρFR[(

(uS)′S + wF

)′S + grad

(

(uS)′S + wF

)

wF

]

· δwF dv+

+

∫

Ω

nFγFR

kFwF · δwF dv −

∫

Ω

p div δwF dv −∫

Ω

ρFR b · δwF dv−

−∫

ΓtF

tF · δwF da = 0 ,

Gp ≡∫

Ω

div (uS)′S δp dv −

∫

Ω

nFwF · grad δp dv +

∫

Γq

q δp da = 0 ,

(3.20)

at times t ∈ [ t0, T ], where tF represents the external load vector of the fluid acting onthe Neumann boundary ΓtF . Since only time-integration methods for PDE of first orderin time are considered in this thesis, the order of the set of PDE needs to be reducedfrom second into first order in time. Therefore, the solid velocity field vS is introduced assecondary variable in the set of unknown field variables uS(t), vS(t), wF (t), p(t) andthe relation vS = (uS)

′S is used in addition to the strong formulations (2.49). This leads

to the supplementary weak formulation

G(vS) ≡∫

Ω

(

(uS)′S − vS

)

· δuS dv = 0 . (3.21)

Moreover, the resulting system of differential equations is discretised using mixed Taylor-Hood elements with a quadratic approximation of the solid displacement field uS and thesolid velocity field vS and a linear approximation of the fluid seepage velocity field wF

and the pore-fluid pressure p. To enable a compact formulation, the nodal unknowns arecollected in a generalised vector of unknowns u ∈ R

N , where N denotes the total numberof degrees of freedom, and the test functions of the primary variables are summarised inan abstract vector δu ∈ R

N , yielding

u := [u1S ... u

nS v1

S ... vnS w1

F ... wnF p1 ... pn ]

T=: [uuuS vvvS wwwF ppp ]T ,

δu := [ δu1S ... δu

nS δw1

F ... δwnF δp1 ... δpn ]

T=: [ δuuuS δwwwF δppp ]T .

(3.22)


Following this, the overall system of the initial-value problem can be given (in analogy tothe previous section) in form of an abstract discrete matrix equation

F (t, y, y) =

[

GGGhu(t, u, u, q)

LLLhq(t, q, q, u)

]

=

[

D(u) u + k(u, q) − f (t)

A q − rev(q, u)

]

!= 0 , (3.23)

where y(t0) = y0 accounts for initial conditions at an initial time t0. Finally, the equilib-rium equation in global form can be written as

I 0 0 0

0 D22 D23 0

0 D32 D33 0

0 D42 D43 0

︸︷︷︸

D

uuuS

vvvS

wwwF

ppp

︸︷︷︸

u(t)

+

−vvvS

k2(u, q)

k3(u, q)

k4(u, q)

︸︷︷︸

k(u(t), q)

=

0

f2(t)

f3(t)

f4(t)

︸︷︷︸

f (t)

,(3.24)

cf. Appendix B.2. An alternative formulation can be found using the fluid velocity fieldvF instead of the seepage velocity field wF in the set of primary variables, cf. Heider [74]or Markert [92] among others. This formulation is less prone to instabilities. However, thedifferent possibilities to choose the set of primary variables will not be further discussedhere.

3.3.2 Drug-infusion model for brain tissue

Simulating drug-infusion processes within brain tissue, cf. Subsection 2.4.2, a distinctionis made between a general brain-tissue model and a simplification thereof. While the setof primary variables is the same, the strong formulations of the balance relations differ forthese two models. Following this, the numerical solution procedure, including the weakformulations and the global systems of equations, is hereinafter presented separately forboth cases.

General model

Following the considerations in Subsection 2.4.2, the primary variables of the multiphasicdrug-infusion model are the solid displacement uS, the effective pore-liquid pressures pBR

and pIR and the molar concentration cDm of the therapeutic agent. Thus, independent testfunctions δuS, δp

BR, δpIR and δcDm are used to convert the governing balance relations


(2.51) from a strong to a weak form, yielding

GuS≡

∫

Ω

(

TSE − (sBpBR + sIpIR) I

)

· grad δuS dv−

−∫

Ω

ρb · δuS dv −∫

Γt

t · δuS da = 0 ,

GpBR ≡∫

Ω

(

(nB)′S + nB div(uS)′S

)

δpBR dv−

−∫

Ω

nB wB · grad δpBR dv +

∫

ΓvB

vB δpBR da = 0 ,

GpIR ≡∫

Ω

(

(nI)′S + nI div(uS)′S

)

δpIR dv−

−∫

Ω

nI wI · grad δpIR dv +

∫

ΓvI

vI δpIR da = 0 ,

GcDm≡

∫

Ω

(

nI (cDm)′S + cDm div(uS)

′S + cDm div(nBwB)

)

δcDm dv−

−∫

Ω

nI cDmwD · grad δcDm dv +

∫

ΓD

D δcDm da = 0 .

(3.25)

Therein, the stress vector t = Tn is acting on the boundary Γt of the overall aggre-gate, where n is the outward-oriented unit surface normal vector. The liquid fluxesvB = nB wB · n and vI = nI wI ·n denote the volumetrical efflux out of the domain andD = nI cDmwD · n is the molar efflux of the therapeutic agent.

Afterwards, the system of weak formulations (3.25) can be written in form of an operatorequation, yielding

Gu(u, u, δu) =

∫

Ω

(

D(u, u, δu) + K(u, δu))

dv −∫

Γ

F(δu) da = 0 , (3.26)

with the vector of unknowns u := [uS pBR pIR cDm ]T containing the primary variables,


the vector of test functions δu := [ δuS δpBR δpIR δcDm ]T and the operators

D(u, u, δu) =

[(

nB(uS, pdif))′

S+ nB(uS, pdif) div(uS)

′S

]

δpBR

︸︷︷︸

DpBR

+

+

[(

nI(uS, pdif))′

S+ nI(uS, pdif) div(uS)

′S

]

δpIR

︸︷︷︸

DpIR

+

+(

nI(uS, pdif)(cDm)

′S + cDm div(uS)

′S

)

δcDm︸︷︷︸

DcDm

,

K(u, δu) =[

TSE(uS) −

(

sB(pdif)pBR + sI(pdif)p

IR)

I]

· grad δuS −

−(

nS(uS) ρSR + nB(uS, pdif) ρ

BR + nI(uS, pdif) ρIR)

b · δuS

︸︷︷︸

KuS

+

+KSB

µBR

(

grad pBR+pdif

sB(pdif)grad sB(pdif)− ρBRb

)

· grad δpBR

︸︷︷︸

KpBR

+

+KSI

µIR( grad pIR − ρIR b ) · grad δpIR

︸︷︷︸

KpIR

−

− cDm div

[KSB

µBR

(

grad pBR+pdif

sB(pdif)grad sB(pdif)− ρBRb

)]

δcDm +

+(

DD grad cDm + cDmKSI

µIR(grad pIR − ρIR b)

)

· grad δcDm ,︸︷︷︸

KcDm

F(δu) = t · δuS

︸︷︷︸

FuS

− vB δpBR

︸︷︷︸

FpBR

− vI δpIR

︸︷︷︸

FpIR

− D δcDm︸︷︷︸

FcDm

.

(3.27)

Therein, some of the quantities are functions of the primary variables. In particular, theextra stress TS

E and the volume fraction nS of the solid are functions of the solid displace-ment field uS, the saturation functions sB and sI are functions of the differential pressurepdif (depending on the pore-liquid pressures pBR and pIR), and the volume fractions nB

and nI of the pore liquids are functions of both the solid displacement field uS and thedifferential pressure pdif . These dependencies are displayed in (3.27) with grey colour to


illustrate the couplings. The system (3.25) is then spatially discretised in all primary un-knowns using mixed finite elements with quadratic shape functions for the approximationof the solid displacement field uS and linear shape functions for the approximation of thepore-liquid pressures pIR and pBR as well as for the concentration cDm of the therapeuticalagent and can be solved in a monolithic manner via

F (t, u, u) = D(u) u + k(u) − f (t) = 0 , (3.28)

written in an abstract semi-discrete setting. In (3.28), all degrees of freedom of the system,namely the nodal unknowns of each primary variable, are summarised in the generalisedvector of unknowns u ∈ R

N , where N denotes the total number of degrees of freedom,yielding

u =[u1S ... u

nS pIR 1 ... pIRn pBR 1 ... pBRn cD 1

m ... cDnm

]T

=:[uuuS pppBR pppIR cccDm

]T.

(3.29)

As in the previous subsection, the numbering of the nodal unknowns pIR j , pBRj andcD jm follows, for the sake of a compact representation, the global numbering of the nodes,although (since different ansatz functions are used) the number of nodal unknowns ofthese quantities is smaller than n, and the nodal unknowns of one primary variable arewritten after another. Following this, the equilibrium equation in global form is given by

0 0 0 0

D21(u) D22(u) D23(u) 0

D31(u) D32(u) D33(u) 0

D41(u) 0 0 D44(u)

︸︷︷︸

D(u(t))

uuuS

pppBR

pppIR

cccDm

︸︷︷︸

u(t)

+

k1(u)

k2(u)

k3(u)

k4(u)

︸︷︷︸

k(u(t))

=

f1(t)

f2(t)

f3(t)

f4(t)

︸︷︷︸

f (t)

.

(3.30)Finally, the occurring time derivatives in the DAE system (3.30) are related to the solidmotion. Furthermore, the system is approximated via a temporal discretisation using animplicit Euler time-integration scheme.

Simplified model

As outlined in Subsection 2.4.2, a drug-infusion model for brain tissue that is as simpleas possible can be of particular interest when the temporal and spatial spreading of anapplied therapeutic agent is simulated. Using the governing (strong) balance equations(2.57) of such a simplified model, the corresponding weak formulations of the balance


equations can be determined as follows:

GuS≡∫

Ω

[

TSE(uS)−

1

1 − nS(uS)

(

nB0S p

BR + nI(uS) pIR)

I]

· grad δuS dv−

−∫

Γt

t · δuS da = 0 ,

GpBR ≡∫

Ω

nB0S div(uS)

′S δp

BR dv +

∫

Ω

KSB

µBRgrad pBR · grad δpBR dv+

+

∫

ΓvB

vB δpBR da = 0 ,

GpIR ≡∫

Ω

(1− nB0S) div (uS)

′S δp

IR dv +

∫

Ω

KSI

µIRgrad pIR · grad δpIR dv+

+

∫

ΓvI

vI δpIR da = 0 ,

GcDm≡∫

Ω

[

nI(uS)(cDm)

′S + cDm div(uS)

′S − cDm div

( KSB

µBRgrad pBR

)]

δcDm dv+

+

∫

Ω

(

DD grad cDm + cDmKSI

µIRgrad pIR

)

· grad δcDm dv+

+

∫

ΓD

D δcDm da = 0 .

(3.31)

After spatially discretising the system in analogy to the general model in the previousparagraph, the equilibrium equation in global form can be written in a compact formula-tion, cf. Appendix B.3, viz.:

0 0 0 0

D21 0 0 0

D31 0 0 0

D41(u) 0 0 D44(u)

︸︷︷︸

D(u(t))

uS

pBR

pIR

cDm

︸︷︷︸

u(t)

+

k1(u)

k2(u)

k3(u)

k4(u)

︸︷︷︸

k(u(t))

=

f1(t)

f2(t)

f3(t)

f4(t)

︸︷︷︸

f (t)

. (3.32)

As in the previous examples, the occurring time derivatives in (3.32) are related to the solidmotion and the system is temporally discretised using an implicit Euler time-integrationscheme. The generalised stiffness matrix K is obtained after the analytical linearisation,where Kts := Kϑtϑs

= ∂kϑt/∂uϑs

=: ∂kt/∂uϑscontains particular blocks. Choosing


an initial-boundary-value problem with the corresponding simplifications and special as-sumptions, the stiffness matrix K and the damping matrix D can be assumed to be(approximately) time-invariant matrices. For such a case, the system (3.32) can be writ-ten as

0 0 0 0

D21 0 0 0

D31 0 0 0

D41 0 0 D44

︸︷︷︸

D

uuuS

pppBR

pppIR

cccDm

︸︷︷︸

u

+

K11 K12 K13 0

0 K22 0 0

0 0 K33 0

0 K42 K43 K44

︸︷︷︸

K

uuuS

pppBR

pppIR

cccDm

︸︷︷︸

u

=

f1

f2

f3

f4

︸︷︷︸

f

. (3.33)

A detailed derivation of the system of equations (3.33), including a determination of theindividual quantities, is provided in Appendix B.3.

3.3.3 Intervertebral-disc model

Specifying the generalised approach of the numerical treatment for the extended biphasicintervertebral-disc model presented in Subsection 2.4.3, the set of unknown field vari-ables uS, P contains as primary variables the hydraulic pressure P and the solid dis-placement field uS. Thus, the governing balance relations (2.62) are weighted by theindependent test functions δuS and δP and are integrated over the domain Ω in order toobtain the corresponding weak formulations

GuS(uS, P, δuS) ≡

∫

Ω

(TSE − P I) · grad δuS dv−

−∫

Ω

(nSρSR + nFρFR) g · δuS dv −∫

Γt

t · δuS da = 0 ,

GP(uS, P, δP) ≡∫

Ω

div (uS)′S δP dv+

+

∫

Ω

( KS

µFR(gradP − ρFRg)

)

· grad δP dv +

∫

Γq

q δP da = 0 .

(3.34)In analogy to the procedure presented in Subsection 3.3.1 for the biphasic quasi-staticporous-soil model (and the corresponding details in Appendix B.1), the system of coupledequations can be solved via

F (t, y, y) =

[

GGGhu(t, u, u, q)

LLLhq(t, q, q, u)

]

=

[

D u + k(u, q) − f (t)

A q − rev(q, u)

]

!= 0 , (3.35)

after spatially discretising the system using mixed finite elements. Therein, the nodalunknowns of the primary variables and the test functions are collected in the generalised


vector of unknowns u ∈ RN and the abstract vector δu ∈ R

N , respectively, where Ndenotes the total number of degrees of freedom and n is the number of nodal points ofthe FE mesh, yielding

u := [u1S ... u

nS P1 ... Pn ]

T=: [uuuS ppp ]T ,

δu := [ δu1S ... δu

nS δP1 ... δPn ]

T=: [ δuuuS δppp ]T .

(3.36)

Again, linear ansatz functions are used to approximate the hydraulic pressure P, whereasthe solid displacement field uS is approximated using quadratic ansatz functions. Finally,the equilibrium equation in global form can be written as

[

0 0

D21 0

]

︸︷︷︸

D

[

uuuS

ppp

]

︸︷︷︸

u(t)

+

[

k1(u, q)

k2(u, q)

]

︸︷︷︸

k(u(t), q)

=

[

f1(t)

f2(t)

]

︸︷︷︸

f (t)

,(3.37)

following the explanations made in Subsection 3.3.1. Therein, due to the assumption ofmaterially incompressible constituents, the generalised damping matrix D can be foundto be time-invariant. As discussed before, system (3.37) becomes a DAE system withindex 1, which suggests the use of an implicit time-integration scheme, as outlined inSection 3.2.

Based on the presented considerations concerning the numerical implementation of thespecified TPM models, the numerical solution time of realistic scenarios is often immensewhen using a full-order system. Therefore, possibilities to reduce the linear system ofequations in (3.19) or (3.33) and the nonlinear systems in (3.23), (3.30) or (3.37) arediscussed, applied and customised in the next two chapters.

Chapter 4:Model-reduction methods

The purpose of this chapter is to review the theoretical basis of the considered model-reduction methods. Therefore, an overview of commonly used projection-based reduc-tion techniques is initially given. Afterwards, the mathematical fundamentals of theconsidered techniques, namely the method of proper orthogonal decomposition and thediscrete-empirical-interpolation method, are presented. In this regard, application-drivenmodifications of the classical reduction processes are pointed out. These modificationsare, inter alia, necessary due to the coupled nature of the equation systems discussedin the previous chapter. Moreover, strategies to indicate the error caused by the use ofthe reduced system are presented. In this context, the utilisation of a-posteriori errorestimators as well as the performance of different sampling processes is discussed.

4.1 Overview

With regard to projection-based model-order-reduction techniques, the so-called Krylov-based methods and the singular-value-decomposition(SVD)-based methods (and addi-tionally methods combining aspects of both of these groups) can be identified as centralapproaches, cf. Figure 4.1 for a schematic overview. A more comprehensive descriptioncan be found in the works of Antoulas [4] and Antoulas & Sorensen [5].

model-reduction methods

Krylov-based

methodsmethods

SVD-based

linear systems nonlinear systems

nonlinearities

...

method

method method

multipoint rationalinterpolation PVL Arnoldi

POD

POD PODHankelapproximation

balancedtruncation

empirical

Gramians

TPWL lookup-tableapproach

hyper-reduction(D)EIM GNATMPE

Gappy- ...

...

... ...

SVD: singular-valuedecomposition

PVL: Padevia Lanczos

TPWL:trajectorypiecewiselinear

(D)EIM: (discrete-)empirical-

interpolationmethod

GNAT: Gauss-Newton withapproximated

tensors

MPE:missing-point

estimation

POD: properorthogonal

decomposition

Figure 4.1: Overview of projection-based model-order-reduction methods.

43

44 4 Model-reduction methods

The Krylov-based methods are approximation methods, which are based on the “momentmatching” of the system response, cf. Freund [59], Grimme [64] or Soppa [116]. In thisregard, the idea of moment matching is to interpolate the response function of the modelby comparing the so-called moments (coefficients of series expansions) of the full andthe reduced system. Thereby, the first moments of the transfer function of the reducedmodel need to match with those of the full system. Advantageously, Krylov subspacesfulfil the moment-matching demand without the need to compute the moments of thesystem. The most widely used Krylov-based methods are the Pade via Lanczos (PVL),the Arnoldi procedures and the multipoint rational interpolation. Classical Krylov-basedmethods were developed for solving linear systems. While Krylov-based methods can beused for systems of very high dimension, their drawbacks are that appropriate reducedsimulations are not necessarily stable and that the error of the reduced system is notguaranteed bounded, cf. Antoulas & Sorensen [5].

As the name implies, SVD-based methods have their roots in the singular-value decom-position. For this purpose, a matrix is broken down into the product of three matricesby means of a singular-value decomposition. This allows to directly read off the singularvalues. Two typical SVD-based methods are the balanced truncation and the methodof proper orthogonal decomposition (POD). The balanced-truncation approach was ba-sically proposed by Moore [96] in the context of realisation theory. The key idea of thismethod is to approximate the system by finding important system invariants and re-moving “unimportant” system modes, cf. Herkt [75]. Following this, the variables aretransformed to a basis which makes it easy to identify system states which require alot of control energy and have a low influence on the output. Therefore, two so-calledLyapunov equations need to be solved to determine two Gramian matrices (the so-calledcontrollability and observability Gramians) before the eigenvalues of the product of thesetwo matrices are computed, cf. Gugercin & Antoulas [65], Rowley [109]. This procedureinduces high computational costs and requires a high memory capacity. Consequently,this method is in general used only for systems with a modest dimension (a few hundreddegrees of freedom). A modification of the method of balanced truncation for nonlinearsystems is given by the method of empirical Gramians, see Lall et al. [90]. In contrast tothe previous methods, the POD method can be seen as a method to approximate a givendata set with a low-dimensional subspace by applying a Galerkin projection to determinean appropriate reduced basis in this manner. In order to generate this “training” data, thestate solutions of the system are stored in pre-computations using the full model and areused as typical “snapshots” of the system. This ensures a high flexibility in application.Following the concept of offline/online decomposition, a time-consuming offline phase (in-cluding the pre-computations and the basis generation) is separated from a time-efficientonline phase with reduced simulations. A computationally intensive offline phase paysoff by the possibility to rapidly produce required simulation results in daily routines orby a sufficient number of individual simulations with varying material and/or simulationparameters using a reduced system. The development of the POD method, which wasoriginally introduced in probability theory as Karhunen-Loeve expansion, cf. Sirovich[115], or even earlier in statistics as Hotelling transformation, cf. Hotelling [79], tracesback to fluid-dynamic applications including turbulence, cf. Berkooz et al. [10]. Beyondthat, the POD method was successfully applied to various problems in fluid flow (cf. Ku-

4.1 Overview 45

nisch & Volkwein [88], Rowley et al. [110]), optimal control (cf. Kunisch & Volkwein [86]),aerodynamics (cf. Bui-Thanh et al. [24], Hall et al. [73]), biomechanics (cf. Radermacher &Reese [102]) and structural mechanics (cf. Herkt et al. [76], Radermacher & Reese [103]),to name only a few. It could be shown that the POD method provides very satisfac-tory results and that the obtained approximations generally are more accurate than theones obtained by other existing reduced-basis methods. The POD approach is thereforeused in this contribution as basis of the performed system reductions. However, sincethe POD-Galerkin approximation does in fact significantly reduce the dimension of theequation system, but not the effort to determine the nonlinear terms, the computationaleffort of nonlinear problems cannot be (sufficiently) reduced when exclusively using thePOD method.

Consequently, additional methods are required for the reduction of the nonlinear terms.In the context of elastoplasticity, Radermacher & Reese [103] presented an approach com-bining the POD method with adaptive sub-structuring. For this purpose, the reductionis only applied in sub-domains with approximately elastic material behaviour. However,the efficiency of this approach is restricted to problems where nonlinearities occur only ina small domain. Moreover, the Gappy-POD and the closely related missing-point estima-tion (MPE) as well as the Gauss-Newton-with-approximated-tensors (GNAT) method,the trajectory-piecewise-linear (TPWL) method and the lookup-table approach are fur-ther existing techniques to deal with the nonlinearities in POD-reduced systems. TheGappy-POD approach, proposed by Everson & Sirovich [56], was one of the first meth-ods calculating the nonlinear terms of a POD-reduced systems in an efficient way andwas initially developed to reconstruct missing data. Based on a sparse sampling scheme,this approach reconstructs all remaining components of the nonlinear terms from specificsampled components. The MPE approach was developed by Astrid et al. [6] and usesthe POD basis to perform POD-Galerkin projections over a restricted spatial domainselected with heuristic methods. In contrast, the nonlinear residual and the Jacobianare optimally approximated using a Gappy-POD approach when performing the GNATmethod, cf. Carlberg [30] and Carlberg et al. [29]. Thereby, the structure of the model isnot necessarily preserved. By applying the TPWL method, linearisations of the systemof equations at chosen states are generated, cf. Rewienski & White [107]. The nonlinearsystem is represented with a piecewise-linear system and then each of the pieces is reducedwith a Krylov projection. Therefore, a given data set of the full system is used (as inthe POD method) and projected on a subspace. Even though good approximation resultscould be obtained for different examples, this method is not suitable for all kinds of non-linear functions and might produce wrong (and unstable) solutions if a new evaluationpoint is not close enough to the linearisation point. Furthermore, the required memorycapacity can be very high. Reducing a nonlinear system with a lookup-table approach,pre-computed values of the nonlinear terms are used to generate a decoupled reducedsystem, cf. Herkt et al. [76] or Herkt [75]. Moreover, the pre-computed system vectorsand system matrices are projected on a low-dimensional subspace. Additionally, the non-linear terms are approximated by a Taylor expansion. Subsequently, the approximatedresults can be used as a lookup table whenever the nonlinear terms or their derivativesare needed. However, the disadvantage of such an approach is that only solutions fromthe lookup-table can be properly represented by the reduced system.


Furthermore, the discrete-empirical-interpolation method (DEIM), which was introducedby Chaturantabut and Sorensen [32], and the approach of hyper-reduction (HR), cf. Ryck-elynck [111] and Ryckelynck & Missoum-Benziane [112], are renowned techniques foran efficient reduction of the nonlinear terms. The DEIM is the discrete variant of theempirical-interpolation method (EIM, cf. Barrault et al. [7]) and similar to the empiricaloperator interpolation (cf. Haasdonk & Ohlberger [68], Haasdonk et al. [71]), while theHR approach is related to the Gappy-POD. Using the DEIM in combination with thePOD method, the nonlinear functions are approximated by combining projection ontoa low-dimensional subspace with interpolation. Consequently, the nonlinear terms onlyneed to be evaluated at the selected nodes (denoted to as DEIM points or magic points)of the FE mesh, while all other entries are interpolated. Besides the initial version of theDEIM, several modified approaches have been developed in recent time. Reducing sys-tems with time- and parameter-dependent system matrices, a matrix version of the DEIM,the so-called matrix-discrete-empirical-interpolation method (MDEIM, cf. Bonomi et al.[19], Negri et al. [97], Wirtz et al. [129]), has been proposed. Therein, the nonlinear ma-trix is transferred in a vector before performing the classical steps of the DEIM. However,the application of the MDEIM leads to a more difficult implementation, a more expensiveoffline computation and a higher storage demand. Furthermore, an unassembled versionof the DEIM (UDEIM) was developed by Tiso et al. [120] and Tiso & Rixen [121] inthe context of the FEM. Therein, unassembled vectors are used, which means that eachselected magic point is linked to only one FE element. Using the UDEIM, the computa-tion time of the online phase can be further reduced in many applications. However, thisinvolves an intervention in the numerical FE solver and higher computational cost of theoffline phase due to a larger size of the snapshots of the unassembled nonlinear vector. Afurther extension was proposed by Peherstorfer et al. [101], presenting a localised variantof the DEIM (LDEIM). Herein, local subspaces related to regions with specific systembehaviours are constructed (instead of projecting the nonlinear function onto one globalsubspace) using machine learning techniques. As many works have shown, the DEIMis able to significantly reduce the numerical effort of complex nonlinear processes. Anefficient reduction of nonlinear systems has been performed by Kellems et al. [85] for amodel of spiking neurons, by Chaturantabut & Sorensen [34] for a model with applicationto non-linear miscible viscous fingering, by Nguyen et al. [98] for reacting flow applica-tions, by Negri et al. [97] for parametrised systems, by Radermacher & Reese [104] forsolid mechanics and by Bonomi et al. [19] for the application to parametrised problemsin cardiac mechanics, just to name a few examples. In Wirtz et al. [129] the authorsintroduced concepts for a-posteriori error estimation for DEIM-reduced systems and il-lustrated their results with further application examples. The DEIM and the HR are(methodologically) compared to each other in Fritzen et al. [60]. Therein, the authorsdemonstrated that both methods can significantly reduce the computational effort of anonlinear heat-conduction problem and evinced the similarity of both approaches. Whilethe DEIM can readily be used within a finite-element solver and its implementation isvery intrusive, the implementation of the HR is far less intrusive. Following this, theDEIM is used in this contribution to further reduce nonlinear systems. Keeping in mindthat strongly coupled systems of equations, based on complex material models, need to bereduced, the main focus in this work lies on an application-driven utilisation of the POD

4.2 Model reduction via proper orthogonal decomposition 47

method, either individually or in combination with the DEIM for occurring nonlinearities.In this regard, modifications of the classical reduction processes are performed dependingon the specific problem.

Additionally, it should be mentioned that the selection of the previously compared model-reduction techniques has been made with respect to the aim to reduce a set of discretisedcoupled partial differential equations, which are approximated using the FEM. Thereis no doubt that various other model-reduction methods exist, customised to differentapplications. However, since it is out of the scope of this work, further methods are notdiscussed here.

4.2 Model reduction via proper orthogonal decom-

position

The following section gives a brief overview of the mathematical fundamentals of themethod of proper orthogonal decomposition, needed to reduce a general system of equa-tions. In this context, the reduction of a linear system of equations is provided. Subse-quently, a modification of the classical POD method is presented, which accounts for the(strongly) coupled nature of the equation systems discussed in Chapter 3. Moreover, theerror caused by the use of a POD-reduced system is characterised. In this context, theuse of a-posteriori error estimators is discussed. Concerning the generation of “training”data in general and the selection of specific snapshots in particular, an appropriate choiceis of particular interest and strongly affects the quality of the reduced simulations. Thus,an extra subsection about this topic will be presented at the end of this section.

4.2.1 Fundamentals of the POD method

The basic idea of the POD method is to approximate a given data set with a low-dimensional subspace. Using the POD method, an optimal global basis is generatedfrom a set of state samples associated with specific parameter configurations (or one par-ticular set of parameter values for systems with fixed parameters, hereinafter referred toas parameter-invariant systems) and a default set of boundary conditions. These sam-ples, which provide the so-called snapshots of the system, are numerical solutions of thediscretised system stored in pre-computations using the initial full system. Thus, theyrepresent values of the vectors of unknowns uii=1,...,m ∈ R

N at different times tki (orti for parameter-invariant systems), which are, for parametrised systems, allocated toa specific parameter configuration µji. Subsequently, the snapshots ui are summarisedin a snapshot matrix U = [u1 u2 ... um] ∈ R

N×m, where N denotes the total numberof degrees of freedom of the discretised (full) system and m represents the number ofsampled snapshots. Starting from a Hilbert space V with dimension N , the projectionui ∈ R

N of a vector ui on the l-dimensional subspace V l with the orthonormal basis


vectors ϕrr=1,...,l ∈ RN is denoted by

ui =l∑

r=1

(ui · ϕr)ϕr , (4.1)

where the notation a · b is used to indicate the inner product of two general columnvectors a and b (often indicated as aT b or 〈a,b〉). Following this, the mean squaredprojection error

PE(U ,V l) =1

m

m∑

i=1

∣∣∣

∣∣∣ui −

l∑

r=1

(ui ·ϕr)ϕr

∣∣∣

∣∣∣

2

2(4.2)

of the data set can be formulated, where ||( · )||2=√

( · ) · ( · ) complies with the 2-norm.Therefore, the linearly independent basis vectors ϕr are chosen such that they minimisethe projection error PE(U ,V l). This leads to the following eigenvalue problem

m∑

i=1

ui (ui · ϕr) = λrϕr . (4.3)

The corresponding non-negative eigenvalues λr are ordered from large to small, so thatλ1 > λ2 > ... > 0. The eigenvalue problem (4.3) can be written in terms of the snapshotmatrix U as

U UT ϕr = λr ϕr for r = 1, ..., N , (4.4)

or alternatively as

UT U vr =: C vr = λr vr for r = 1, ..., m , (4.5)

with the corresponding eigenvectors vr ∈ Rm of the positive semi-definite Gramian

matrix C = UTU ∈ Rm×m, which usually contains fewer entries (namely the amount of

snapshots) than the basis vectors ϕr. In (4.5), the basis vectors ϕr and the eigenvec-tors vr are related to each other via the relation ϕr = Uvr. In order to generate thesubspace matrix Φu = [ϕ1 norm ...ϕl norm] ∈ R

N×l, the normalised basis vectors ϕr norm aredetermined via

ϕr norm =Uvr

||Uvr ||2=

m∑

i=1

vri ui

∣∣∣

∣∣∣

m∑

i=1

vri ui

∣∣∣

∣∣∣2

, (4.6)

where vri represent the m entries of the corresponding eigenvector vr. It should benoted that there are further developing approaches performing the POD method with aweighted inner product to perform a “goal-oriented” POD, cf., e. g. , Carlberg & Farhat[27, 28], Haasdonk [66], Hinze & Volkwein [78]. Thereby, the Gramian matrix is definedin the form C = UTGU , where G ∈ R

N×N is a symmetric, positive definite weightingmatrix. This enables the reduced basis to optimally represent specific outputs of interest.In conclusion, the projection error (4.2) can consequently be computed via

PE(U ,V l) =1

m

m∑

r=l+1

λr . (4.7)


Pre-defining a certain tolerance for the projection error, the minimum required dimensionl of the reduced system can be specified. Following this, only the basis vectors ϕrr=1,...,l

of the l largest considered eigenvalues λr are included into the basis, whereas the basisvectors of small eigenvalues are neglected. Finally, the approximation u of the vector ofunknowns u is related to the reduced vector of unknowns ured ∈ R

l (corresponding to thecoefficient vector) via the subspace matrix Φu, yielding

u ≈ u = Φuured . (4.8)

Following these approaches, the reduced system of a general equilibrium equation of theform D(u) u+ k(u) = f with initial degree N can be formulated, viz.:

ΦTuD(Φuured)Φu

︸︷︷︸

D(Φuured)

ured + ΦTu k(Φuured)︸︷︷︸

k(Φuured)

= ΦTu f︸︷︷︸

f

. (4.9)

Herein, the vector of unknowns u is initially replaced by its approximation u = Φuured.Afterwards, using a standard POD-Galerkin approach, the resulting equation (the so-called residual) is multiplied (from the left side) by ΦT

u to require the residual to beorthogonal to the subspace V l. Finally, the reduced system

D(Φuured) ured + k(Φuured) = f (4.10)

with degree l ≪ N can be formulated. Therein, D ∈ Rl×l denotes the reduced system

matrix and k ∈ Rl and f ∈ R

l are the reduced stiffness and force vectors. Note in passingthat it is also possible to use a second subspace matrix Υ for the multiplication of thesystem of equations from left side with ΥT (Petrov-Galerkin projection). Particularly forproblems where numerical oscillations occur, the Petrov-Galerkin projection can providemore accurate and stable results. However, in most applications the (Bubnov-)Galerkinprojection defined by Υ = Φu is used and provides an adequate basis for the approxima-tion. For systems with symmetric positive definite Jacobians, the Galerkin projection iseven the optimal choice.

Considering simulations based on a linear system of equations (implying time-invariantsystem matrices D and K), such as system (3.19) or (3.33), with initial degree N , thereduced system with degree l ≪ N can be obtained as

ΦTuDΦu

︸︷︷︸

D

ured + ΦTuKΦu

︸︷︷︸

K

ured = ΦTu f︸︷︷︸

f

, (4.11)

where K ∈ Rl×l denotes the reduced stiffness matrix. Taking a closer look at the reduced

system (4.10), it becomes obvious that the dimension of the linearised system of equations,which is solved in each iteration step with the Newton-iteration scheme, is considerablysmaller when the reduced system is used instead of the full system. Following this,the POD method reduces the effort to solve the linearised system of equations in eachiteration step. Moreover, the reduced model (4.11) for linear systems is set up in the


first step and can be used through the whole computation. However, due to the subspaceconstruction via a snapshot-based data set, it is obvious that the reduced system canonly describe processes properly, which are as well represented within this data set. Forthis reason, the generation of appropriate snapshots is of particular interest and stronglyaffects the quality of the reduced simulations. Hence, a more detailed discussion of thistopic is conducted at the end of this section. To summarise, a reduced model constructedfrom a rich enough set of snapshots may be used to obtain approximated solutions fora variety of simulations using different initial or boundary conditions and/or differentparameter values. For nonlinear systems, additional methods are required to deal withthe nonlinearities, which will be discussed in Section 4.3. The POD method withoutcombination of other additional methods is only useful when most of the computationalcost lies in the solution of the linearised problem.

Furthermore, it is necessary to incorporate another important character of the consideredporous-media models. On the one hand, the vector of unknowns u contains primaryvariables, which physically exhibit a different behaviour in time, on the other hand, theentries of u have very huge differences in their absolute values. Considering the couplednature of the associated equation systems, this may lead to problems, when applying thePOD method by default. To overcome these problems, a modification of the classicalPOD method is presented in the next subsection.

4.2.2 Modified POD method

When simulating porous-media problems, the vector of unknowns u contains the nodalunknowns of the primary variables ϑs, such as the nodal values of the solid displacementfield uS and the fluid-pore pressure p, respectively. Another example are the nodal un-knowns of the molar concentration cDm within the addressed brain-tissue model. Physically,these primary variables exhibit a different behaviour in time. To obtain computationallyefficient simulation results, the block structure of the (strongly) coupled equation systemneeds to be preserved while considering the different temporal (physical) behaviour ofthe primary variables. Moreover, the entries of the vector u have very huge differencesin their absolute values. This may lead to numerical problems when applying the PODmethod by default because a relatively large change in an entry with small absolute valueis comparably small if compared to a relatively small change in an entry with a signif-icantly larger absolute value. As a consequence, rounding errors may occur which arenot negligible due to the strongly coupled form of the equation system. Following this,a modified POD method is presented in this subsection. Similar modification processeshave been applied in Chaturantabut & Sorensen [34] for the dimension reduction of non-linear miscible viscous fingering or in Zhou [130], reducing nonlinear dynamical systemsfor models of, e. g., a turbo reactor.

In order to obtain a modified reduction matrix Φmod, which preserves the block structureof the coupled equation system and is not influenced by huge differences in the absolutevalues of the primary variables, the snapshot matrix U is separated into smaller snapshotmatrices for each primary variable ϑs, yielding

Uϑ1 = [uϑ1 1 uϑ1 2 ... uϑ1 m ] , Uϑ2 = [uϑ2 1 uϑ2 2 ... uϑ2 m ] , ... . (4.12)


Therein, the vector uϑs ii=1,...,m ∈ RNϑs contains all nodal values of the snapshot ui,

which are allocated to the primary variable ϑs, leading to a total number of Nϑsentries

with totally N = (Nϑ1 +Nϑ2 + ...) degrees of freedom. Since extended Taylor-Hood el-ements with different ansatz functions for the different primary variables are used forreasons of numerical stability of the full system, the number of nodes differs for these pri-mary variables. In the next step, the eigenvalue problem (4.5) is solved for each of thesesnapshot matrices separately. Following this, individually obtained subspace matrices Φϑs

are generated and summarised in an unsorted reduction matrix

Φunsort =

Φϑ1 0 0

0 Φϑ2 0

0 0. . .

. (4.13)

For each subspace matrix Φϑs, the basis vectors corresponding to the lϑs

largest consideredeigenvalues of the respective Gramian matrix Cϑs

are included into the basis. Thus, thetotal dimension l ≪ N of the reduced system yields l = lϑ1+ lϑ2+ ..., which corresponds tothe reduced number of DOF. Finally, the reduction matrixΦmod is obtained by rearrangingthe lines of Φunsort. Depending on the way the nodal unknowns are written into thevector of unknowns u, the structure of an assembled snapshot ui unsort = [ui ϑ1 ui ϑ2 ...]

T

may differ from the structure of u. Therefore, the lines of Φunsort need to be resortedin the same way as uiunsort would be resorted in order to obtain the structure of u.After generating the modified reduction matrix Φmod, the reduction can be performedin analogy to the approach presented in Subsection 4.2.1 for the classical POD method,using the reduction matrix Φu := Φmod. This yields the approximation

u ≈ u = Φmodured . (4.14)

The results obtained by using either the modified or the classical POD method are com-pared in Chapter 5, on the one hand, for linear problems and, on the other hand, fornonlinear problems using the POD method in combination with the DEIM, which is anal-ysed in Subsection 4.3.1. A detailed derivation and the final formulation of the reducedsystems customised to the specific problems can be found in Appendix C.

4.2.3 Error bounds and error estimation for efficient reduced-

order models

In the context of projection-based model reduction in general, and the application ofthe POD approach in particular, the use of a reduced system is in general numericallyefficient, but also erroneous. Thus, an appropriate error indicator can certify the reducedmodel and provides a useful tool in the context of sampling strategies for the snapshotsas well as for the estimation of the necessary dimension of the reduced system. Beforedifferent sampling schemes (with and without the use of a-posteriori error estimation) arediscussed in the next subsection, a brief overview of different error estimators and errorbounds is provided in the following. For this purpose, the error between the full and thereduced system is initially characterised and associated with a-posteriori error indicators.


Characterisation of the error between the full and the reduced system

Comparing the solution vector u(t) of the full discretised system of equations and itsapproximation u(t) = Φuured(t), the error ε(t) between these two solutions at a specifictime t is defined as

ε(t) := u(t) − Φuured(t) . (4.15)

Therein, the error caused by the discretisation of the FEM is not considered, since thefocus is on the reduction of the discretised system of equations. In practice, the solutionof the full system is unknown or not available or its determination is computationallytoo expensive. Therefore, the use of a residual-based formulation of the error is com-mon practice, cf., e. g. , Amsallem & Hetmaniuk [3], Haasdonk & Ohlberger [69, 70] orPaul-Dubois-Taine & Amsallem [100]. Substituting the full-order solution u(t) in thediscretised system D u(t) + k(u(t))− f (t) = 0 with its approximation u(t) leads to theformulation

DΦuured(t) + k(Φuured(t)) − f (t) = r(t) , (4.16)

where the residual vector r(t) is introduced. Due to the enforced orthogonality of r(t)to Φu, the residual vector does not appear in the reduced system of equations (4.9).Subtracting (4.16) from the full-order system and restricting on problems with a time-invariant system matrix D, the residual vector can be formulated as

r(t) = D (Φuured(t)− u(t)) + k(Φuured(t)− u(t)) . (4.17)

Inserting relation (4.15) into equation (4.17), it can be shown that the error ε(t) satisfiesthe differential equation

D ε(t) + k(ε(t)) + r(t) = 0 . (4.18)

Considering problems with a time-invariant stiffness matrix K = ∂k(u(t))/∂u(t), theerror ε(t) can be derived as

ε(t) = e−D−1K t ε0 −t∫

t=0

e−D−1K (t−s)D−1 r(s) ds , (4.19)

where ε0 = ε(t = 0) denotes the initial error. However, the discretised coupled problemsdiscussed in this work result in systems of DAE with differential index of 1, where thegeneralised system matrixD does not possess the full rank and consequently is a singularmatrix and therefore is non-invertible. Thus, a residual-based formulation of the erroras outlined above cannot be achieved without further assumptions and reformulations.Therefore, an adapted formulation of the residual-based error is presented in Subsection5.1.3, exemplarily for the biphasic porous-soil model under the assumption of quasi-staticconditions.

Since a determination of the exact error is not always available, an alternative approachcan be found in the use of error indicators. This subject is further addressed in thefollowing by discussing concepts for (a-posteriori) error estimation.


Error bounds and (a-posteriori) error estimates

Performing a simulation for a specific set of parameter and boundary conditions on thebasis of a POD-reduced system of equations with dimension l, leads to an error betweenthe full and the approximated solution as well as to an error between an output of interestand its approximation computed within the reduced simulation. These errors depend, inparticular, on two factors: the dimension l of the reduced system and the choice of thesampled snapshots. Besides, a weighting of the snapshots can, where appropriate, have animpact. Moreover, these errors can exceed a maximum acceptable error εtol, but can alsobe considerably below such a value with the consequence that the number l of consideredPOD modes is larger than required, which leads to the result that the computationalefficiency is unnecessarily restricted. A-posteriori error estimators can help assessingthe error introduced by the reduced-basis approximation. Moreover, they can devisean efficient procedure for selecting the snapshots. Particularly as a part of a Greedyprocedure, a-posteriori error estimators are of particular interest, which will be discussedin the next subsection. Applying the POD method, the reduced basis is constructed suchthat it is optimal with respect to the minimisation of the approximation error concerningthe sampled snapshots. Following this, the dimension of the reduced system is determinedby pre-defining a certain tolerance for the projection error PE(U ,V l), defined in (4.7).This error corresponds to the minimum 2-norm of the error between the solution vectorssampled as snapshots and their approximation using a POD basis. However, this errorprovides no information about the error ε between the full and the approximated solutionvector when performing a simulation where the parameter and/or loading conditions differfrom the ones used for the pre-computations.

As mentioned before, the determination of an exact error ε, resulting from the reducedformulation of the system of equations, is not always available or leads to far too highmemory requirements or computational costs. Therefore, problem-specific error bounds

||ε||G≤ εbound , (4.20)

where G is a symmetric positive definite matrix defining a suitable vector norm||v||G=

√

v · (Gv), are used in many applications to estimate the error. The choice of thespecific norm depends on the respective problem. In the context of reduced parametrisedlinear dynamical systems, Haasdonk & Ohlberger [70] derived an error bound and pre-sented a residual-based error analysis. Since they compute the exact residual normsduring the reduced simulations in order to obtain such an error bound, an a-priori errorestimate is turned into an a-posteriori error estimator. Restricting on first-order linearsystems with time-invariant system matrices, Amsallem & Hetmaniuk [3] determined amore generalised error bound and proposed an a-posteriori error indicator based on ananalytical representation of the error. Another approach is used in the work of Bhattet al. [11], deriving an error estimator on the basis of a logarithmic Lipschitz constant ofa function (or finally on the basis of the gradient of this function). In the context of non-linear dynamical systems, Wirtz et al. [129] introduced extended concepts for a-posteriorierror estimation, also on the basis of logarithmic Lipschitz constants, and formulated thecorresponding error between the full and the reduced system. All these approaches enablea full decomposition of the computation into an offline and an online stage.


4.2.4 Selection of snapshots

Applying the POD method to general parametrised systems, the solution vectoru = u(t,µ) of such a system is time- and parameter-dependent. Consequently, m par-ticular solutions ui = u(tki,µji)i=1,...,m, allocated to a specific pre-defined parameterconfiguration µj and a specific time tk during a pre-computation, need to be constructed.Due to the subspace construction via a snapshot-based data set, it is obvious that a POD-reduced system can only describe processes properly, which are as well represented withinthe underlying snapshots. Thus, an appropriate strategy to sample the “training” datawithin the online phase, respectively to select the parameter configurations µj (materialparameters and/or loading scenarios) for the pre-computations, is fundamental in orderto capture the physical behaviour of the model in the entire parameter domain. Moreover,the specific choice of the snapshots strongly affects the quality of the reduced simulations.On the one hand, too many (especially the consideration of irrelevant) snapshots candiminish the results and the necessary number of POD modes. On the other hand, onlystates and phenomena which are represented by the snapshot data set can be properlyrepresented. Note in passing that, with respect to the time domain, the snapshots aresampled in this contribution at each time-step of the performed pre-computations. Butthere exist also approaches in which the choice of these time instances is optimised, suchas, for example, in the work of Kunisch & Volkwein [89], or concepts in which the snap-shots are weighted using a weighted version of the Gramian matrix, cf., e. g. , Haasdonk[66], Hinze & Volkwein [78]. Furthermore, it should be noted that Carlberg & Farhat[27, 28] proposed a “goal-oriented compact POD” that, in a similar way, weights thesnapshots of a static parametrised system via an inner product that enables the reducedbasis to optimally represent specific outputs of interest.

With respect to the selection of the parameter configurations µj, either physical intuitionor a uniform distribution can be a good choice if only few quantities need to be variedand/or less information about their distribution is available. However, a systematic andautomatic strategy to generate the snapshot data set is in general preferable. This isparticularly the case if many material parameters or loading conditions need to be varied,resulting in a large dimension of the parameter domain. For model-reduction methodsbased on the POD method, Greedy approaches are popular and widely used, cf. Buffaet al. [23], Bui-Thanh et al. [25, 26], Grepl & Patera [63], Haasdonk [66], Haasdonk et al.[67], Patera & Rozza [99] and Veroy & Patera [125] among others. Applying a Greedy pro-cedure, the reduced model is determined iteratively. In each iteration step, that parameterconfiguration, at which the current reduced model leads to the largest error, is investi-gated. Subsequently, this parameter configuration is used to sample additional snapshotsand recalculate the reduced basis for the next iteration step. Computing an error boundinstead of using the exact error (comparing the solution vector of the full system, which isnot always available, with the one from the reduced system), an alternative approach canbe found in the use of a-posteriori error indicators within the Greedy procedure, cf. Am-sallem & Hetmaniuk [3], Carlberg et al. [29], Grepl & Patera [63], Haasdonk & Ohlberger[69, 70], Paul-Dubois-Taine & Amsallem [100], Veroy & Patera [125]. Furthermore, itshould be mentioned that Reese et al. [105] recently presented an approach combiningthe POD method with an hierarchical tensor approximation (HTA, cf. Grasedyck et al.


[62]) in the context of uncertain parameters. Thereby, adaptive projection matrices aredeveloped during the simulation and the HTA is used for uncertainty quantification.

Sampling via physical intuition or standard sampling schemes

Considering problems with a very small number of varied parameters, such as loadingscenarios with a varying loading function or simulations, where only a handful of materialparameters are varied within given limits, a standard sampling scheme or a sampling viaphysical intuition provides a good basis to generate the parameter configurations of thepre-computations.

Possible standard sampling schemes are, for example, uniform sampling (uniform grid-ding within default limits) or random sampling. If no or only little information aboutthe variable distribution is available, a uniform distribution may be the best choice tocover the whole value ranges of the varying parameters. However, uniform sampling willrapidly become computationally expensive if the number of varied parameters increases.In contrast, random sampling is generally computationally less expensive, while it tendsto miss important ranges of the parameter values. When using physical intuition to selectthe parameter and/or the loading scenarios for the pre-computations and thus to producethe “training” data, the focus is usually on the minimum and maximum values of theparameters as well as on particularly important values or value ranges. Consequently, theresulting reduced systems usually provide sufficiently precise simulation results for prob-lems with a small number of varied parameters. Therefore, physical intuition is hereinafterused for such types of problems. However, such an approach is neither automatic nor suit-able for a high number of varied parameters since a variety of combinatorial samples isneeded to cover the value ranges of all these parameters.

Sampling via the Greedy procedure

A systematic and automatic sampling procedure is given by the Greedy sampling method,which was introduced in Grepl & Patera [63], Veroy & Patera [125], Veroy et al. [126].Using the standard Greedy procedure, the samples are chosen adaptively and the reducedmodel is determined iteratively by investigating in each iteration from a set of c previously(randomly) selected candidate parameter configurations µNc

Nc=1,...,c that configuration

∗µNj

Nj=1,...,j, at which the current reduced model leads to the largest error

εj = maxε(µNc)Nc=1,...,c = ε(

∗µNj

) = u(∗µNj

)− u(∗µNj

) . (4.21)

More precisely, that configuration, at which a specific norm of the error provides thelargest value, is determined. In this regard, the norm of the error is given by

|||ε(µ)|||G= |||u(µ)− u(µ)|||G=

T∫

t=t0

||u(t,µ) − u(t,µ)||G dt , (4.22)

where u(t,µ) and u(t,µ) = Φuured(t,µ) are the solution vectors of the full and the re-duced discretised system of equations at a specific time t for a specific parameter configu-ration µ. Following this, full-order simulations for all candidate parameter configurations


need to be performed beforehand in order to determine the respective errors. Thus, the

parameter configuration∗µNj

is chosen in each iteration of the Greedy procedure as themaximiser

∗µNj

= argmaxµ∈µ1,...,µc

|||ε(µ)|||G . (4.23)

It should be noticed that in optimisation problems the norm of the error between an out-put of interest and its approximation is often used to determine the relevant parameter

configurations, cf., e. g. , Bui-Thanh et al. [25]. After the parameter configuration∗µNj

ofthe current iteration step is determined, the corresponding state samples of the respec-tive full-order simulation for this parameter configuration are added to the snapshot set.Afterwards, the reduced basis is firstly recalculated with the upgraded training data andthen used for the next iteration step. This procedure is repeated until the norm of theerror reaches a prescribed value εtol or a maximum number of iteration steps has beenperformed, cf. Algorithm 1.

Algorithm 1 Greedy procedure using the norm |||ε(µ)|||G1 : Select c candidate parameter configurations µNc

Nc=1,...,c .2 : For all µNc

Nc=1,...,c perform full-order simulations and sample thesolution vectors u(µNc

) .

3 : Select an initial candidate parameter configuration∗µ1∈ µ1, ...,µc

and add the corresponding state samples u(∗µ1) to the snapshot set .

4 : Generate the initial reduced basis Φ1u and set Nj = 2 .

5 : For all µNcNc=1,...,c perform reduced-order simulations, sample the

solution vectors u(µNc) and determine the respective error norms

|||ε(µNc)|||G .

6 : Select the parameter configuration∗µNj

= argmaxµ∈µ1,...,µc |||ε(µ)|||Gand add the corresponding state samples u(

∗µNj

) to the snapshot set .

7 : Generate the extended reduced basis ΦNju and set Nj = Nj + 1 .

8 : Repeat steps 5− 8 until maxµ∈µ1,...,µc |||ε(µ)|||G≤ εtol or until Nj = j .

For a general problem, a good error norm may not be available or the determination ofthe full-order solution u(µ) leads to far too high memory requirements or computationalcosts. Thus, the error norm is in practice replaced by an error indicator providing an

upper bound εbound(µ) for the error. Consequently, the parameter configuration∗µNj

ischosen in each iteration of the Greedy procedure as the maximiser

∗µNj

= argmaxµ∈µ1,...,µc

εbound(µ) . (4.24)

A closer look at the derived formulation (4.19) motivates the use of the residualr(·,µ) = r(t,Φuured(t,µ),Φuured(t,µ)), defined in (4.16), to indicate the error. Andalso the use of a vector norm ||ε(µ)||D, supposing a symmetric positive definite systemmatrix D, as applied in Amsallem & Hetmaniuk [3] or Paul-Dubois-Taine & Amsallem


[100], is motivated by an expression of the error in the form (4.19). An alternative pop-ular error indicator is the 2-norm ||v||2= √

v · v of the residual r(t,µ), yielding the errorindicator

|||ε(µ)|||2≤ Cε ||ε(t0,µ)||2+Cε

T∫

t=t0

||r(t,µ)||2 dt =: εbound(µ) , (4.25)

cf., e. g. , Haasdonk & Ohlberger [69, 70]. Herein, the sum of the norm of the initial errorε(t0,µ) (which is equal to zero when assuming zero-initial conditions for the solutionvector) and the norm |||r(µ)|||2, weighted with a possibly parameter-dependent constantCε, serves as upper bound for the exact error. However, the given formulation (4.25) isrestricted to problems where the system matrices fulfil certain properties (such as positivedefiniteness). Note in passing that Haasdonk et al. [72] are using the term “strong”Greedy procedure to characterise that the true projection error is used as error indicator,as done, e. g. , in Buffa et al. [23], Haasdonk [66], Haasdonk et al. [72]. In contrast,the use of a-posteriori error estimators to determine such an error indicator is referred toas “weak” Greedy procedure. An approach similar to the aforementioned residual-basedGreedy-POD method has been proposed by Hinze & Kunkel [77]. Therein, a weightedresidual of a nonlinear system of differential equations is used to estimate the error withoutproviding an upper bound for the error. If the norm of this weighted residual is smallerthan a pre-defined tolerance, the sampling process is completed - otherwise, the parameter

configuration with the highest value of this norm is chosen as parameter configuration∗µNj

of the current iteration step. Afterwards, a simulation using the full system is performedfor this parameter configuration and the corresponding state samples are added to thesnapshot set. After the reduced basis is recalculated, the next iteration step is performed,cf. Algorithm 2.

Algorithm 2 Greedy procedure using an error indicator with upper bound εbound(µ)

1 : Select c candidate parameter configurations µNcNc=1,...,c .

2 : Perform a full-order simulation with an initial candidate parameter

configuration∗µ1∈ µ1, ...,µc and sample the snapshots u(

∗µ1) .

3 : Generate the initial reduced basis Φ1u and set Nj = 2 .

4 : For all µNcNc=1,...,c perform reduced-order simulations and determine

the respective error indicator εbound(µNc) .

5 : Select the parameter configuration∗µNj

= argmaxµ∈µ1,...,µc εbound(µ) .

6 : Perform a full-order simulation with the parameter configuration∗µNj

and sample the snapshots u(∗µNj

) .

7 : Generate the extended reduced basis ΦNju and set Nj = Nj + 1 .

8 : Repeat steps 4− 8 until maxµ∈µ1,...,µc εbound(µ) ≤ εtol or until Nj = j .

However, as mentioned in the previous subsection, the problem with a residual-basedformulation of the error - and such with an error estimation on the basis of a (weighted)residual - with respect to the discretised problems discussed in this work, is the singularityof the system matrix. Therefore, adapted approaches for a Greedy-POD method on the


basis of an error indicator are discussed in Section 5.4 for the extended biphasic IVDmodel.

An efficient extension of the standard Greedy approach is given by an adaptive Greedyprocedure (cf. Haasdonk et al. [67], Paul-Dubois-Taine & Amsallem [100]) in which theset of candidate parameter configurations is adaptively enlarged during the sampling pro-cess. This extension addresses two main problems accompanied by the standard Greedyprocedure: the issue of over-fitting (while a small training set is fitted very well, this isnot the case for independent test parameter configurations) and the problem of excessivetraining times in case of large training sets. However, since it is out of the scope of thiswork, an extension to an adaptive approach is not further discussed in this monograph.

4.3 Model reduction of nonlinear systems

Using the porous-media models presented in Section 2.4 and numerically discretised inSection 3.3 with all their complexity, it is crucial to account for nonlinearities. Whenapplying the POD-Galerkin approximation to the corresponding equation systems, thedimension of these systems is significantly reduced. However, the computational costs todetermine the nonlinear terms still scale with the dimension of the full system. Conse-quently, the computational effort of nonlinear problems cannot be (sufficiently) reducedwhen exclusively using the POD method. In contrast, additional methods provide an ap-proach to reduce the evaluation costs of the nonlinear terms considerably. As the DEIMhas been proven to be a method, which efficiently extends the POD method for nonlinearproblems, all necessary fundamentals of the DEIM are presented within this section.

4.3.1 Discrete-empirical-interpolation method

As many works have shown, the DEIM is able to significantly improve the efficiency incomputing the projected nonlinear terms in the reduced system. Chaturantabut andSorensen [32] introduced the DEIM, which is similar to the empirical operator interpola-tion (cf. Haasdonk & Ohlberger [68], Haasdonk et al. [71]), as the discrete variant of theEIM, cf. Barrault et al. [7]. Based on this work, Wirtz et al. [129] introduced conceptsfor a-posteriori error estimation for DEIM-reduced systems. Initially, the DEIM variantof the EIM was developed in order to apply to arbitrary systems of ordinary differen-tial equations, regardless of the applied numerical approximation technique (such as theFEM, the finite-difference method or the finite-volume method). Thus, it can readily beused within the finite-element solver PANDAS. In the following, the basic idea of theDEIM is first briefly illustrated. Afterwards, the mathematical fundamentals and thefinal formulation of an appropriate reduced system of equations are presented.

Basic idea of the DEIM

According to the derivations in Subsection 4.2.1, a reduced formulation of a general equa-tion system D(u) u + k(u) = f , with an initial degree N and with the nonlinear termsD(u(t)) and k(u(t)), is given by (4.10) as D(Φuured) ured + k(Φuured) = f with a degree

4.3 Model reduction of nonlinear systems 59

l ≪ N . Therein, the reduced nonlinear terms

D(Φuured) := ΦTu

︸︷︷︸

∈ Rl×N

D(Φuured)︸︷︷︸

∈ RN×N

Φu

︸︷︷︸

∈ RN×l

and k(Φuured) := ΦTu

︸︷︷︸

∈ Rl×N

k(Φuured)︸︷︷︸

∈ RN

(4.26)

have a computational complexity that still depends on the full number N of DOF. Thus,solving this system might be still as costly as solving the original system. Applying theDEIM in combination with the POD method (POD-DEIM) to construct a reduced-ordermodel of a complex nonlinear process, the nonlinear functions are approximated by com-bining projection with interpolation, cf. Chaturantabut & Sorensen [32, 33, 34]. Whilea dimension reduction of the discretised system is performed via the POD method usinga Galerkin projection, the nonlinear terms are additionally interpolated performing theDEIM. Therefore, specially selected interpolation indices are constructed to enable aninterpolation-based projection that provides a nearly 2-norm-optimal subspace approxi-mation of the nonlinear functions. These interpolation indices refer to important entriesof the nonlinear terms (and thus to selected nodes of the FE grid, cf. Figure 4.2) andare denoted to as DEIM points or magic points. Consequently, a nonlinear function onlyneeds to be determined at the selected nodes. Applying the DEIM to a FE framework, therespective value of the nonlinear term is obtained by assembling the partial values of allneighbouring FE elements. Thus, only the corresponding elements need to be consideredduring the simulation for the composition of the respective nonlinear terms via interpola-tion, cf. Figure 4.2. In most cases, the selected magic points are clustered close to spatialregions, where the nonlinear function increases sharply, or rather close to regions with highnonlinear material behaviour. Following the approaches of Chaturantabut & Sorensen [32]and Chaturantabut [31], the interpolation indices are selected using a POD basis as in-put, although, in general, different types of bases are possible. Thus, two different setsof POD bases are used, related, on the one hand, to the state variables and, on the otherhand, to the nonlinear terms. Note in passing that there is also an unassembled versionof the DEIM, operating with unassembled vectors. As a result, each selected magic pointis linked with just one FE element, cf. Tiso et al. [120], Tiso & Rixen [121]. In this way,the computation time of the online phase can be further reduced. However, the UDEIMrequires an intervention in the numerical FE solver and higher computational costs of theoffline phase, due to a larger size of the snapshots of the unassembled nonlinear vector.

q(t)loaded,undrained

free,drained

free, undrained

Figure 4.2: First 10 selected magic points (red nodes) and neighbouring FE elements (high-lighted grey) for an exemplary model of a porous material undergoing large deformations withloading and boundary conditions and FE mesh of the initial-boundary-value problem.


Fundamentals of the DEIM

For a further reduction of a POD-reduced nonlinear system of equations (4.9), the non-linear functions D(Φuured)Φu ured and k(Φuured) are approximated via a projection to asubspace Vk, spanned by a basis of dimension k ≪ N . Thus, the approximation w ∈ R

N

of the composed nonlinear term

w(Φuured) := D(Φuured)Φu ured + k(Φuured) (4.27)

is of the form

w(Φuured) ≈ w(Φuured) = Ψw c . (4.28)

Therein, Ψw = [ψ1 ...ψk] ∈ RN×k represents the subspace matrix with the orthonormal

basis vectors ψrr=1, ..., k ∈ RN and c(t) ∈ R

k is the corresponding coefficient vector. Theprojection basis ψ1 ...ψk is constructed by applying the POD method to the nonlinearsnapshots wi(ui) = Di(ui) ˙ui + ki(ui), obtained from pre-computations using the POD-reduced system. This leads to significantly more precise results than storing the nonlinearterms directly in the pre-computations of the full system. In contrast to the snapshots ofthe state variables, the snapshotswi are stored not only at the converged states, but also atall non-converged states within the iterative solution process, which provides considerablyimproved results. Following this, the nonlinear snapshots are the sets wi(ui)i=1, ...,w withthe approximated vectors of unknowns ui at each Newton step (with totally w Newtonsteps). To determine the coefficient vector c, k rows from the overdetermined systemw(Φuured) = Ψw c (and thus important entries of the nonlinear terms) are selected.These entries (also referred to as DEIM points or magic points) correspond to the DOFon which the nonlinear terms are computed in the reduced simulation. Therefore, amatrix P = [ep1 ... epk ] ∈ R

N×k is considered, where each vector epr complies with thepr-th column of the identity matrix I ∈ R

N×N . The interpolation indices pr can bedetermined by Algorithm 3, cf. Chaturantabut & Sorensen [34].

Algorithm 3 DEIM

1 : [ |ρ|, p1 ] = max |ψ1|2 : Ψw = [ψ1 ], P = [ ep1 ], p = [ p1 ]3 : for r = 2 to k do4 : Solve (P T Ψw) cr = P T ψr for cr5 : rr = ψr − Ψw cr6 : [ |ρ|, pr ] = max |rr|7 : Ψw = [ψ1 ...ψr], P = [ ep1 ... epr ], p = [ p1 ... pr ]

T

8 : end for

Therein, the first interpolation index p1 corresponds to the entry of the first basis vector ψ1

with the largest magnitude. The other interpolation indices pr with r = 2, ..., k corre-spond to the entries with the largest magnitude of the vector rr = ψr − Ψw cr, whichcan be interpreted as a residual representing the “error” between the basis vector ψr andits approximation Ψwcr = Ψw (P TΨw)

−1P T ψr from interpolating the projection basisψ1 ...ψr−1 at the interpolation indices p1 ... pr−1. Following this, the coefficient vector

4.3 Model reduction of nonlinear systems 61

c can be determined from

P Twsel(Φuured) = (P TΨw) c . (4.29)

Inserting the result into relation (4.28), leads to the final approximation

w(Φuured) = Ψw c = Ψw (P TΨw)−1P Twsel(Φuured) . (4.30)

Thus, the nonlinear term is approximated by w in such a way that only k ≪ N entries ofthe nonlinear term w need to be calculated in the reduced system (represented by wsel).Nevertheless, these entries still depend on the full-dimensional approximation Φuured,whose N entries hence need to be determined in each iteration step. For this reason,extended approaches have been developed, which, for example, replace P Twsel(Φuured) bywsel(P

TΦuured) when the nonlinear functionw is evaluated component-wise at u or whichuse a sparse evaluation procedure when general nonlinear vector-valued functions aretreated, cf. Chaturantabut & Sorensen [32] discussing different nonlinear PDEs. However,the determination (and storage) of the nodal values of the primary variables, collectedin u or rather in Φuured, is for the applications, presented in this monograph, requiredanyway, since the visualisation of the temporary behaviour of the primary variables withinthe domain during the simulation process is desired. Therefore, a further reduction ofthe number of numerical operations will not be further discussed here. Consequenlty, thisis not a complete offline/online decomposition in the sense of an online phase, which iscompletely independent of the full number N of DOF.

Again, resulting from the different physical time behaviour of the primary variables, thesnapshot matrix W = [w1 w2 ... ww] ∈ R

N×w needs to be divided into smaller snapshotmatrices Wϑs

for each primary variable ϑs. Following this, reduction matrices Ψϑsare

computed and summarised in the reduction matrix Ψw = blkdiag [Ψϑ1 , Ψϑ2 , ...]. Therein,the term blkdiag( · ) denotes that a block diagonal matrix is constructed from the inputarguments. The matrix P contains the information on the selected DOF and can beallocated to matrices Pϑs

, containing the information on the selected DOF correspondingto the respective primary variable ϑs. Inserting the results into the reduced system (4.10),the following POD-DEIM-reduced system can be found:

D(Φuured)︷︸︸︷

ΦTu Ψw (P TΨw)

−1P TDsel(Φuured)Φu

︸︷︷︸

D(Φuured)

ured +

k(Φuured)︷︸︸︷

ΦTu Ψw (P TΨw)

−1P Tksel(Φuured)︸︷︷︸

k(Φuured)

=

f︷︸︸︷

ΦTu f .

(4.31)Finally, the reduced system

ΦTu ξ(

Dsel(Φuured)Φu ured + ksel(Φuured))

= ΦTu f (4.32)

with degree l ≪ N , in which only k ≪ N entries of the nonlinear terms need to be deter-mined during a reduced simulation (represented by ( · )sel), can be formulated. Thus, acertain coefficient matrix ξ = Ψw (P TΨw)

−1P T is pre-computed for the composite non-linear term w(u) and can be used through the whole reduced simulation. Alternatively,


separated snapshots and reduction matrices for the nonlinear terms k(u) andD(u)u canbe computed in order to achieve more precise results. Furthermore, the nonlinear matrixD(u) can be treated independently from the vector u by transferring the matrix into avector. For this purpose, a matrix version of the DEIM, the so-called MDEIM, has beenproposed for the efficient reduction of a set of discretised partial differential equations, cf.[19, 97, 129]. Nevertheless, the implementation would become considerably more difficult,the offline computation would be more expensive and the storage demand would increase.

A detailed derivation and the final formulation of the specific POD-DEIM-reduced sys-tems, customised to the respective problem, can be found in Appendix C.

Chapter 5:Numerical examples with application toselected porous materials

This chapter discusses the application of the model-reduction methods presented in theprevious chapter to the specified TPM models derived in Chapter 2, as well as their numer-ical implementation. Therefore, it is necessary to provide further information on the corre-sponding problem settings such as the particularly chosen initial-boundary-value problem,the respective reduced system or the underlying processes to sample the snapshots.

Starting with the relatively simple porous-soil model with linear-elastic material be-haviour, which results in a system of equations with (approximately) time-invariant sys-tem matrices (hereinafter referred to as linear equation systems), the suitability of asystem reduction using the (modified) POD method is demonstrated for different prob-lem settings. Afterwards, the transition to time-efficient simulations of a nonlinear porousmaterial undergoing large deformations, modelled with a Neo-Hookean approach, is dis-cussed. In this context, the DEIM is used as an additional method in combination withthe POD method to deal with the occurring nonlinearities. Next, a reduction of dif-ferent biomechanical problems is further examined. Therefore, reduced simulations ofdrug-infusion processes in the human brain are initially performed. In particular, thePOD method is first applied to the simplified brain-tissue model with a linear equationsystem, implying isotropic permeability conditions and the further assumptions of thesimplified model derived in Subsection 3.3.2. Then, the POD-DEIM approach is usedto reduce the system of equations of the general (nonlinear) brain-tissue model underanisotropic permeability conditions, investigating a variation of the material parameters.Subsequently, the POD-DEIM approach is applied to the intervertebral-disc model spec-ified in Subsection 3.3.3, studying the simulation of different deformation states. In thisregard, the selection of specific snapshots is extensively investigated, since an appropriatechoice strongly affects the quality of the reduced simulations. The main focus in all thenumerical examples is on the investigation of the accuracy of the reduced models, as wellas on the examination of the resulting time savings due to the system reduction. Finally,a generalised approach for a reduction of a coupled system of equations using the evolvedmodifications is presented in order to enable a systematic adaptation to other models.

For all intents, the numerical implementation is realised with the coupled FE solverPANDAS. All computations were performed on a single core of an Intel i5-4590 with32 GB of memory running at clock speed of 3.30 GHz. While the FE meshes for sim-ple geometrical problems are directly generated in the FE solver, the program packageCUBIT1 is used to define FE meshes for more complex geometries. Moreover, the deter-mination of the reduced basis and the required reduction matrices is performed with thenumerical computing environment MATLAB2.

1Geometry and mesh generation toolkit, cf. http://www.cubit.sandia.gov2Matrix laboratory, cf. http://www.mathworks.com

63

64 5 Numerical examples with application to selected porous materials

5.1 Application of the POD approach to porous-soil

models with linear systems of equations

In order to generally validate the suitability of the POD approach applied to porous-mediamodels, the quasi-static two-dimensional (2-d) porous-soil model with linear-elastic ma-terial behaviour, which is presented in Subsection 2.4.1 and which can be described witha linear system of equations, is used as a first numerical example. Thereby, the resultsof simulations using the initial full system and simulations using either the classical orthe modified POD-reduced system are compared to each other. In a first step, the sameproblem setting as in the pre-computation performed to collect the required snapshotsis utilised. In the next step, a test-computation with varying boundary conditions (inparticular, the loading function) is realised. The intention is to investigate the mainte-nance of the accuracy and to examine the computational effort when using the reducedmodel to simulate various loading scenarios. Afterwards, dynamic conditions are assumedinstead of quasi-static conditions under small deformations, leading to additional primaryvariables. Finally, a three-dimensional (3-d) problem is considered in order to investigateif a change in the number of necessary POD modes or in the saving of the computationtime is observed. Moreover, the suitability of the reduced system for simulations withvarying material parameters is studied. In this regard, an appropriate determination ofthe reduced system matrices is necessary in order to ensure time-efficient simulations.Apart from this, an error indicator is formulated in order to certify the reduced model.

5.1.1 Quasi-static 2-d porous-soil model

In this subsection, the biphasic porous-soil model derived in Subsection 2.4.1 with ge-ometrically and materially linear behaviour and under the assumption of quasi-staticconditions is simulated. On the one hand, the full discretised system (3.19) and, on theother hand, different reduced formulations on the basis of the (modified) POD methodare used. Furthermore, the problem is spatially discretised with two-dimensional FE ele-ments as specified below, and the material parameters are given in advance and remainunchanged. The loading conditions, however, are varied in this first numerical example.

Problem setting and reduced-order system

For the investigation of the suitability of the POD-reduced system, a simple rectangulargeometry with the dimensions 10m × 10m is spatially discretised with two-dimensionalTaylor-Hood elements. The loading is applied on the top via a loading function q(t).A schematic representation of the geometry and the corresponding boundary conditionsof the initial-boundary-value problem is shown in Figure 5.1. In order to mechanicallysupport the domain, the right, the bottom and the left side of the geometry are fixed in per-pendicular direction to the boundary (Dirichlet boundary condition for the solid displace-ment). In addition, an efflux of the fluid over the top surface is possible (Neumann bound-ary condition for the pore pressure). The chosen simulation and material parameters arecollected in Table 5.1. For further details regarding the selection of the material param-eters, the interested reader is referred to the work of Heider [74] and citations therein.

5.1 Application of the POD approach to linear porous-soil models 65

e1

e2A

q(t) loaded, drained

fixed perpendicular,undrained

Figure 5.1: Geometry of a (2-d) fully saturated porous-soil model with loading and boundaryconditions (left) and FE mesh (right) of the initial-boundary-value problem.

symbol value unit description

nS0S 0.67 [ - ] Initial solidity

ρSR 2.0×103 [kg/m3] Effective solid density

ρFR 1.0×103 [kg/m3] Effective fluid density (water)

λS 8.375×106 [N/m2] First elastic Lame constant

µS 5.5833×106 [N/m2] Second elastic Lame constant

kF 1.0×10−4 [m/s] Darcy permeability

γFR 1.0×104 [N/m3] Effective specific weight of the fluid (water)

Table 5.1: Material parameters of the biphasic porous-soil model according to [74] and citationstherein.

In order to reduce the descriptive set of governing equations with the POD method, therequired snapshots are firstly collected in a pre-computation using the initial full systemwith N = 5771 degrees of freedom. 784 8-nodular Taylor-Hood elements with a quadraticapproach for the solid displacement vector uS and a linear approach for the pore pressurep are used. More precisely, a constant loading q = 10 kN/m2 is applied on the top surfaceof the geometry in this pre-computation within m = 100 time steps tii=1,...,m. Thevalues of the vector of unknowns are stored in each of these time steps and are writtenin the snapshot matrix. Moreover, the time-invariant system matrices are stored in thefirst time step of the pre-computation. Afterwards, the eigenvalue problem (4.5) is solvedand the normalised basis vectors (4.6) are collected in the reduction matrix. The offlinephase is completed by a computation of the reduced system matrices.

Numerical results

For a comparison of the numerical solutions obtained from the full system and the reducedsystem, simulations using different numbers of considered POD modes (which correspondto the reduced number of unknowns) are, in a first step, performed with the identicalproblem setting, material parameters and boundary conditions as in the pre-computation(hereinafter referred to as reproduction setting). Therefore, the classical POD method is


in the first instance applied without any modifications. Additionally, the determination ofthe reduced basis and the required reduction matrices is performed with MATLAB usingthe usual accuracy of 16 decimal digits. The derivation and the final formulation of thecorresponding reduced system can be found in Appendix C.1. In order to evaluate thesuitability of the reduced system, a normalised root-mean-square (NRMS) error

εNRMS =1

3nm

√√√√√

m∑

i=1

n∑

j=1

(

ujS1 i − ujS1 iuS1max − uS1min

)2

+

(

ujS2 i − ujS2 iuS2max − uS2min

)2

+

(

pji − pjipmax − pmin

)2

(5.1)is computed by comparing the n nodal values of each primary variable stored in all mtime steps of simulations, using, on the one hand, the full system and, on the otherhand, the reduced system. Therein, ujS1 i, u

jS2 i and pji are the entries of the vector of

unknowns ui obtained at time step ti of the simulation of the full system and allocatedto the jth node of the FE mesh, while ujS1 i, u

jS2 i and pji are the entries of the vector

ui = Φu ured i, calculated from the reduced vector of unknowns ured i during the reducedsimulation. Moreover, the differences of the nodal values are divided by the value rangeof the respective primary variable.

Comparing the resulting error εNRMS, it can be pointed out that a reduced system withtoo few considered POD modes cannot rebuild the results obtained from a simulationusing the full system, cf. Figure 5.2 (a). However, the results are sufficiently exact whenenough (here at least 5) POD modes are utilised. Using the classical POD method withstandard accuracy for the computation of the reduction matrices, problems arise when us-ing considerably more than 10 POD modes, even though the projection error PE(U ,V l)(representing the error between the solution vectors sampled as snapshots and their ap-proximation using the POD basis) decreases steadily, cf. Figure 5.2 (b). This is due tothe fact that only the first 10 normalised singular values (divided through the maximumsingular value) are larger than the calculation accuracy and that the absolute values ofthe solid displacement field uS are in another range as the absolute values of the fluid-porepressure p. Following this, rounding errors may occur, which are not negligible. Moreover,the solid displacement and the fluid-pore pressure physically exhibit a different behaviour

(a)

10−11

10−08

10−04

10+00

0 10 20 30 40 50 60 70 80 90 100

errorε N

RMS[-]

number of POD modes [-] (b) number of POD modes [-]

errorPE

[-]

0 20 40 60 80 100

10+00

10−04

10−08

10−12

10−16

Figure 5.2: (a) Normalised error εNRMS obtained from simulation results using the full andthe POD-reduced system for different numbers of considered POD modes (reproduction setting)and (b) corresponding normalised projection error PE(U ,V l) of the respective POD basis.


in time. Without any modification, the NRMS error first decreases, as expected, if thenumber of considered POD modes increases. However, if the number of considered PODmodes is too high, the error again increases (and the calculation is interrupted resultingfrom convergence problems - indicated in Figure 5.2 (a) with setting the error εNRMS to 1).

One possibility to deal with this problem is to increase the calculation accuracy to a higherprecision than the previously chosen 16 decimal digits. This can be done, for instance,by using GMP3 floating point numbers or by using the Symbolic Math Toolbox4 withinMATLAB, both providing variable-precision arithmetic. With a significantly increasedcalculation accuracy (more than 50 decimal digits), the number of considered POD modescan be increased without any problems and it can be shown, as expected, that the errordecreases with an increasing number of considered POD modes, cf. Figure 5.3. However,an increase of the calculation accuracy is not always possible and usually leads to anincrease of the computation time in the offline phase, in order to determine the reductionmatrix.

10−11

10−08

10−04

10+00

0 10 20 30 40 50 60 70 80 90 100

errorε N

RMS[-]

number of POD modes [-]

Figure 5.3: Normalised error obtained from the POD-reduced system with an increase of thecalculation accuracy for different numbers of considered POD modes (reproduction setting).

Another possibility to deal with the convergence problems for a high number of consid-ered POD modes is to use the modified POD method as presented in Subsection 4.2.2.Therefore, two separate snapshot matrices Uϑs

and two reduction matrices Φϑsare de-

termined and afterwards summarised in the reduction matrix Φu = blkdiag [ΦuS, Φp] in

accordance to (4.13). The corresponding reduced system is given in Appendix C.1. Usingthe modified POD method, it is possible to deal with the problem of different ranges ofthe absolute values of the primary variables without the need to increase the calculationaccuracy, as shown in Figure 5.4 (a). For demonstration purposes (to be able to set thenumber of considered modes to a certain value), the number of modes luS

and lp is setequal for both primary variables. This assumption is also supported by the fact that thedecrease of the singular values is similar for both primary variables, cf. Figure 5.4 (b).However, when the number of necessary modes is determined via an error estimator (aswill be shown later in this contribution), the number of modes does not have to be thesame for the different primary variables. Unfortunately, a disadvantage of this method isthe increasing number of necessary POD modes compared to the classical POD method.In the discussed numerical example, at least 8 POD modes (4 for each primary variable)

3GNU Multi Precision library, cf. https://gmplib.org4Toolbox software developed by MathWorks, cf. http://www.mathworks.com


(a)

10−14

10−12

10−08

10−04

10+00

0 10 20 30 40 50 60 70 80 90 100

errorε N

RMS[-]

number of POD modes [-] (b) number of POD modes [-]

errorPE

[-]

0 20 40 60 80 100

10+00

10−04

10−08

10−12

10−16

Figure 5.4: (a) Normalised error εNRMS obtained from simulation results using the full and themodified POD-reduced system for different numbers of considered POD modes (reproductionsetting) and (b) corresponding normalised projection errors PE(UuS

,V luS ) (red graph) andPE(Up,V lp) (blue graph) of the respective POD basis.

are sufficient to obtain satisfactory results. The necessary increase may arise from thefact that the reduction matrix is a block sparse matrix compared to the reduction matrixof the classical POD method. However, the NRMS error using the modified POD methodconverges against a smaller value in comparison to the results using the classical PODapproach. Of course, usage of the modified POD method in combination with an increaseof the calculation accuracy is also possible and leads to a steadily decreasing error for anincreasing number of consider POD modes, cf. Figure 5.5.

10−14

10−12

10−08

10−04

10+00

0 10 20 30 40 50 60 70 80 90 100

errorε N

RMS[-]


Figure 5.5: Normalised error obtained from the modified POD-reduced system with an in-crease of the calculation accuracy for different numbers of considered POD modes (reproductionsetting).

Having demonstrated that the sophisticated theoretical basis of the porous-soil modelis preserved using the reduced system, the geometry is in the next step loaded with atime-dependent force q(t) in order to investigate if the reduced model (under the re-usageof the previously calculated subspace matrix) is able to predict various loading scenariossufficiently well (hereinafter referred to as generalisation setting). More precisely, loadingfunctions q(t) = 10 sin(π t/400 s) kN/m2 for t ≤ 200 s, q(t) = 10 kN/m2 for 200 s≤ t ≤1400 s, q(t) = 10 cos(π(t − 1400 s)/400 s) kN/m2 for 1400 s≤ t ≤ 1600 s, q(t) = 0 for1600 s≤ t ≤3000 s, q(t) = 0.05 (t− 3000 s) kN/m2 for 3000 s≤ t ≤3100 s, q(t) = 5 kN/m2

for 3100 s≤ t ≤ 3800 s, q(t) = (5 − 0.02 (t − 3800 s)) kN/m2 for 3800 s≤ t ≤ 4050 s and


q(t > 4050 s) = 0 are used. Since a variation of the value q(t) is considered in the forcevector and thus has no influence in the formulation of the system matrices, the accuracyof the reduced system should be preserved in such a simulation, provided that no otherdynamics are triggered. To prove this issue, the NRMS error between the computationin the full system and the reduced computations are determined and compared in Figure5.6 for the application of either the classical or the modified POD method.

10−16

10−12

10−08

10−04

10+00

0 10 20 30 40 50

classical POD method

modified POD method

errorε N

RMS[-]


Figure 5.6: Normalised error obtained from the classical (red graph) or the modified (bluegraph) POD-reduced system for different numbers of considered POD modes (generalisationsetting).

For demonstration purposes, the reduction matrix and the reduced system matrices arein both cases computed with high calculation accuracy to be able to perform simulationswith many considered POD modes. As expected, a variation of the loading function doesnot affect the accuracy of the reduced system. As before, the error converges againsta smaller value, while a higher number of considered POD modes is necessary, whenthe modified POD method is used in order to obtain the same error in comparison tothe results obtained by the classical POD method. Obviously, a too small number ofconsidered POD modes causes inaccurate results. This is demonstrated by comparing theresulting solid displacement uAS2 in e2-direction and the effective pore-fluid pressure p atpoint A (cf. Figure 5.1) for a simulation using the full system and simulations using thereduced systems with different numbers of considered POD modes, cf. Figure 5.7 for anapplication of the classical POD method and Figure 5.8 for an application of the modifiedPOD method.

Comparing the obtained results, the accuracy of both reduced systems is ensured (accom-panied by a NRMS error of 10−5) if at least 5 POD modes are considered for the classicalPOD method or at least 6 POD modes (3 modes for each primary variable, respectively)for the modified POD method. In both cases, the time to solve the equation system can bereduced from 6 s to 2 s. Following this, different loading scenarios can be simulated usingthe reduced system without the need to recalculate the reduction matrix and the reducedsystem matrices. Furthermore, it becomes obvious that the compared values yield a muchsmaller error with the modified POD-reduced system when only a small number of modesis taken into account.


0

0

−1

−2

−3

8

4

−8

−4

0 01000 1000 20002000 30003000 40004000 50005000 60006000

time t [s]time t [s]

porepressurepA

[kN/m

2]

soliddisplacementuA S2[m

m]

full system - 5771 DOFred. system - 1 POD mode

red. system - 2 POD modesred. system - 3 POD modes


Figure 5.7: Values of the solid displacement uAS2 in e2-direction (left) and the effective pore-fluid pressure pA (right) at node A obtained from the full system (crosses) and the reducedsystem (classical POD method) for different numbers of considered POD modes (solid dots).

0

0

−1

−2

−3

8

4

−8

−4

0 01000 1000 20002000 30003000 40004000 50005000 60006000


porepressurepA

[kN/m

2]


m]

full system - 5771 DOFred. system - 2 POD modes (1 for ΦuS

, 1 for Φp)

red. system - 4 POD modes (2 for ΦuS, 2 for Φp)

red. system - 6 POD modes (3 for ΦuS, 3 for Φp)

Figure 5.8: Values of the solid displacement uAS2 in e2-direction (left) and the effective pore-fluid pressure pA (right) at node A obtained from the full system (crosses) and the reducedsystem (modified POD method) for different numbers of considered POD modes (solid dots).

5.1.2 Dynamic porous-soil model

In this subsection, a fully saturated porous soil is modelled with geometrically and mate-rially linear behaviour assuming dynamic instead of a quasi-static conditions. Followingthis, the vector of unknowns contains the values of the solid displacement field uj

S, thesolid velocity field vj

S, the fluid seepage velocity field wjF and the pore pressures pj at

each nodal point Pj of the FE grid, as described in Subsection 2.4.1. Consequently, thenumber of degrees of freedom, and therefore also the computing times using the full-ordermodel, are considerably higher than in the discretised quasi-static model. Therefore, theneed of a reduced model to minimise the computational effort is even more desirable. Asin the previous example, the prediction of various loading scenarios is investigated.


Problem setting

As before, a simple rectangular geometry (cf. Figure 5.1) is discretised with 784 8-nodularTaylor-Hood elements, leading to a system with N = 12 383 degrees of freedom due tothe quadratic approach for the solid displacement field uS and the solid velocity field vS

and the linear approach for the fluid seepage velocity field wF and the pore pressure p.In order to reduce the system (3.24), a pre-computation using the initial full system isperformed in the offline phase, storing the state variables in all time steps (resulting in250 snapshots) and the system matrices in the first time step. In contrast to the previousexample, only the right quarter of the top surface of the geometry is loaded with a time-dependent force specified for the pre-computation as q(t) = 10 sin(π t/0.4 s) kN/m2 fort ≤ 0.4 s and q(t >0.4 s) = 0. The boundary conditions and the material parametersare chosen in analogy to the previous example (cf. Figure 5.1 and Table 5.1), except forthe Darcy permeability kF , which is changed to 1.0×10−2 m/s. This high permeabilitycase is studied in order to examine a model with (approximately) time-invariant systemmatrices. The corresponding reduced system is given in Appendix C.2.

Numerical results

In order to investigate if the reduced model is able to predict various loading scenarios,the geometry is loaded with a varied time-dependent force q(t). For a comparison of thenumerical solutions obtained from the full system and the reduced system, the geometryis loaded four times with different loading functions qi(t) = q0i [1− cos(2 π t/tcycle i)] spec-ified by q0i = 10, 13, 20, 5 kN/m2 and tcycle i = 2, 0.5, 2, 1 s, whereby the loading cyclesare starting at t0 i = 0, 3, 4, 7 s and the force is set to zero in between the loading cycles.For demonstration purposes, the reduction matrix and the reduced system matrices arecomputed with high calculation accuracy to be able to perform simulations with many con-sidered POD modes. Comparing the resulting error εNRMS, it can be pointed out that theresults are sufficiently exact when enough POD modes (here at least 25 modes using theclassical or 44 modes using the modified POD method) are taken into account, cf. Figure5.9. However, for both variants, significantly more POD modes are necessary comparedto simulations under quasi-static conditions. This is due to the fact that the vector of


modified POD method


errorε N

RMS[-]

0 20 40 60 80 100 120 140

10+00

10−02

10−04

Figure 5.9: Normalised error obtained from the classical (red graph) or the modified (bluegraph) POD-reduced system for different numbers of considered POD modes (generalisationsetting).


unknowns contains more primary variables, which have a different physical behaviour.As expected, due to the separated reduction matrices for all primary variables, a highernumber of POD modes needs to be considered when using the modified POD method.

Looking at the resulting values of the primary variables at point A (cf. Figure 5.1) forsimulations using either the full or the reduced system, it can be pointed out that bothreduced systems exhibit results with a high degree of alignment between the full and thereduced model (such that the crosses in Figure 5.10 almost coincide with the correspondingdots and circles), when the dimension of the reduced system is sufficiently large, cf. Figure5.10. This can also be seen in Figure 5.11 for the spatial distribution of the values ofthe primary variables within the domain (exemplarily displayed for t = 3.5 s), given forsimulations obtained from the full system and the reduced system using 25 POD modes.Therein, the deformed grid is displayed to visualise the deformation process, which issuperelevated by a factor of fifty.

In terms of the numerical costs within the online phase, the results of the full system arecalculated in 530 s, whereas the computation time is reduced to 278 s using the classicalPOD-reduced system, cf. Table 5.2. This results in a markedly time saving and reducesthe computational costs of the online phase to less than 53 %. In particular, the timeto solve the equation system is reduced to less than 8 %. In contrast, the time to writeout specific data, e. g. the results of the primary variables or the stresses at all nodes ofthe FE grid, naturally remains the same. However, it is not always necessary to writeout all this data or it is sufficient to write out selected results only at individual nodes.Consequently, the time to write out specific data may become significantly smaller (forboth, the simulations using the full or the reduced system).

time t [s]

0 2 4 6 8 10

qualitativevalues

ofthe

primaryvariab

lesat

nodeA

full system -12383 DOF

classical POD-red. system - 25 POD modesmodified POD-red. system - 44 POD modes

Figure 5.10: Values of the solid displacement uAS2 (bottom), the solid velocity vAS2 (secondfrom the bottom), the fluid seepage velocity wA

F2 (second from the top) in e2-direction and theeffective pore-fluid pressure pA (top) at node A obtained from the full system (red crosses),the classical POD-reduced system using 25 POD modes (blue circles) and the modified POD-reduced system using 44 POD modes (green solid dots).


vS2 [ m/s ] wF1 [ m/s ] p [ kN/m2 ]−0.08 0.12 −0.06 0.3 −90 5.2

Figure 5.11: Values of the solid velocity vS2 in e2-direction (left), the seepage velocity wF1

in e1-direction (middle) and the effective pore-fluid pressure p (right) during the simulation attime t = 3.5 s visualised within the deformed grid (superelevated by a factor of fifty) obtainedfrom the full system (top) and the POD-reduced system using 25 POD modes (bottom).

time solving eq. system time write out data total CPU time

full system 274 s 256 s 530 s

reduced system 20 s 258 s 278 s

Table 5.2: Computing time on a single core of an Intel i5-4590 with 32 GB of memory runningat clock speed of 3.30 GHz (time to solve the equation system and to write out special datasummarised for all time steps and total computing time) of the online phase, obtained from thefull and the reduced system with 25 POD modes.

Even though nearly two times as many POD modes are necessary for the simulationsusing the modified POD-reduced system, the computation time of 304 s (34 s to solvethe equation system and 270 s to write out special data) is very close to the computationtime using the classical POD-reduced system. Note in passing that the time to collect thesnapshots in the pre-computation (121 s) and the time to compute the reduction matrixand the reduced system matrices in the offline phase (2075 s) are not considered here.Following this, a time saving is only obtained if several simulations (here more than 9) areperformed with the reduced system. For example, in the present case this was requiredfor the study of various loading scenarios. Beside, a fast online phase enables a rapidproduction of simulation results when they are actually needed.


5.1.3 Quasi-static 3-d porous-soil model

In order to investigate if a change in the number of necessary POD modes or in thesaving of the computation time is observed, a three-dimensional problem is considered inthe next step. Moreover, simulations with varying material parameters are performed toprove the suitability of the reduced model with respect to changing system matrices. Inaddition, an adapted formulation of a residual-based error estimator is presented at theend of this subsection.

Problem setting

In the present example, a cubic geometry with the dimensions 1m × 1m × 1m, spatiallydiscretised with three-dimensional FE elements, is studied. The simulation and materialparameters are initially chosen in accordance to the first numerical example in Subsec-tion 5.1.1, cf. Table 5.1. Again, the porous soil is modelled with linear-elastic materialbehaviour assuming quasi-static conditions and small deformations. In order to representa consolidation process in a porous soil, the fluid-saturated porous sample is loaded at aquarter of the top surface with a force q(t), see Figure 5.12. Moreover, the outer surfacesof the geometry are fixed in perpendicular direction (Dirichlet boundary condition forthe solid displacement). In addition, an efflux of the fluid over the unloaded part of thetop surface is possible (Neumann boundary condition for the pore pressure). The fullsystem has N = 16 214 degrees of freedom, using 1000 20-nodular Taylor-Hood elementswith a quadratic approach for the solid displacement field uS and a linear approach forthe fluid-pore pressure p.

e1

e2

e3

free,drained loaded,

undrained

fixed perpendicular,undrained

q(t)

A

Figure 5.12: Geometry of a (3-d) cubic porous-soil model with loading and boundary conditions(left) and FE mesh (right) of the initial-boundary-value problem.

Numerical results for a variation of the loading

With the aim of performing reduced simulations for various loading scenarios, the snap-shots of the state variables are stored in all time steps of a pre-computation using thepreviously described problem setting, where a force of q = 50 kN/m2 is applied within 10 sand is subsequently kept constant. Additionally, the system matrices are stored in the firsttime step. Afterwards, the offline phase is completed by a determination of the reductionmatrix and the reduced system matrices for different numbers of considered POD modes.


Based on these data, reduced simulations are performed in the online phase with a var-ied loading function q(t). More precisely, forces qi = 300, 200, 500 kN/m2 are appliedsuccessively within short time ranges and the loading function is set to q(t) = const. inbetween the loading cycles. Thus, the fluid-saturated porous sample is initially loadedwith a force of q1 = 300 kN/m2. In order to examine the accuracy of the reduced modelin the case of both a load removal and a load increase, the force is afterwards reduced att = 200 s to q2 = 200 kN/m2, before finally increasing it at t = 450 s to q3 = 500 kN/m2.

The resulting errors εNRMS between the simulation using the full system and the reducedsimulations can be found in Figure 5.13. It can be pointed out that, analogously tothe simulation of a two-dimensional porous-soil model, the results are sufficiently exactwhen at least 6 POD modes (for both methods) are utilised. Following this, basicallyno difference is observed concerning the necessary dimension of the reduced basis for thetwo-dimensional or the three-dimensional model. This is due to the fact that the physicalbehaviour is the same in both cases. Using the modified POD method, the error convergesagainst a smaller value, while a higher number of considered POD modes (apart from thecase where less than 6 modes are considered) is necessary to obtain the same error as forthe classical POD method. In terms of the numerical costs of the online phase, the time tosolve the equation system (summarised for all time steps) is reduced to less than 5 % whenconsidering 6 POD modes. In contrast, this time can only be reduced to around 13 %when the system matrices are determined with full degrees of freedom in the first timestep of the reduced simulation before applying the reduction matrix (instead of readingin the previously calculated reduced system matrices).

10−16

10−12

10−08

10−04

10+00

0 10 20 30 40 50


modified POD method

errorε N

RMS[-]


Figure 5.13: Normalised error obtained from the classical (red graph) or the modified (bluegraph) POD-reduced system for different numbers of considered POD modes.

Numerical results for material parameter variations

In the next step, the performance of simulations with varying material parameters isinvestigated, while the loading function q(t) = 50 kN/m2 is kept constant. Here, the solidskeleton stiffness is exemplarily varied by means of the elastic material constants µS andλS. Thus, the time-invariant, but parameter-dependent, system matrices differ for each


parameter variation. Specifically, the particular block

K11 =

∫

Ωh

1

2grad (NNNuS

I)4

De

[(

grad (NNNuSI)T)

23T

+[(

grad (NNNuSI)T)

23T ]

12T]

dv (5.2)

in the block matrix K depends on the fourth-order elasticity tensor

4

De = 2µS (I⊗ I)23T + λS (I⊗ I) ,

and thus depends linearly on the elastic material constants µS and λS, while the quanti-tiesD21,K12 andK22 are (approximately) time- and parameter-invariant with respect tothe Lame constants, cf. Appendix B.1. Consequently, the particular block K11 using themodified POD method (or the entire reduced system matrix K using the classical PODmethod), cf. Appendix C.1, needs to be recalculated for each parameter variation. Asmentioned before, the time to solve the equation system can be reduced considerably less,when the system matrices need to be determined with full degrees of freedom in the firsttime step of the reduced simulation. To counteract this, the appropriate reduced systemmatrices are pre-calculated before each reduced simulation. Therefore, the constant re-duced system matrix D and the constant matrices Kconst, KµS and KλS are determinedin the offline phase, while the reduced stiffness matrix

K = Kconst + µSKµS + λSKλS

is calculated in the online phase before performing a reduced simulation. In order todetermine the matrices KµS and KλS and therefore to be able to predict various materialparameter scenarios in the online phase with simulations using the reduced system, mul-tiple pre-computations need to be performed using different parameter values. In thesepre-computations, the corresponding stiffness matricesK(µS, λS) need to be stored in onetime-step. Additionally, as in the previous subsection with constant material parameters,the values of the vector of unknowns are stored in each time step and the system matrixD in the first time step of one of these pre-computations. The offline phase is completedby a determination of the reduction matrix and the reduced matrices D, Kconst, KµS and

KλS . It should be mentioned that appropriate adaptations are necessary, if other materialparameters than the elastic material constants are varied. However, this preliminary workenables time-efficient simulations in the online phase.

After completing the offline phase, different test simulations with varying values of theelastic material constants are performed to prove the usefulness of the reduced system.The results obtained from the full system and the reduced system (when considering 10POD modes using the modified POD method) for different values of the elastic materialconstants µS and λS are shown in Figure 5.14 for the solid displacement uAS2 in e2-directionand the pore pressure pA at point A (cf. Figure 5.12). It can be shown that, as a resultof the extended snapshot data set for the stiffness matrix, the reduced model is ableto produce accurate results for different parameter settings. Furthermore, the time tosolve the equation system summarised over all time steps can in all test simulations bereduced to less than 3 % (compared to simulations using the full system). Moreover, the


00 1010 2020 3030 4040 5050

0

0

−3

−6

−910

20

30

40


pA[kN/m

2]

uA S2[m

m]

full syst.

full syst.

full syst.

full syst.µS = 56250 kN/m2, λS = 37500 kN/m2 :

µS = 5583.3 kN/m2, λS = 8375 kN/m2 :

µS = 1492 kN/m2, λS = 5714 kN/m2 :

µS = 1000 kN/m2, λS = 1000 kN/m2 :red. syst.

red. syst.

red. syst.

red. syst.

Figure 5.14: Values of the solid displacement uAS2 in e2-direction (left) and the effective porepressure pA (right) at node A for different values of the elastic material constants obtained fromthe full system (solid dots) and the reduced system using 10 POD modes (circles).

NRMS error between the computations in the full system and the reduced computationsis between 1.4× 10−3 and 6.4× 10−5 for the different parameter settings, which indicatessufficiently accurate simulation results.

Error indication and error bounds

As already discussed in Subsection 4.2.3, the use of a reduced system is, in general,numerically efficient, but also erroneous. Therefore, it is desirable to certify the reducedmodel with an appropriate error indicator. Ideally, such an error indicator should becalculated directly within the reduced-order simulation (in contrast to the “true” NRMSerror εNRMS, which additionally needs the results of the full-order simulation). In thefollowing, two formulations of an error indicator using different error (semi)norms arepresented and compared to each other, exemplarily for the biphasic porous-soil modelunder the assumption of quasi-static conditions. In particular, adapted formulations of aresidual-based error and the appropriate error estimators using either a 2-norm or a Gseminorm5 of the error are discussed.

Due to the fact that the generalised damping matrix D within the index-1 DAE system(3.19) does not possess the full rank and consequently is a singular matrix, an adaptedformulation is used. Therefore, the algebraic equation

K11 uuuS + K12 ppp − f1 ext = 0 (5.3)

and the differential equation

D21 uuuS + K22 ppp − f2 ext = 0 (5.4)

5In contrast to a norm, a seminorm may have ||v||= 0 with v 6= 0. A seminorm is a norm if ||v||= 0 isequivalent to v = 0.


are considered separately. Thus, also the residual r, formulated in equation (4.17), canbe separated into two parts, yielding r = [r1 r2]

T with

r1 = K11ΦuSuuuS red + K12Φp pppred − f1 ext

= K11 (ΦuSuuuS red − uuuS) + K12 (Φp pppred − ppp) ,

r2 = D21 ΦuSuuuS red + K22Φp pppred − f2 ext

= D21 (ΦuSuuuS red − uuuS) + K22 (Φp pppred − ppp) .

(5.5)

Moreover, the error (4.15) is given by

ε = u − Φuured =

[

uuuS − ΦuSuuuS red

ppp − Φp pppred

]

=:

[

εuS

εp

]

. (5.6)

Inserting (5.6) in (5.5) shows that the errors εuSand εp satisfy the residual-based equa-

tionsK11 εuS

+ K12 εp + r1 = 0 ,

D21 εuS+ K22 εp + r2 = 0 .

(5.7)

Afterwards, equation (5.7)1 is solved for εp. Furthermore, equation (5.7)2 is solved for εpand the result is inserted in equation (5.7)1, before solving for εuS

(utilising the solutionwith minimum 2-norm ||Ax− b||2 of a system Ax = b), yielding

εp = −K+12K11 εuS

− K+12 r1 ,

εuS= D+

21K22K+12K11 εuS

+ D+21K22K

+12 r1 − D+

21 r2 ,(5.8)

where the (not generally valid) relation (K12K−122 D21)

+ = D+21K22K

+12 is used (the

matrix K12K−122 D21 does not possess full rank). Therein, the inverses K+

12 and D+21

are interpreted as generalised inverses (pseudo-inverses), since the matrices K12 and D21

are not square. Due to the specific form of the matrices K12 and D21, the relationsK+

12K12 = I and D21D+21 = I can be assumed. Solving the differential equation (5.8)2 for

εuSand inserting the result in equation (5.8)1, the errors εuS

(t) and εp(t) can finally beformulated as

εuS(t) = eD

+21K22K

+12K11 t εuS

(t = 0)+

+

t∫

t=0

eD+21K22K

+12K11 (t−s)D+

21

(

K22K+12r1(s)− r2(s)

)

ds ,

εp(t) = −K+12K11 εuS

(t) − K+12r1(t) .

(5.9)

It should be mentioned at this point, that the initial error εuS(t = 0) becomes zero when

zero-initial conditions are assumed for the solution vector. In the next step, an errorbound

||ε(t)||2=√

(||εuS(t)||2)2 + (||εp(t)||2)2 ≤ εbound(t) (5.10)


is used to estimate the error of the reduced system under consideration of a 2-norm. Basedon the argumentation given in Amsallem & Hetmaniuk [3] and Haasdonk & Ohlberger[70], the 2-norm of the error εuS

(t) is bounded as

||εuS(t)||2 ≤

∣∣∣

∣∣∣eD

+21K22K

+12K11 t

∣∣∣

∣∣∣2||εuS

(t = 0)||2 +

+

t∫

0

( ∣∣∣

∣∣∣eD

+21K22K

+12K11 (t−s)

∣∣∣

∣∣∣2||D+

21K22K+12||2 ||r1(s)||2 +

+∣∣∣

∣∣∣eD

+21K22K

+12K11 (t−s)

∣∣∣

∣∣∣2||D+

21||2 ||r2(s)||2)

ds

≤ sups∈[0,t]

∣∣∣

∣∣∣eD

+21K22K

+12K11 s

∣∣∣

∣∣∣2

(

||εuS(t = 0)||2+

+

t∫

0

(

||D+21K22K

+12||2 ||r1(s)||2+ ||D+

21||2 ||r2(s)||2)

ds)

≤ C0

(

||εuS(t = 0)||2+

t∫

0

(C1 ||r1(s)||2+C2 ||r2(s)||2 ) ds)

=: εuS bound(t) ,

(5.11)

assuming that there are computable (possibly parameter-dependent) constants C0, C1

and C2, such that

sups∈[0,t]

∣∣∣

∣∣∣eD

+21K22K

+12K11 s

∣∣∣

∣∣∣2≤ C0 , ||D+

21K22K+12||2 ≤ C1 , ||D+

21||2 ≤ C2 . (5.12)

Analogously, the 2-norm of the error εp is bounded as

||εp(t)||2 ≤ ||K+12K11||2 ||εuS

(t)||2 + ||K+12||2 ||r1(t)||2

≤ C3 εuS bound(t) + C4 ||r1(t)||2 =: εpbound(t) ,(5.13)

assuming that there are computable constants C3 ≥ ||K+12K11||2 and C4 ≥ ||K+

12||2.For an evaluation of the derived error bounds, the previously described problem setting fora test simulation with varying values of the elastic material constants (µS = 5583.3 kN/m2,λS = 8375 kN/m2) is used. Comparing the time history of the relative 2-norms of the trueerrors and the estimated errors, cf. Figure 5.15, it becomes obvious that the derived boundsclearly overestimate the norms of the true errors. In this regard, relative means that thevector norms are divided through the 2-norms of uuuS and ppp, respectively. Moreover, dueto the bounded constants Ci > 0, the estimators are monotonically increasing, whereasthe norms of the true errors can increase, as well as decrease.

In order to improve the error estimator, a symmetric positive definite matrix G ∈ RN×N

is used to induce a vector norm ||v||G=√

v · (Gv) on RN and a corresponding matrix

norm ||A||G= sup||v||G=1||Av||G for A ∈ RN×N . In the following, the matrix G is chosen

as G = blkdiag [DT21D21, K

T12D

T21D21K12] =: blkdiag [GuS

, Gp]. Thus, an error bound

||ε(t)||G=√

(||εuS(t)||DT

21D21)2 + (||εp(t)||KT

12DT21D21K12

)2 ≤ εbound(t) (5.14)


10+0410+02

10+0010+00

10−0210−04

10−04

10−08

10−12

00 1010 2020 3030 4040 5050time t [s] time t [s]

errornorm

ofε p

[-]

errornorm

ofε u

S[-]

Figure 5.15: Time behaviour of the relative error norm estimators (blue) εuS bound(t)/||uS ||2 andεp bound(t)/||ppp||2 and the corresponding relative 2-norms (red) ||εuS

(t)||2/||uuuS ||2 and ||εp(t)||2/||ppp||2of the true errors.

is used, where the GuSseminorm of the error εuS

(t) is bounded as

||εuS(t)||DT

21D21≤∣∣∣

∣∣∣eD

+21K22K

+12K11 t

∣∣∣

∣∣∣DT

21D21

||εuS(t = 0)||DT

21D21+

+

t∫

0

( ∣∣∣

∣∣∣eD

+21K22K

+12K11 (t−s)

∣∣∣

∣∣∣DT

21D21

||D+21K22K

+12||DT

21D21||r1(s)||DT

21D21+

+∣∣∣

∣∣∣eD

+21K22K

+12K11 (t−s)

∣∣∣

∣∣∣DT

21D21

||D+21 r2(s)||DT

21D21

)

ds

≤ sups∈[0,t]

∣∣∣

∣∣∣eD

+21K22K

+12K11 s

∣∣∣

∣∣∣DT

21D21

(

||εuS(t = 0)||DT

21D21+

+

t∫

0

(||D+21K22K

+12||DT

21D21||r1(s)||DT

21D21+ ||r2(s)||(D+

21)TDT

21) ds)

≤ C0

(

||εuS(t = 0)||DT

21D21+

t∫

0

(C1 ||r1(s)||DT21D21

+ ||r2(s)||2 ) ds)

=: εuS bound(t) , (5.15)assuming that there are computable constants C0 and C1, such that

sups∈[0,t]

∣∣∣

∣∣∣eD

+21K22K

+12K11 s

∣∣∣

∣∣∣DT

21D21

≤ C0 , ||D+21K22K

+12||DT

21D21≤ C1 . (5.16)

Analogously, the Gp norm of the error εp is bounded as

||εp(t)||KT12D

T21D21K12

= ||K12 εp(t)||DT21D21

= ||−K11 εuS(t) − r1(t)||DT

21D21

≤ ||K11||DT21D21

||εuS(t)||DT

21D21+ ||r1(t)||DT

21D21

≤ C2 εuS bound(t) + ||r1(t)||DT21D21

=: εpbound(t) ,

(5.17)

assuming that there is a computable constant C2 ≥ ||K11||DT21D21

. Comparing the timehistory of these modified vector (semi)norms of the true errors and the estimated errors(using the same problem setting as above), cf. Figure 5.16, one recognises that the derived

5.2 Application of the POD-DEIM to a biphasic model undergoing large deformations 81

bounds can considerably be improved in comparison to the above utilised 2-norm. Asbefore, the estimators are monotonically increasing due to the bounded constants Ci > 0.

10+0410+02

10+0010+00

10−0210−04

10−04

10−08

10−12

00 1010 2020 3030 4040 5050time t [s] time t [s]

errornorm

ofε p

[-]

errornorm

ofε u

S[-]

Figure 5.16: Time behaviour of the relative error norm estimators (blue)εuS bound(t)/||uuuS ||DT

21D21and εp bound(t)/||ppp||KT

12DT21D21K12

and the corresponding relative

vector (semi)norms (red) ||εuS(t)||DT

21D21/||uuuS ||DT

21D21and ||εp(t)||KT

12DT21D21K12

/||ppp||KT12D

T21D21K12

of the true errors.

Taking a closer look at the obtained error bounds, it becomes obvious that the errorestimators can be determined directly during reduced-order simulations. Therefore, theresiduals r1 and r2 need to be sampled in all time steps during the online computation,before computing their respective norm. Comparing the results, one can observe that theresidual-based formulation using an appropriate matrix G to define a problem-specificseminorm ||ε(t)||G significantly improves the error indicator and is hence preferable to theuse of a simple 2-norm. However, in both cases the derived bounds are monotonicallyincreasing and overestimate the norms of the true errors. Consequently, the error esti-mators cannot really estimate the respective norm of the true errors. Nevertheless, it ispossible to compare the estimated errors of different reduced simulations under varyingparameters and, as a result, to identify the parameters with the highest true error. Thiscan be used to determine decisive parameter configurations during a sampling process,as will be discussed in Section 5.4 for the extended biphasic IVD model. It should alsobe noted that some new approaches have recently been developed to further improve thistype of error indicators. In this regard, Fehr et al. [57] proposed different strategies toapproximate a matrix norm ||A||G in order to derive an overall speedup of the error es-timation routine. Such an approach may reduce the computation time for determiningthe constants Ci. Furthermore, Schmidt et al. [113] presented an error estimation pro-cedure for nonlinear, parameter-dependent problems aiming at an improvement of theaccuracy of error predictions. For specific problems, such an approach can counteract theoverestimation that occurs when using standard error bounding techniques.

5.2 Application of the POD-DEIM to a biphasic

model undergoing large deformations

In this section, a two-dimensional initial-boundary-value problem of a porous materialundergoing large deformations is studied. The used biphasic model is equivalent to theporous-soil model under quasi-static conditions derived in Subsection 2.4.1, with the dif-


ference that the solid skeleton is described with a Neo-Hookean model. Therefore, pos-sibilities to reduce the resulting nonlinear system of equations (3.18) are discussed in thefollowing.

5.2.1 Problem setting

For a comparison of the numerical solutions obtained from the full system and an appro-priate reduced system, a geometry with the dimensions 1m × 0.1m is used and spatiallydiscretised with 2560 8-nodular (2-d) Taylor-Hood elements with a quadratic approach forthe solid displacement field uS and a linear approach for the fluid-pore pressure p, leadingto a full system with N = 18 803 degrees of freedom. A schematic representation of themodel with the corresponding boundary conditions is shown in Figure 5.17. The fluid-saturated porous material is loaded at one half of the top surface with a time-dependentforce q(t). In order to mechanically support the domain, the bottom side of the geometryis spatially fixed (Dirichlet boundary condition for the solid displacement). In addition,an efflux of the fluid over the unloaded part of the top surface is possible (Neumannboundary condition for the pore pressure). The simulation and material parameters arechosen in accordance with the values given in Table 5.1.

e1

e2

q(t)

loaded, undrained

free, drained

free, undrained

free, undrained

spatially fixed, undrained

A

Figure 5.17: Geometry of a (2-d) porous-media model with loading and boundary conditionsof the initial-boundary-value problem.

5.2.2 Reduced-order system and numerical results

As it is described in Chapter 4, the application of a POD-Galerkin approximation doesin fact significantly reduce the dimension of the equation system and the effort to solvethe linearised system of equations in each iteration step. However, the computation timeand costs to determine the nonlinear terms in a general system of equations (3.18) are notsignificantly reduced. Following this, additional methods are necessary for the reductionof the nonlinear terms in the POD-reduced system (4.10). In particular, the numericalresults obtained from simulations using the initial full system, the POD-reduced systemand the POD-DEIM-reduced system are compared to each other in this section. In thisregard, the application of the classical and the modified POD method are discussed. Thecorresponding reduced systems in all used formulations are given in Appendix C.3.


In a first step, the suitability and, in particular, the computational effort using the POD-reduced system are examined. Therefore, snapshots of the state variables are stored in alltime steps of a pre-computation in which the geometry is loaded with a linear increasingforce q(t), that remains constant when a specific value qmax is obtained. The offline phaseis completed by a computation of reduction matrices for different numbers of consideredPOD modes. In the online phase, the simulation is performed with the reduced systemusing, in the first instance, the same problem setting as in the pre-computation. Figure5.18 shows the resulting solid displacement uAS2 in e2-direction and the resulting porepressure pA at node A (cf. Figure 5.17), on the one hand, for a computation on the basisof the full system and, on the other hand, for simulations using either the classical orthe modified POD-reduced system with different numbers of considered POD modes.For demonstration purposes, the number of modes within the modified approach is hereidentical for each primary variable to be able to set the number of considered modes to acertain value. Moreover, the NRMS error between the computation in the full system andthe reduced computations and the corresponding computation time to solve the equationsystem (summarised for all time steps and divided by the time obtained from the fullsystem) are determined and compared in Figure 5.19 for the application of either theclassical or the modified POD method.

0

0

0

0

−2

−2

−4

−4

−6

−6

00

00

44

44

88

88

1212

1212

1616

1616

2020

2020

200

200

400

400

600

600

time t [s] time t [s]


porepressurepA

[N/m

2]

porepressurepA

[N/m

2]


m]


m]

full system - 18803 DOFred. system - 2 POD mode



Figure 5.18: Values of the solid displacement uAS2 in e2-direction (left) and the effective pore-fluid pressure pA (right) at nodeA obtained from the full system (crosses) and the reduced system(top: classical POD method, bottom: modified POD method) performed with the reproductionsetting for different numbers of considered POD modes (solid dots).


0 0

classical POD

classical POD

modified POD

modified POD

0

1

2

3

4

1010 2020 3030 40 50

number of POD modes [-]number of POD modes [-]

errorε N

RMS[-]

normalised

CPU

time[-]

10+00

10−02

10−04

10−06

10−08

Figure 5.19: Normalised error (left) and normalised computation time to solve the equationsystem (right) obtained from the classical (red graph) or the modified (blue graph) POD-reducedsystem for different numbers of considered POD modes (reproduction setting).

Therein, a computation time with a value of 1 indicates that the time is equal to thetime obtained from the full system. It can be pointed out, that 7-10 POD modes (ac-companied by a NRMS error lower than 10−4) are sufficient to ensure the accuracy of thereduced systems. However, using more than 7 POD modes, no time benefit is obtainedin comparison to the simulation using the full system. This is due to the fact, that onlythe time to solve the linearised problem can be reduced when the POD method is usedwithout a combination of additional methods. Consequently, the time saving is very smallor non-existent. Following this, additional methods are needed to reduce the nonlinearterms while the POD method without a combination of additional methods is only usefulwhen most of the computational cost lies in the solution of the linearised problem, whichis solved in each iteration step with a Newton-iteration scheme.

In order to furthermore reduce the computational effort to determine the nonlinear terms,the system of equations is in the next step reduced with the POD-DEIM approach. There-fore, the offline phase is extended by an additional pre-computation using the POD-reduced system to generate snapshots of the nonlinear terms (stored at each Newtonstep). With this data, the DEIM algorithm is performed and the additional reductionmatrices are determined. Afterwards, the simulation is performed in the online phase us-ing the POD-DEIM-reduced system. Thereby, a distinction is made between two differentapproaches. While in the first case the POD-DEIM is applied in a classical way withouta separation of the snapshot matrices, both snapshot matrices (of the vector of unknownsand of the nonlinear terms) are separated into smaller snapshot matrices for each primaryvariable in the second case, where the modified POD-DEIM approach is used. As before,the number of modes within the modified approach is the same for each primary variable.To enable a direct comparison, 10 POD modes are used for both variants. Consideringdifferent numbers of DEIM modes, the NRMS error between the computation in the fullsystem and the reduced computations, as well as the corresponding computation timeto solve the equation system (summarised for all time steps and divided by the time ob-tained from the full system), are determined and compared in Figure 5.20 using either theclassical or the modified POD-DEIM approach. Therein, an NRMS error of 1 indicatesthat the simulation has been interrupted due to convergence issues.


0 0

classical POD-DEIM

classical POD-DEIM

modified POD-DEIM

modified POD-DEIM

0

0.1

0.2

0.3

2020 4040 6060 8080 100100

number of DEIM modes [-]number of DEIM modes [-]

errorε N

RMS[-]

normalised

CPU

time[-]

10+00

10−01

10−02

10−03

10−04

10−05

Figure 5.20: Normalised error (left) and normalised computation time to solve the equationsystem (right) obtained from the classical (red graph) or the modified (blue graph) POD-DEIM-reduced system for 10 POD modes and different numbers of considered DEIM modes (repro-duction setting).

It can be pointed out, that a system reduced with the classical POD-DEIM approach isnot able to produce accurate simulation results (particularly at the end of the simulation),not even if the number of considered DEIM modes is further increased. This can also beseen in Figure 5.21 concerning the spatial distribution of the values of the pore pressurewithin the domain at time t = 8 s and at the end of the simulation. Therein, the deformedgrid is displayed to visualise the deformation process. Note that the results obtained bythe classical POD-DEIM-reduced system are generated using 28 DEIM modes, whichcorresponds to the lowest achieved error, cf. Figure 5.20. However, using the modifiedapproach with separated snapshot and reduction matrices, the error converges against

t = 8 s t = 20 s

p [ N/m2 ] p [ N/m2 ]

0 0

130 3500

Figure 5.21: Values of the pore-fluid pressure p during the simulation at time t = 8 s andat the end of the simulation visualised within the deformed grid obtained from the full system(left), the classical POD-DEIM-reduced system using 10 POD and 28 DEIM modes (middle)and the modified POD-DEIM-reduced system using 10 POD and 30 DEIM modes (right).


the error obtained from a simulation using the modified POD-reduced system when asufficiently high number of DEIM modes (here at least 30) is used, cf. Figure 5.20, andthus provides significantly improved simulation results compared to the classical approach,cf. Figure 5.21. Furthermore, the time to solve the equation system, summarised over alltime steps, can be reduced from 73 s using the full system to 6 s using the modifiedPOD-DEIM-reduced system with 10 POD modes and 30 DEIM modes. This results in amarkedly time saving and reduces the computational costs for solving the equation systemwithin the online phase to less than 9%.

Summarising the above, it could be demonstrated that it is necessary to use the modifiedapproach to reduce a nonlinear coupled system of equations containing primary variableswith a different physical behaviour in time. A significant reduction of the computationtime can be achieved, while the accuracy of the simulation results is maintained. Conse-quently, the modified POD-DEIM approach hereinafter serves as the basis for a reductionof the used equation systems. Moreover, the necessary number of considered POD andDEIM modes is determined via suitably chosen error estimators.

Since the primary objective of this thesis is to enable fast simulations for models withvarying boundary and/or parameter conditions, when they are urgently needed, the suit-ability of the reduced system within different loading scenarios with varying materialparameters is examined in the following. Therefore, an altered loading function q(t) inform of a combination of linear and constant functions is used while the solid skeleton stiff-ness is exemplarily varied by means of the elastic material constants µS and λS. In a firststep, the previously determined reduction matrices are used for these simulations and arecompared to simulations based on the full system. Figure 5.22 shows the resulting soliddisplacement uAS2 in e2-direction and the resulting pore pressure pA at node A (cf. Figure5.17) using, on the one hand, the same material parameters as in the pre-computation(µS = 5583.3 kN/m2, λS = 8375 kN/m2) and, on the other hand, varied elastic materialconstants (µS = 8000 kN/m2, λS = 9000 kN/m2).

0

0

−2

−4

−6

0 025 2550 5075 75100 100125 125150 150

200

100

−100


porepressurepA

[N/m

2]


m]

full system

full system

µS = 5583.3 kN/m2, λS = 8375 kN/m2 :

µS = 8000 kN/m2, λS = 9000 kN/m2 :

red. system

red. system red. system (50 DEIM modes)

Figure 5.22: Values of the solid displacement uAS2 in e2-direction (left) and the effective pore-fluid pressure pA (right) at node A for different values of the elastic material constants (general-isation setting) obtained from the full system (solid dots) and the POD-DEIM-reduced systemusing 10 POD and 30 DEIM modes (circles) or an extension to 50 DEIM modes (squares).


While the reduced model is able to produce simulation results with high accuracy for ascenario with an altered loading function q(t) and stable material parameter conditions,the chosen pre-computation is not an adequate basis for different parameter scenarios.Despite a distinct improvement is obtained by an extension to 50 DEIM modes, an in-creasing number of considered DEIM (or POD) modes does not provide sufficiently exactsimulation results. This is due to the fact that the subspace is constructed via a snapshot-based data set, such that the reduced system can only describe processes properly, whichare well represented within this data set. As long as only one set of material parame-ters is used within the snapshot sampling process, the reduced system is not suitable forvariations of these material parameters.

Consequently, pre-computations with very small (µS = λS = 1 × 106 N/m2) and veryhigh (µS = λS = 10× 106 N/m2) values of the elastic material constants are additionallyperformed to cover the expected value range of the varied material parameters within thesampled snapshots. While these values are at this point chosen with physical intuition,a more systematic and automatic strategy is in general preferable for the selection of thesnapshots and is therefore discussed in detail in Section 5.4. On the basis of the modifiedsnapshot data set, the reduction matrices are determined using normalised projection er-rors PE(U ,V l) and PE(W ,Vk) of 10−10 for the specification of the number of consideredPOD and DEIM modes. This error estimation leads to a consideration of 23 POD modesand 99 DEIM modes for the reduced basis. It can be shown that, as a result of the ex-tended snapshot data set, the reduced model is now able to produce accurate results fordifferent loading scenarios and for different parameter settings, cf. Figure 5.23. Further-more, the time to solve the equation system can be reduced from around 205 s, when usingthe full system, to around 50 s, when using the modified POD-DEIM-reduced system.

−8

0

0

−2

−4

−6

0 025 2550 5075 75100 100125 125150 150

200

100

−100


porepressurepA

[N/m

2]


m]

full system

full system

full systemµS = 5583.3 kN/m2, λS = 8375 kN/m2 :

µS = 8000 kN/m2, λS = 9000 kN/m2 :

µS = 9000 kN/m2, λS = 9000 kN/m2 :

red. system (23 POD and 99 DEIM modes)



Figure 5.23: Values of the solid displacement uAS2 in e2-direction (left) and the effective pore-fluid pressure pA (right) at node A for different values of the elastic material constants (general-isation setting) obtained from the full system (solid dots) and the POD-DEIM-reduced systemusing 23 POD and 99 DEIM modes (circles).


5.3 Reduced simulations of drug-infusion processes

within a brain-tissue model

The used thermodynamically-consistent drug-infusion model for the brain tissue has beenstudied in Subsection 2.4.2. In this section, the focus is placed on the application of themodel-reduction techniques described in Chapter 4 to the simplified model with a simplerectangular geometry and further assumptions which have been discussed in Subsection3.3.2 and to the complex model without these simplifications. Therefore, customisedapproaches to reduce the coupled linear system of equations in (3.33) with the PODmethod and the general nonlinear system of equations in (3.30) with the POD-DEIM arepresented and the respective simulations are performed and discussed in the following.

5.3.1 Application of the POD method for the simplified brain-tissue model with linear system of equations

When simulating the convection-enhanced drug-delivery procedure within brain tissue, thetemporal and spatial spreading of the applied therapeutic agent is of particular interest.Restricting on the evaluation of these results (and thus not being interested in a detailedknowledge of the blood flow, among other things), the simplified model, obtained inSubsection 2.4.2 under the assumptions of quasi-static conditions, negligible gravitationalforces, geometrically linear material behaviour, isotropic permeability conditions and aconstant blood volume fraction, provides a multi-component brain-tissue model that isas simple as possible. However, a consideration of anisotropic permeability conditions, aswell as using the full range of the described theoretical brain-tissue model, is in generalfeasible and will be presented in the next subsection. For a further reduction of thecomputational effort, the resulting linear system of equations (3.33) is reduced with thePOD method. In order to investigate the suitability of the reduced model and its abilityto simulate various infusion scenarios, simulations based on the full (simplified) modeland the reduced model are performed and the required results are hereinafter comparedwith each other.

Reduced-order system

In order to determine the reduced system (4.11), the modified POD method presented inSubsection 4.2.2 is used since the vector of unknowns contains as primary variables ϑs thesolid displacement field uS, the effective pore-liquid pressures pIR and pBR and the molarconcentration cDm of the therapeutic agent. Physically, these primary variables exhibit adifferent behaviour in time. Moreover, the primary variables and such the entries of thegeneralised vector of unknowns u have very huge differences in their absolute values. Toovercome resulting problems when applying the POD method by default, four separatedsnapshot matrices Uϑs

are generated, containing the nodal values of the FE-grid for thedifferent primary variables at each time step. Since extended Taylor-Hood elements withdifferent ansatz functions, namely quadratic ones for uS and linear ones for pIR, pBR andcDm, are used for reasons of numerical stability of the full system, this leads to a total


number of N = NuS+NpBR +NpIR +NcDm

degrees of freedom. Afterwards, the eigenvalueproblem (4.5) is solved for each of these snapshot matrices and the individually obtainedsubspace matrices Φϑs

(which can have different numbers lϑsof considered POD modes)

are summarised in accordance to (4.13) in the reduction matrix Φu. Finally, the reductioncan be performed according to (4.11). A detailed derivation and the final formulation ofthe corresponding reduced system can be found in Appendix C.4.1.

In conclusion, a required task is to perform a pre-computation in PANDAS using the initialfull system (3.33). In this pre-computation, the state variables are stored in each timestep. In addition, the storage of the system matrices D and K within the first time stepis performed. Afterwards, the reduction matrix Φu and the reduced system matrices Dand K are generated using MATLAB. This completes the offline phase. Based on thisdata, the simulation can be performed in an online phase for the reduced system (4.11).

Problem setting

For a comparison of the numerical solutions obtained from simulations using either thefull system or the reduced system, a simple rectangular geometry (a thin slice with thedimensions 10 mm × 10 mm × 0.25 mm, spatially discretised with 160 three-dimensionalTaylor-Hood elements) is studied, cf. Figure 5.24.

12catheter

e1

e2

e3

Figure 5.24: Geometry of a horizontal cut through a human brain (left), rectangular geometryand mesh of the initial-boundary-value problem (middle) and evaluated points 1 and 2 withinthe domain (right).

In order to represent the CED procedure within brain tissue, a catheter is placed inthe centre of the geometry and a therapeutic solution is applied via an infusion at thesurface of the infusion site. In particular, a solution influx (Neumann boundary condi-tion) of vI = 4.24 · 10−6 m3/m2s is combined with a constant therapeutic concentration(Dirichlet boundary condition) of cD0m = 3.7 · 10−3 mol/l, which corresponds to a typicalapplication dose of Q = 10µl/min. In addition, an efflux of the interstitial fluid and thetherapeutic agents over the surfaces at the outside of the rectangular tissue sample ispossible (Neumann boundary condition for the pore pressures and the concentration). Inorder to mechanically support the domain, the outer surfaces of the geometry are spatiallyfixed (Dirichlet boundary condition for the solid displacement). The chosen simulationand material parameters are collected in Table 5.3, a comprehensive discussion can befound in Wagner [127].


symbol value unit description/reference

nB0S 0.05 [ - ] Initial blood volume fraction, cf. [9]

nI0S 0.20 [ - ] Initial interstitial fluid volume fraction, cf. [118]

nS0S 0.75 [ - ] Initial solidity arising as a result of (2.53)

ρBR 1.035×103 [kg/m3] Effective density of blood, cf. [127]

ρIR 0.993×103 [kg/m3] Effective density of interstitial fluid (water at 37C)

µBR 3.5×10−3 [Ns/m2] Dynamic viscosity of blood, cf. [127]

µIR 0.7×10−3 [Ns/m2] Dynamic viscosity of interstitial fluid (water at 37C)

λS0 5.0×103 [N/m2] (First and second) elastic Lame constants,

µS0 1.0×103 [N/m2] according to [35] and citations therein

µS1 8.0×101 [N/m2] Material parameters governing the anisotropic

γS1 1.0×101 [ - ] material behaviour (assumed low fibre stiffness)

αB 200 [N/m2] Material parameter in relation (2.53)

βB 3.75 [ - ] Material parameter in relation (2.53)

DDij 10−9 - 10−10 [m2/s] Order of magnitude of the spatially varying

diffusion coefficient, cf. [9] and citations therein

KIij 10−7 - 10−8 [m/s] Order of magnitude of the spatially varying Darcy

permeability of the interstitial fluid, cf. [80]

KBii 3.0×10−5 [m/s] Isotropic Darcy permeability coefficient, cf. [117]

Table 5.3: Material parameters of the multi-component brain-tissue model for the numericalsimulation of CED procedures according to Fink et al. [58] and Wagner [127].

Numerical results

In a first step, the CED procedure is simulated using the full-order system (4782 degreesof freedom) with the previously outlined boundary conditions and material parameters.This simulation is used as pre-computation, storing all snapshots and the system matri-ces. Afterwards, the offline phase is completed by a computation of the reduction matrices(and the corresponding reduced system matrices) for different numbers of considered PODmodes. To demonstrate that the reduced system is able to conserve the theoretical basisof the simplified brain-tissue model, it is initially used to perform in the online phase asimulation with the identical problem setting, material parameters and boundary condi-tions as in the pre-computation. Subsequently, the numerical results obtained from thefull system and the reduced system with different numbers of POD modes are compared.In view of the results of the normalised therapeutic concentration cDm/c

D0m at point 1 (cf.

Figure 5.24), given in Figure 5.25, it can easily be concluded that the results are adequatewhen considering at least 20 POD modes. However, in order to quantitatively judge theaccuracy of the results, the NRMS error εNRMS, determined using the nodal values of all


0.4

0.2

0.0

norm

alisedconcentrationcD m

/cD 0

m[-]

applied infusion time [hours]0

0.6

0.8

1.0

6 12 18 24 30 36

8 POD modes

12 POD modes

16 POD modes

20 POD modes

24 POD modes

full system

Figure 5.25: Normalised concentration of the therapeutic agent at point 1, obtained from thefull system (red crosses) and the reduced system for different numbers of POD modes (circles).

primary variables, and additionally the NRMS errors

εϑS

NRMS =1

nm

√√√√√

m∑

i=1

n∑

j=1

(

ϑjS i − ϑjS i

ϑSmax − ϑSmin

)2

, (5.18)

which are determined decoupled for each primary variable ϑS by comparing the n nodalvalues of the respective primary variable stored in all m time steps of simulations using,on the one hand, the full system and, on the other hand, the reduced system, are displayedin Figure 5.26 and can be used to estimate the required dimensions of the POD basis. Ifmore than 24 POD modes are considered, the errors of all primary variables are in theorder of their minimal values. Looking at the therapeutic concentration, the minimal erroris reached using more than 20 POD modes. Note in passing that for small dimensions ofthe reduced basis, individual runaways may occur, e. g., due to rounding errors. This canbe seen in Figure 5.26 at the non-smooth convergence of the solid-displacement error aswell as the pore-liquid error, e. g., at 20 POD modes. However, for increasing numbers of

εNRMS

εuS1NRMS

εpIR

NRMS

εcDmNRMS


error[-]

0 10 20 30 40 50

10+00

10−01

10−02

10−03

10−04

Figure 5.26: Normalised (mean-value) error εNRMS (red) and errors εϑSNRMS of the sagittal solid

displacement (blue), the pore-liquid pressure (green) and the normalised concentration (orange)between the full and the reduced simulation averaged over all degrees of freedom and time steps.


POD modes the error converges to its final value. Furthermore, it can be recognised thatthe minimal error is still relatively large. This is due to the assumption of time-invariantsystem matrices. Consequently, a further reduction of the simulation error can only beachieved when temporal changing system matrices are used within the reduced model.However, for the chosen application area the error caused by the use of the reduced modelis tolerable when 24 POD modes are used. This can also be demonstrated by the spatialtherapeutic spreading within the domain at the end of the infusion process, cf. Figure 5.27.

cD m/cD 0m

[-]

(cD m

full−cD m

red)/cD 0m

[-]

00

1 8.3×10−03

Figure 5.27: Therapeutical spreading, computed with the full system (left) and the reducedsystem using 24 POD modes (middle), and corresponding error (right) at the end of an infusionprocess under isotropic conditions.

In terms of the numerical costs of the online phase, the results of the full system arecalculated in 504 s, whereas the computation time is reduced to 145 s when using thePOD-reduced system, cf. Table 5.4. This results in an enormous time saving and reducesthe computational costs of the online phase to less than 30%. In particular, the time tosolve the equation system is reduced significantly to about 1%. In contrast, the timeto write out specific data, e. g. the results of the primary variables or the stresses at allnodes of the FE grid, naturally remains the same. In conclusion, the results are verypromising concerning the overall time reduction. Moreover, it was demonstrated that thesophisticated theoretical basis of the simplified brain-tissue model is conserved by therealised reduced simulation. Note in passing that the time to collect the snapshots in thepre-computation and the time to compute the reduction matrix in the offline phase arenot considered here. Following this, a time saving is only obtained if several (here at leasttwo or more) simulations are performed with the reduced system. However, this is oftenrequired such as in the study of various infusion scenarios as it is discussed in the following.

computing solving write out totaltime [s] eq. system data CPU time

full system 364 140 504

red. system 4 141 145

Table 5.4: Computing time on a single core of an Intel i5-4590 with 32 GB of memory runningat clock speed of 3.30 GHz (time to solve the equation system and to write out special datasummarised for all time steps and total computing time) of the online phase, obtained from thefull and the reduced system with 24 POD modes.


Since the prediction of various infusion scenarios is of particular interest in the planningphase of a scheduled clinical procedure, the application dose is in the next step varied fordifferent test cases. Therefore, a sufficiently well performance with the reduced system,under the re-usage of the previously calculated subspace matrix Φu, is required for aneffective time benefit. In Figure 5.28, the simulation results for the evolution of thetherapeutic concentration at points 1 and 2 (cf. Figure 5.24) are shown for a variation ofthe applied initial values cD0m using, on the one hand, the full system and, on the otherhand, the reduced system when considering 24 POD modes.

molarconcentrationcD m

[10−3mol/l]

applied infusion time [hours]

0

1

2

3

4

5

0 6 12 18 24 30 36

test case 1: cD0m = 0.001mol/l



full system: point 1



red. system: point 1



point 2

point 2

point 2

point 2

point 2

point 2

Figure 5.28: Therapeutic concentration at evaluated points for a variation in the applied initialvalues cD0m, obtained from the full system (solid dots and squares) and the reduced system (circlesand unfilled squares).

Comparing the results with the reference solution of the full system, the accuracy of thetherapeutic concentration is ensured. Naturally, the resulting therapeutic concentrationincreases during the infusion, where the points closer to the infusion site reach a higher(therapeutically more effective) level. Moreover, a higher therapeutic concentration isachieved when the applied initial value cD0m is increased. Varying the applied solutioninflux vI , the accuracy of the effective pore-liquid pressure pIR is not effected when thereduced system is used, cf. Figure 5.29.

effectivepore-liquid

pressure

pIR

[N/m

2]


0

10

20

30

40

50

0 6 12 18 24 30 36

test case 1: vI = 1×10−6 m3/m2s









point 2

point 2

point 2

point 2

point 2

point 2

Figure 5.29: Effective pore-liquid pressure at evaluated points for a variation in the appliedsolution influx vI , obtained from the full system (solid dots and squares) and the reduced system(circles and unfilled squares).


However, the results for the normalised concentration (plotted in Figure 5.30) differslightly for the moderately chosen application doses which result a diffusion-dominatedspreading. In particular, for an increase of the solution influx, the concentration increasesmoderately when using the full model. In contrast, it remains unchanged when using thereduced system. Nevertheless, the difference is negligibly small and the reduced simulationseems to be sufficient.

molarconcentrationcD m

[10−3mol/l]


0

1

2

3

4

0 6 12 18 24 30 36










point 2

point 2

point 2

point 2

point 2

point 2

Figure 5.30: Therapeutic concentration at evaluated points for a variation in the appliedsolution influx vI , obtained from the full system (solid dots and squares) and the reducedsystem (circles and unfilled squares).

5.3.2 Application of the POD-DEIM approach for thegeneral brain-tissue model

In the previous subsection, a simplified model was used with emphasis on the drug spread-ing. However, the use of the general brain-tissue model (cf. Subsection 2.4.2) with all itscomplexity may be necessary to obtain sufficiently accurate results of different quantities(e. g., required for the description of solid displacements or stresses). As there are variousappearing nonlinearities resulting especially from nonlinear terms in the saturation func-tion (and, thus, in the volume fractions) and from the geometrically nonlinear materialbehaviour, the application of the POD-DEIM is required to reduce the nonlinear systemand thus the computation time of simulations based on the general brain-tissue model inan appropriate manner.

Reduced-order system

Using the general brain-tissue model, it is crucial to account for nonlinearities. Therefore,the POD-DEIM approach is applied to the global equilibrium equation (3.30) leading tothe reduced system (4.31). As before (for the simplified model), the different physical timebehaviour of the primary variables needs to be considered. Therefore, snapshot matricesUϑS for the state variables and WϑS for the nonlinear terms are generated separatelyfor each primary variable ϑS to compute separated reduction matrices ΦϑS and ΨϑS ,respectively. A detailed derivation and the final formulation of the corresponding reducedsystem can be found in Appendix C.4.2.


Summarising the required tasks, two different groups of pre-computations need to beperformed. First, the state variables uii=1,...,m are stored in each time step ti of pre-computations with varying parameter conditions (depending on the expected scope ofapplication) using the initial full system (3.30), followed by a determination of the re-duction matrix Φu. Afterwards, a second set of pre-computations is performed with thePOD-reduced system (4.10) using the identical problem setting and the same varyingparameter conditions as in the pre-computations of the full system. Within these simu-lations, the nonlinear terms wii=1,...,w are stored in each Newton step. Moreover, thereduction matrix Ψw and the resulting coefficient matrix ξ = Ψw (P T Ψw)

−1P T are gen-erated. This completes the offline phase. Based on this data, simulations with varyingparameter conditions can be performed in an effective way in the online phase using thePOD-DEIM-reduced system (C.20). It should be noted that the necessary number ofconsidered POD and DEIM modes is determined via an error estimation by means of theprojection errors PE(U ,V l) or PE(W ,Vk), respectively. Even if these errors are deter-mined with respect to the snapshot data sets and do not represent the true simulationerrors, they can give a good indication of the necessary number of considered modes.

Problem setting

For a comparison of various aspects based on the general nonlinear model, a thin brain slice(horizontal cut, discretised with one three-dimensional element in thickness direction) isstudied, cf. Figure 5.31. Thereby, the geometric parts used for the examples correspond totypical anatomically-based geometries of the brain, cf., e. g., Dutta-Roy et al. [38], Taylor& Miller [119]. Moreover, a therapeutic solution is applied according to the problemsetting in Subsection 5.3.1 while the mechanical support is provided due to the spatiallyfixed inner surfaces. The simulation and material parameters are chosen in accordancewith the previous example (collected in Table 5.3), whereby the solid skeleton stiffness isexemplarily varied by means of the elastic material constants µS

0 and λS0 . In order to keepthe snapshot selection as short and simple as possible, the material parameters used in thepre-computations are chosen with physical intuition as the minimum (µS

0 = 1000 N/m2

and λS0 = 5000 N/m2) and maximum (µS0 = 10 000 N/m2 and λS0 = 50 000 N/m2) values

of the assumed value range and combinations thereof to cover the expected value rangeof these parameters. In contrast to Subsection 5.3.1, anisotropic permeability conditions

1

2

3

e1

e2

e3

Figure 5.31: Rectangular geometry and mesh of the initial-boundary-value problem (left) andevaluated points within the domain (right).


are considered for the entire domain of the brain-tissue model. Therefore, the anisotropicpermeability tensors KSB and KSI and analogously the diffusivity DD of the therapeuticagent are evaluated from (patient-specific) medical imaging data. Utilising 2179 Taylor-Hood elements, the initial full system yields 61 065 degrees of freedom.

Numerical results

Initially, all pre-computations are performed with the parameter conditions as statedabove. Moreover, the reduction matrices are determined using normalised projectionerrors PE(U ,V l) and PE(W ,Vk) less than 10−5 for the specification of the number ofconsidered POD and DEIM modes. This error estimation leads to a consideration of 18POD modes and 47 DEIM modes. Regarding the evaluated DEIM points (DOF of theFE grid, where the nonlinear terms are computed), it becomes obvious that most of themare near the position of the catheter and, thus, in the area of interest, cf. Figure 5.32.Following this, there are many elements for which the values of the nonlinear term areapproximated with the reduction matrix Ψw instead of computing them individually.

Figure 5.32: Evaluated DEIM points within the domain: all 47 corresponding nodes of theselected DOF (left) and zoom into the area of interest (right).

After the offline phase is completed, the drug-infusion process is, in a first step, simu-lated using all the material parameters as given in Table 5.3 (which comply with one ofthe parameter settings used within the pre-computations) to prove the accuracy of thereduced system. Therefore, the evolution of the pore pressures of the interstitial fluidand the normalised concentration during the applied infusion at the evaluated points 1-3(cf. Figure 5.31) are compared to each other for simulations using either the full or thereduced system, cf. Figure 5.33. It is obvious that the displayed results are sufficiently

full system, point 1red. system, point 1



applied infusion time [min.]applied infusion time [min.]

pIR

[N/m

2]

cD m/cD 0

m[%

]

00

00

20

40

60

30

60

90

4040 8080 120120

120

160160

Figure 5.33: Values of the effective pore-liquid pressure and the normalised molar concentrationat selected points 1-3 obtained from the full system (solid dots) and the reduced system (circles).


exact when using the reduced system. Furthermore, the accuracy of the spatial spreadingof the normalised therapeutic concentration and the total stress σ11 (and all other stressesanalogously) within the domain at the end of the infusion process is ensured, cf. Figure5.34. It is obvious that due to the anisotropic properties of the brain-tissue material, thespreading is not uniform.

cD m/cD 0

m[-

]σ11[N

/m

2]

(cD m

full−

cD mred)/cD 0

m[-

](σ

11full−

σ11

red)/σ11

max[-

]

0 0

00

1

135

9×10−03

2×10−02

Figure 5.34: Therapeutical spreading (top) and stress σ11 (bottom) at the end of an infusionprocess under anisotropic conditions, displayed in the region of interest and computed with thefull system (left) and the reduced system (middle), and corresponding error (right).

Since the majority of the nonlinear terms are approximated with the reduction matrixΨw instead of computing them individually, a significant reduction of the computing timeto less than 7 % is obtained in the online phase, cf. Table 5.5.

computing solving write out totaltime [min] eq. system data CPU time

full system 156 8 164

red. system 3 8 11

Table 5.5: Computing time on a single core of an Intel i5-4590 with 32 GB of memory runningat clock speed of 3.30 GHz (online phase), obtained from the full and the reduced system with18 POD modes and 47 DEIM modes.

In a next step, the reduced model is investigated in terms of the accuracy for changingmaterial parameters. Therefore, the simulation is performed for two test cases with differ-ent values of the elastic material constants (test case 1: µS

0 = 5000 N/m2 and λS0 = 30 000N/m2, test case 2: µS

0 = 8000 N/m2 and λS0 = 10 000 N/m2) to prove the usefulness of


1

testcase 2

testcase

full system: point 1 full system: point 1

red. system: point 1 red. system: point 1

point 2point 2

point 2point 2

point 3point 3

point 3point 3

applied infusion time [min.] applied infusion time [min.]

applied infusion time [min.]applied infusion time [min.]

uS1[m

m]

pBR[N

/m

2]

pIR

[N/m

2]

cD m/cD 0

m[%

]

0

0

00

20

40

60

30

60

90

120

0

0

0

0

40

40

40

40

80

80

80

80

120

120

120

120

160

160

160

160

-2

-4

-60.01

0.02

0.03

0.04

Figure 5.35: Values of different primary variables at selected points 1-3 for a variation in thematerial parameters (test case 1: squares, test case 2: circles), obtained from the full system(small solid dots) and the reduced system (large unfilled dots).

the reduced system. In Figure 5.35, the simulation results of the primary variables areshown for both test cases at selected points 1-3 (cf. Figure 5.31). While a variation of theelastic material constants strongly influences the solid displacement, the influence on thepore-liquid pressures is only weak and the concentration is almost unaffected. Comparingthe results with the reference solutions of the full system, the accuracy of the primaryvariables is ensured. The same applies for the corresponding stresses. A determination ofthe corresponding NRMS errors results in εNRMS = 8 - 9×10−4 for the different simulations.Therefore, it can be concluded that an error estimation using the normalised projectionerrors PE(U ,V l) and PE(W ,Vk) delivers sufficiently accurate results for simulationsusing the POD-DEIM-reduced system.

Regarding the computational effort, the time for the generation of the reduced system(offline phase) occurs particularly from pre-computations in the full system (here around140min per simulation), the pre-computations in the POD-reduced system (here around75min per simulation), the saving of the snapshots for the POD method and the DEIM(here less than 10min) and the computation of the reduction matrices (here around47min). The time to solve the equation system for one simulation with a given setof material parameters in the online phase is reduced from around 180min (full system)to around 3min (reduced system). Additionally, both kind of simulations need around5min to read in the patient specific data and 8min to write out all necessary data. Fol-lowing this, performing more than five reduced simulations leads already to a time saving(sum up the times of the offline and the online phase) which grows significantly when per-forming more reduced simulations, see Table 5.6. As the computational effort is split ina costly offline phase that is performed only once and an inexpensive online phase that is

5.4 Application of the POD-DEIM approach to an intervertebral-disc model 99

offline phase online phase

number of simulations - 1 2 5 10 50

full system [min] - 193 386 965 1930 9650

reduced system [min] 917 16 32 80 160 800

Table 5.6: Computing time on a single core of an Intel i5-4590 with 32 GB of memory runningat clock speed of 3.30 GHz for different numbers of performed simulations, obtained from thefull and the reduced system when taking into account the time for the offline phase.

performed in every reduced simulation, fast simulations in daily clinical practice becomespossible. In this context, the computing time of the offline phase plays only a minor role.Furthermore, multiple simulations with varying material parameters or infusion boundaryconditions become increasingly time efficient.

5.4 Application of the POD-DEIM approach to an

intervertebral-disc model

A possibility to simulate the mechanical behaviour of the human spine is given by mod-elling the stiffer structures, i. e., the vertebrae, as a discrete multi-body system (MBS),whereas the softer connecting intervertebral discs are represented in a continuum-mecha-nical sense using the FEM. The mechanical behaviour of the IVD can be included intothe MBS by a co-simulation of a MBS and a FEM. For further details on the coupling ofthe MBS and the FE model, the interested reader is referred to the work of Karajan et al.[84] and citations therein. Regarding the co-simulation of the MBS and the FE model, areaction force or moment is activated whenever a relative displacement or an orientationchange is detected between two vertebrae. Following this, the resulting deformation state,consisting of displacements and rotations, is applied at the IVD’s centre of gravity (COG)and the respective deformation behaviour is computed by a FE simulation. Finally, a ho-mogenisation step needs to be applied to obtain the discrete mechanical response (i. e.,the resulting forces and moments) of the IVD.

However, an accompanying use of the FE model in form of a co-simulation with the MBSis typically not feasible due to the high computation time for realistic scenarios. Thisdrawback motivates the application of model-reduction techniques, aiming to reduce therequired computation time. Therefore, the nonlinear system of the intervertebral-discmodel, derived in Subsection 2.4.3, is reduced in this section using the POD-DEIM toenable numerically efficient simulations. In this context, the selection of specific snapshotsis extensively investigated in an extra subsection since an appropriate choice stronglyaffects the quality of the reduced simulations. Note in passing that another possibilityto include the mechanical behaviour of the IVD into the MBS is an offline simulationin a pre-computation step, where a representation of the discrete mechanical response ofthe IVD is defined in form of a polynomial in terms of the applied degrees of freedom(deformation states) of the MBS, see Karajan et al. [83].


5.4.1 Reduced-order system

Simulating the anisotropic IVD with the extended biphasic model derived in Subsection2.4.3, a nonlinear system (3.37) with an approximately time-invariant system matrix Dand with a nonlinear term k(u, q) is used to describe the respective set of equations. Inorder to determine the reduced system (4.10), the modified POD method, presented inSubsection 4.2.2, is used, since the vector of unknowns u contains as primary variablesthe solid displacement vector uS and the hydraulic pressure P, which (physically) exhibita different behaviour in time. Moreover, in order to account for the nonlinearities, thePOD-DEIM approach is applied to the global equilibrium equation (3.37) in accordanceto Appendix C.3, leading to the reduced system (C.16).

Recapitulating the required tasks, two different groups of pre-computations need to beperformed. First, the state variables ui are stored in each time step ti of a number ofpre-computations, which are performed with a default parameter setting and pre-defineddeformation states, using the initial full system (3.37). Additionally, the system matrixD is stored in the first time step of one of these pre-computations. Subsequently, thereduction matrix Φu and the reduced system matrix D are determined. Afterwards, asecond set of pre-computations is performed with the POD-reduced system (4.10) usingthe identical problem setting as in the pre-computations of the full system and storingthe nonlinear terms wi in each Newton step. Finally, the reduction matrix Ψw andthe resulting coefficient matrix ξ = Ψw (P TΨw)

−1P T are determined. This completesthe offline phase and enables time-efficient simulations within an online phase using thereduced system (C.16).

5.4.2 Problem setting

Following the problem setting described in Karajan et al. [84] for a co-simulation of theFE model of the IVD with a MBS, a simplified representation of a cylindrical IVD withhomogeneous material properties is considered in this numerical example. Furthermore,the cylindrical geometry with a radius of 25 mm and a height of 15 mm is spatially dis-cretised with 640 three-dimensional Taylor-Hood elements, cf. Figure 5.36, leading to asystem with a total of 10 095 DOF. In particular, the IVD is modelled with a homoge-neous distribution of the collagen fibres in the circular AF, using a uniform fibre angleof φS

0 = 57 with respect to the axial direction e3, combined with constant anisotropic

e1

e2e3

free, drained

loaded, undrained

fixed, undrained

Figure 5.36: Geometry of a (3-d) intervertebral-disc model with boundary conditions (left)and FE mesh (right) of the initial-boundary-value problem.


material parameters referring to the inner AF. The material parameters are chosen inaccordance to the initial-boundary-value problem presented in Karajan et al. [83, 84].A detailed derivation of the nonlinear material laws and their material parameters canbe found in Ehlers et al. [50] or Karajan [81, 82]. In order to mechanically support thedomain, the lower side of the geometry is fixed in space (Dirichlet boundary conditionfor the solid displacement), while the upper side of the geometry is displacement-drivenaccording to the applied discrete deformation states (DDS), using a rigid bearing plate.In addition, an efflux of the interstitial fluid over the sidewall of the cylindrical geometryis possible (Neumann boundary condition for the hydraulic pressure). Resulting fromthe rotational symmetry of the cylindrical geometry, the possible deformation states aresimplified to allow only movements in the sagittal plane. In particular, these are the ro-tation ϕ1 and the two displacements u2 and u3. In order to capture the viscous materialbehaviour stemming from dissipative interactions inside the IVD, the rates of these de-formation variables add another three DDS. Note that the inhomogeneous wedge-shapedIVD exhibits a total of three translations and three rotations as well as their rates asDDS. As presented in Figure 5.37, the discrete rotation ϕ1 is parameterised with respectto the top surface of the IVD, yielding a conversion into the nodal displacements uS ofthe discretised FE system. In the case of only a single rotation in the sagittal plane,the parameterisation of the discrete angle ϕ1 and the discrete displacements u2 and u3 isgiven by

uS =(

u2 + (cosϕ1 − 1)X2 − sinϕ1 X3

)

︸︷︷︸

uS2

e2+(

u3 + sinϕ1X2 + (cosϕ1 − 1)X3

)

︸︷︷︸

uS3

e3 . (5.19)

As a next step, the surface traction vector t, which is the reaction stress of the IVD dueto the applied deformation uS, needs to be converted into a discrete force and momentby applying a homogenisation procedure (cf. Figure 5.37) to capture the integral responseof the IVD. Following Karajan et al. [84], the discrete mechanical reaction of the IVD,in terms of a force R and a moment M, is computed via an integration of the surfacetraction vector t, yielding

R =

∫

∂Ω

Tn da =

∫

∂Ω

t da and M =

∫

∂Ω

r×Tn da =

∫

∂Ω

r× t da , (5.20)

e3

e2e1 COG

COG

x =

XS

uS QXS

t

r

da

Figure 5.37: Schematic drawing of a cut IVD with the parameterisation of the Dirichletboundary conditions (left) and quantities for the homogenisation procedure to the resultingsurface traction vector t (right), cf. Karajan et al. [83].


where r indicates the corresponding lever arm of the surface traction with regard to theCOG of the IVD. Thus, a reaction force or moment can be activated in the MBS whenevera relative displacement or orientation change is detected between two vertebrae.

5.4.3 Complexity of the snapshot choice

Proceeding from the intended application of the simplified cylindrical IVD, shown inFigure 5.36, within the co-simulation with the MBS, different deformation states mayoccur, representing movements of the IVD in the sagittal plane. In particular, these are therotation ϕ1, the two displacements u2 and u3 and, in order to capture the viscous materialbehaviour, the rates ϕ1, u2 and u3 of these deformation variables. Since only states andphenomena, which are represented by the snapshot data set, can be properly representedby a (POD-)reduced system, an appropriate choice of the snapshots is important to be ableto simulate these different deformation states and can significantly improve the reducedsimulation results. Furthermore, too many (especially the consideration of irrelevant)snapshots can diminish the results and the necessary number of POD modes. Regardingthe selection of the snapshots with respect to a parameter variation, a uniform distributioncan be a good choice if only few parameters need to be varied and/or less informationabout the variable distribution is available. However, a systematic and automatic strategyis in general preferable for the selection of the snapshots, especially if many parametersare varied or if many different deformation states are simulated.

Following this, a preferably automatic generation of suitable deformation states, whichare likely to occur due to the statistical distribution of all possible deformation states,should be performed for the IVD model in the offline phase. Note that physiological limitshave to be satisfied and thus, the applied deformation states need to lie within the limitsgiven in Table 5.7, which are based on the range of motion of the L4 and L5 vertebrae,cf. Monteiro [95] among others.

deformation ϕ1 [deg] u2 [mm] u3 [mm]

min. limit -6.0 -2.5 -2.5

max. limit 6.0 2.5 2.5

Table 5.7: Limiting deformation states embracing the range of motion for the displacementsu2 and u3 and the rotation ϕ1 of L4 and L5 vertebrae, based on values given in Monteiro [95].

Moreover, care must be taken regarding the applied deformation rate to reach the desireddeformation states. Following Karajan et al. [84], the underlying TPM model yields aquasi-static response if the loading rates are sufficiently small, i. e., they need to be at thelower end of

0.0001 deg/s ≤ ϕ1 ≤ 1.0 deg/s ,

0.00003mm/s ≤ u2 ≤ 1.0mm/s ,

0.00005mm/s ≤ u3 ≤ 1.0mm/s .

(5.21)


Following this, the lower limit is identified as purely elastic response, whereas an appli-cation of a loading rate within these intervals leads to viscous effects (taken into accountby an incorporation of the loading rates ϕ1, u2 and u3) until a “dynamic” limit evolvesdue to the limitations of the underlying quasi-static TPM model, cf. Figure 5.38. In thisregard, it should be considered that small values of the deformation rates (and also lowdeformations) occur more likely than large values (deformations near the physiologicallimits). Furthermore, it should be ensured that sufficient very small values of ϕ1, u2 andu3 are recognised when using randomly distributed values of the deformation rates. Thiscannot be assured when uniformly distributed values are generated. Therefore, a log-normal distribution is used in the following whenever random values of the deformationrates are required.

Ω = Ωel + Ωvisc

Ωmax = Ωel + Ωdyn

Ωel

Ωdyn

Ωvisc

Figure 5.38: Additive split of a response Ω (e. g. forces and moments) in a purely elastic partΩel and a viscous part Ωvisc and corresponding response limits (lower limit Ωel and upper limitΩmax = Ωel+Ωdyn, corresponding to the “dynamic” limit) of an IVD with viscoelastic properties.

While an appropriate choice of the snapshots can be performed comparatively easy whenonly one deformation state is applied, the sampling process of the snapshots becomesconsiderably more complex when all discrete deformation states are applied simultane-ously and, where appropriate, with different loading rates. However, this is essential tobe able to simulate the variety of typical movements of the IVD in the sagittal plane.Consequently, all these different types of loading states need to be represented by thesnapshot data set. In the following numerical examples, these snapshots are generatedusing the different types of sampling described in Subsection 4.2.4. In particular, sam-pling via physical intuition, random sampling and sampling via the Greedy procedure arediscussed and compared to each other.

5.4.4 Numerical results

In a first step, a purely elastic behaviour of the cylindrical IVD (cf. Figure 5.36) is in-vestigated by using the lower limits of (5.21) to apply different deformation states ϕ1,u2 and u3 with sufficiently small loading rates. Already for the elastic consideration, anappropriate choice of the snapshots is important to be able to simulate different defor-mation scenarios (particular with regard to simultaneously applied deformation states)within the limits given in Table 5.7 with a system, which is reduced using the modifiedPOD-DEIM approach. In order to capture the viscous effects of the IVD, the loading rates


ϕ1, u2 and u3 are in a next step incorporated as quantities, which are varied within thepre-computations in the offline phase by applying the deformation states within accord-ingly adjusted time periods. Since this increases the complexity of the underlying modelas well as the number of possible deformation scenarios, an appropriate snapshot choicebecomes even more important. Therefore, different possibilities to select the snapshotsare discussed with the aim to find a preferably systematic and automatic strategy, whichleads to sufficiently accurate (comparing the results using the full system with the oneusing the reduced system) simulation results.

Purely elastic behaviour of the IVD

Initially, multiple pre-computations are performed with the problem setting described inSubsection 5.4.2 for different values of the deformation states ϕ1, u2 and u3 using thefull system with 10 095 DOF. Thereby, the discrete deformation states are, in a first step,considered separately. Since this results in a relatively straightforward set of potentialdeformation scenarios, the respective deformation states for the pre-computations can bechosen with physical intuition as a combination of very high (limits given in Table 5.7) andlow (ϕ1 = 2deg, u2 = 0.5mm and u3 = 0.5mm) absolute values. Moreover, the reductionmatrices are determined using normalised projection errors PE(U ,V l) and PE(W ,Vk)of at most 10−7 to specify the number of considered POD and DEIM modes. This leadsto a consideration of 26 POD and 154 DEIM modes. Afterwards, 29 test simulationsare performed varying the deformation states ϕ1, u2 and u3 within their physiologicallimits. In order to demonstrate that the reduced system of the IVD model with purelyelastic behaviour is able to produce accurate simulation results for different deformationscenarios, the total stress σ11 (all other stresses show an analogous behaviour) withinthe domain at the end of three exemplary deformation processes, obtained from the fullsystem and the POD-DEIM-reduced system, is compared in Figure 5.39.

σ11[N

/m

2]

σ11[N

/m

2]

σ11[N

/m

2]

error[-]

error[-]

error[-]

−0.26

−0.16

−0.33

0.28

0.23

0.04

0

0

0

2×10−03

2×10−03

7×10−03

Figure 5.39: Stress σ11 at the end of three exemplary deformation processes (top: ϕ1 = 3 deg,middle: u2 = 1.25 mm and bottom: u3 = −1.25 mm), visualised within the deformed grid(superelevated by a factor of two) and computed with the full system (left) and the reducedsystem (middle), and corresponding errors (σ11 full − σ11 red)/σ11max (right).


Since the prediction of the integral response of the IVD (discrete forces and moments)is of particular interest for a co-simulation of the IVD model with a MBS, the resultinghomogenised moment and force coefficients M1, R2 and R3 are presented in Figure 5.40,respectively. Note that due to the symmetry of the investigated problem, the integralresponse given in (5.20) is of the form M = M1 e1 and R = R2 e2 + R3 e3. The resultsof the reduced system show a very good agreement with the results of the full system.Moreover, several couplings become obvious, where an integral response may be sensitiveto multiple deformation states. For instance, the force coefficient R3 depends directly onu3 and is indirectly coupled to u2 and ϕ1.

full system: M1 full system: M1full system: M1red.system: M1 red.system: M1red.system: M1

full system: R2 full system: R2 full system: R2red.system: R2 red.system: R2 red.system: R2full system: R3 full system: R3 full system: R3red.system: R3 red.system: R3 red.system: R3

momentM

1[N

m]

momentM

1[N

m]

momentM

1[N

m]

forces

R2/R

3[N

]

forces

R2/R

3[N

]

forces

R2/R

3[N

]

rotation ϕ1 [deg]

rotation ϕ1 [deg]

displacement u2 [mm]




000

000

00

0

000

11

11

222

222

4

4

6

6

4

8

-1-1

-1-1

-2-2-2

-2-2-2

-4

-4

-6

-6

-4

-8

0.30.3

0.60.6

-0.3-0.3

-0.6-0.6

10

200

400

500

1000

-10

-20

-30

-40

-200

-400

-500

-1000

Figure 5.40: Dependence of the resulting moment M1 (top) and the resulting forces R2 andR3 (bottom) on the applied deformation states ϕ1 (left), u2 (middle) and u3 (right) using eitherthe full or the POD-DEIM-reduced system.

Regarding the computational effort, the time to solve the equation system is, for anapplication of a deformation state ϕ1, reduced from 6-33min (full system) to around1min (reduced system) per test simulation. Applying a deformation state u2, this timeis reduced from 11-28min to 1-4min, while for a specific applied deformation state u3,this time is reduced from 1-11min to around 1min. Thus, the entire time to solve theequation system (averaged over all test simulations) can be reduced to less than 8 %.Additionally, 3-6min per test simulation are needed to write out all necessary data (forboth, the full and the reduced system). Consequently, the total CPU time required toperform the test simulations can be reduced to less than 23 %, see Table 5.8. Furthermoreshould be mentioned that the time to write out data can be significantly reduced whenonly the resulting homogenised moment and force coefficients M1, R2 and R3 are writtenout.


offline phase online phase

CPU time [min] in total solving eq. system write out data in total

full system - 521 117 638

red. system 214 40 107 147

Table 5.8: Computing time on a single core of an Intel i5-4590 with 32 GB of memory runningat clock speed of 3.30 GHz for 29 test simulations, obtained from the full and the reduced system,when taking into account the time for the offline phase (including 7 pre-computations).

In the next step, the discrete deformation states are applied simultaneously. Obviously,this increases the number of possible deformation scenarios. Therefore, different samplingstrategies are performed to examine the respective advantages and disadvantages. In par-ticular, the following four strategies are used to generate combinations of the deformationstates ϕ1, u2 and u3, which are afterwards applied at the IVD in several pre-computa-tions. Fist, a sampling via physical intuition is performed. Therefore, 35 combinationsof ϕ1, u2 and u3 are chosen with physical intuition as a combination of very high andlow absolute values. Moreover, the values are applied separately as well as two or threesimultaneously. Furthermore, the symmetry of the investigated problem is considered byusing only positive values for ϕ1 (2 deg or 6 deg) and u2 (0.5 mm or 2.5 mm), while forthe displacement u3 also the lower limit is used (0.5 mm, 2.5 mm or -2.5 mm when ap-plied separately and -1.5 mm when applied simultaneously, respectively). In this regard,consideration has been given to the fact that extreme deformations may result in unphysi-ological high stresses and associated discrete forces and moments. In particular, this is thecase for unfavourable (unphysiological) combinations of ϕ1, u2 and u3 (here a combinationof u3 = −2.5mm with an additional deformation by ϕ1 or u2), which should therefore beavoided. Secondly, a random sampling scheme is used choosing 35 uniformly distributedrandom values of ϕ1, u2 and u3, respectively, within the physiological limits given in Table5.7, which are randomly combined. In both approaches, 35 full-order simulations need tobe performed, where in each simulation the snapshots are sampled. Within the last twosampling strategies, the Greedy procedure is performed using either the true projectionerror or the residual as error indicator. In both cases, the 35 uniformly distributed randomvalues of the deformation states, obtained from the random sampling scheme, are used asstarting point. From these 35 values, those 10 values with the lowest error/error indicatorare selected with the Greedy approach. When the true projection error is used as iden-tifier, simulations based on the full system have to be performed for all 35 deformationscenarios (whereby the snapshots are sampled only in the 10 selected pre-computations)to be able to determine the true error. In contrast, only 10 full-order simulations arenecessary when the residual (determined within the reduced-order simulations) is used aserror indicator. For the residual-based formulation of an error indicator, the error bound(5.14) is used. While the generalised damping matrix actually is a time-invariant matrix,the generalised stiffness matrix results only after the analytical linearisation. However,due to the restriction of purely elastic behaviour, the stiffness matrix changes only slightlyduring the simulations. Thus, the error bound is determined for this application on thebasis of the stiffness matrix arising at the first time step of the simulation.


In order to reduce the system, the offline phase (including the whole sampling process) isinitially performed for all four sampling strategies. Thereby, the reduction matrices aredetermined using normalised projection errors PE(U ,V l) of at most 10−5 and PE(W ,Vk)of at most 10−5 for the first two sampling variants and 10−7 for the Greedy approaches,respectively. The resulting number l of considered POD modes and the number k ofDEIM points for the different sampling strategies can be found in Table 5.9.

sampling strategy physical int. random sampl. Greedy (error) Greedy (res.)

number POD modes 24 27 21 22

number DEIM points 209 228 213 225

CPU time offline phase 17 h 24min 22 h 55min 26 h 11min 18 h 36min

CPU time test sim. 1 h 14min 1 h 28min 1 h 10min 1 h 14min

averaged NRMS error 1.43×10−2 0.47×10−2 0.56×10−2 0.63×10−2

Table 5.9: Characteristic values of the different sampling strategies: number l of consideredPOD modes, number k of DEIM points, total computing time of the offline phase, accumulatedcomputing time of the 20 test simulations and averaged NRMS error of the test simulations.

Afterwards, 20 test simulations with randomly determined deformation states are per-formed in each case. As a consequence of the reduced number of snapshots, which aresampled only in 10 instead of 35 pre-computations, the number of POD modes and thusalso the computing time of the test simulations is slightly smaller when using a Greedy-sampling approach. However, this has only a small influence on the accumulated com-puting time of the 20 test simulations, performed with the POD-DEIM-reduced system,which is for all cases significantly lower (reduction to 18-22%) than the accumulated timeof 6 h 34min for appropriate full-order simulations, cf. Table 5.9. Moreover, the averagedNRMS error of the test simulations is not so much influenced by the number of PODmodes and DEIM points than by the specific choice of the sampled snapshots. In thisregard, the best results are obtained for the reduced simulations on the basis of a ran-dom sampling scheme and a Greedy procedure using the true projection error as errorindicator, respectively. But also the residual-based Greedy-sampling approach provides alow NRMS error. In contrast, the sampling via physical intuition delivers a much highererror in comparison to the other approaches. At this point, it should also be noted thatthe use of only 10 different deformation scenarios for the pre-computations (as used forthe Greedy approaches) would produce very imprecise simulation results when perform-ing the sampling via physical intuition or random sampling. Taking a closer look at thecomputing time required to perform the offline phase, it can be noted that due to thenecessarily performance of the 35 full-order simulations in combination with all the re-peated reduced-order simulations, the offline phase of the Greedy procedure under use ofthe true projection error is far more time-consuming than the offline phase of all othersampling variants, cf. Table 5.10. In this regard, the residual-based error indicator sub-stantially improves the computational effort of the offline phase and also enables a fasterperformance than using the random-sampling approach. Only the performance of a sam-pling via physical intuition provides an even lower computing time of the offline phase.


sampling strategy physical int. random sampl. Greedy (error) Greedy (res.)

number full-order sim. 35 35 36 10

number POD-red. sim. 35 35 55 55

number POD-DEIM-red. sim. - - 315 315

time full-order sim. 10 h 36min 13 h 3min 13 h 4min 2 h 42min

time determination of Φu 12min 13min 1 h 19min 1 h 20min

time POD-red. sim. and6 h 36min 9 h 39min 6 h 35min 7 h 45mindetermination of Ψw

time POD-DEIM-red. sim. - - 5 h 13min 6 h 49min

Table 5.10: Factors influencing the total computing time of the offline phase: number of sim-ulations, performed with the full-order, the POD-reduced and the POD-DEIM-reduced system,total computing time of all full-order simulations, total time for the determination of all requiredPOD reduction matrices, total computing time needed to perform all POD-reduced simulationsplus to determine the respective DEIM reduction matrices and total computing time of all POD-DEIM-reduced simulations.

This is because the included pre-computations, in which only a discrete deformation u3is applied, are performed very fast. However, the computing time of the offline phaseplays in many applications only a minor role since the main focus is on the provisionof time-efficient simulations when they are actually needed. Instead, the efficiency andthe accuracy of the reduced model are the most important aspects. In this regard, it isobvious that the error of a test parameter set is considerably larger than the ones for thetraining parameter configurations.

In order to further evaluate the accuracy of the reduced models, particularly with regardto a co-simulation of the IVD model with a MBS, the resulting homogenised moment andforce coefficients M1, R2 and R3 are compared in Figure 5.41. Therein, the respectiveintegral response is plotted over that deformation state with the strongest influence onthis quantity to enable a comparison of the results. Obviously, also the other deformationstates have an influence on the resulting integral responses. Thus, the red lines in Figure5.41 exclusively serve to identify the results related to the full-order simulation and donot provide any information about the values of the moment and forces in between thedetermined values. Additionally, the deviations of the integral responses of the reducedsimulations from the full-order simulations are plotted with their corresponding meanvalues. Taking a closer look at the integral responses, it can be seen that particularly formoderate values of the deformation states, all reduced models can provide satisfactoryresults. At this point, it is worth remembering that in most cases small deformationsoccur and that extreme deformations may result in unphysiological high stresses andshould therefore be avoided anyway. However, not all sampling variants yield equallygood results. Comparing the resulting deviations of the integral responses, the momentM1 has generally the largest and the force R2 the smallest deviations. While the samplingvia physical intuition, followed by the Greedy procedure under use of the true projectionerror, performs worst, the random-sampling approach provides the best results. And alsothe residual-based Greedy-sampling approach yields only small deviations.


momentM

1[N

m]

forceR

2[N

]forceR

3[N

]

deviationmomentM

1[N

m]

deviationforceR

2[N

]deviationforceR

3[N

]

rotation ϕ1 [deg]rotation ϕ1 [deg]

displacement u2 [mm]displacement u2 [mm]


full-order simulations



sampling via physical int.






random sampling

random sampling

random sampling

random sampling

random sampling

random sampling

Greedy appr. (error-based)






Greedy appr. (residual-based)






MW 0.74MW 0.19MW 0.47MW 0.29

MW 17.98MW 18.23MW 20.20MW 26.13

MW 39.04MW 21.76MW 52.00MW 42.29

00

00

00

11

11

22

22

22 44 66

−1−1

−1−1

−2−2

−2−2

−2−2 −4−4 −6−6

0

0

0

0

0

0

1

2

3

4

4

8

12

16

−4

−8

−12

100

100

200

200

200

300

300

400

400

500

1000

−200

−400

−600

−500

−1000

Figure 5.41: Left: resulting moment M1 (top) and resulting forces R2 (middle) and R3 (bot-tom) plotted over that deformation state with the strongest influence and determined usingeither the full or the POD-DEIM-reduced system with different sampling schemes, right: corre-sponding deviations of the integral responses with associated mean values (MW).

Summarising the above, it can be stated that for this application the most suitable sam-pling approaches are the random sampling and the residual-based Greedy procedure.While the first one produces the most satisfactory simulation results, the second one canbe performed with less computational effort (for both, the total offline phase and the re-duced simulations). Comparing the two Greedy-sampling approaches, the residual-based


error indicator is clearly preferred to the use of the true projection error as error indicator.This approach substantially improves the computational effort of the offline phase withoutloosing much accuracy. However, an appropriate residual-based error indicator had firstto be formulated. Moreover, the sampling via physical intuition is the most unsuitableapproach. On the one hand, the specific choice of the discrete deformation states is notautomated and requires much preliminary work to identify the deformation states for thesampling process. On the other hand, the simulation results are significantly worse incomparison to the other approaches. This cannot be adequately compensated by the lowcomputing times of the offline phase and the reduced simulations. Furthermore, it canbe determined that for the presented application the consideration of all 35 uniformlydistributed random values of the deformation states does not diminish the simulationresults. The use of only the 10 most appropriate ones, as performed within the Greedyprocedure, does not lead to more accurate results. Only the number of considered PODmodes and thus also the computing time of a test simulation can be reduced - even ifthis is a moderate time reduction for this example. Consequently, the random-samplingapproach is preferable to the Greedy procedure using the true projection error as errorindicator. However, when a significantly higher number of pre-computations is necessary,for example due to an increased number of varied parameters, a random-sampling ap-proach can become inefficient which would prefer a (residual-based, as far as possible)Greedy approach.

Consideration of viscous effects

As was already stated, loading rates ϕ1, u2 and u3 need to be incorporated as additionalDDS, by applying the deformation states ϕ1, u2 and u3 within accordingly adjusted timeperiods, in order to capture the viscous effects of the IVD. Thus, the discussed cylindricalIVD exhibits a total of six DDS, which have to lie within the limits given in Table 5.7 andthe values given in (5.21). As before, a separate consideration of the deformation statesϕ1, u2 and u3 is discussed in a first step. For the sake of illustration, the applicationof u2 in connection with a varied loading rate u2 is presented hereinafter. In particu-lar, following the argumentation of the previous paragraph, a random sampling and aresidual-based Greedy procedure are performed to generate 15 (random sampling) andthe five most important of theses 15 (Greedy procedure) different combinations of u2 andu2, respectively. While a uniform distribution is used for u2, the values of u2 are generatedutilising a log-normal distribution. Afterwards, the reduced models are determined usingnormalised projection errors PE(U ,V l) of at most 10−6 and PE(W ,Vk) of less than10−9. With regard to the performed residual-based Greedy approach, no appropriate er-ror indicator for the respective nonlinear DAE system with singular system matrix canbe provided at the present state. Therefore, the residual-based error indicator, discussedin the previous paragraph for the simulation of the purely elastic behaviour, is reused toidentify the decisive combinations of u2 and u2 from the set of candidate parameter config-urations. Finally, test simulations with 10 randomly determined combinations of u2 andu2 are performed. The number of considered POD modes and DEIM points, the resultingcomputation times of the offline and the online phase, and the averaged NRMS errors ofthe test simulations can be found in Table 5.11. In total, the performance of the 10 testsimulations requires a computation time of 5 h 7min under use of the full system. This


sampling strategy random sampling Greedy approach

number POD modes 23 20

number DEIM points 187 175

CPU time offline phase 11 h 36min 7 h 25min

CPU time test sim. 1 h 15min 1 h 9min

averaged NRMS error 1.71×10−3 4.99×10−3

Table 5.11: Characteristic values of the different sampling strategies: number l of consideredPOD modes, number k of DEIM points, total computing time of the offline phase, accumulatedcomputing time of the 10 test simulations and averaged NRMS error of the test simulations.

time can be decreased to 1 h 15min (random sampling) and 1 h 9min (Greedy approach),respectively, when using the reduced system. While the total time for the performance ofthe offline phase is significantly smaller when using the residual-based Greedy-samplingapproach, the accuracy of the simulation results is higher when the snapshots are sam-pled within pre-computations of all 15 randomly determined combinations of u2 and u2.Nevertheless, the reduced model obtained by using the Greedy procedure also providessatisfactory simulation results. As before, the resulting homogenised moment and forcecoefficients M1, R2 and R3 are compared in order to further evaluate the accuracy of thereduced models, cf. Figure 5.42. As in the previous example, the coloured lines in Figure

momentM

1[N

m]

momentM

1[N

m]

forces

R2/R

3[N

]forces

R2/R

3[N

]

full system: M1

full system: M1

red.system: M1

red.system: M1

full system: R2

full system: R2

red.system: R2

red.system: R2

full system: R3

full system: R3

red.system: R3

red.system: R3

displacement u2 [mm] displacement u2 [mm]


rate

u2[m

m/s]

00

00

11

11

22

22

−1−1

−1−1

−2−2

−2−2

0

0

0

0

0.3

0.3

−0.3

−0.3

−0.6

−0.6

200

200

−200

−200

−400

−400

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 5.42: Dependence of the resulting moment M1 (left) and the resulting forces R2 and R3

(right) on the applied deformation state u2 and the deformation rate u2 for the use of either thefull or the POD-DEIM-reduced system with different sampling schemes (top: random sampling,bottom: residual-based Greedy approach).


5.42 exclusively serve to identify the results related to the full-order simulation and donot provide precise information about the values of the moment and forces in betweenthe determined values, as these values also depend on the applied loading rate u2. Inorder to provide additional information on the respective loading rates, the colour of thedots and circles are associated with the values of u2. Taking a closer look at the resultingforces, it can be noted that both reduced models provide very good results. In terms ofthe resulting moment, the performance of the random-sampling approach yields betterresults, particularly for moderate values of u2. Hence, as long as the computation timeof the offline phase only plays a subordinate role and the number of necessary parameterconfigurations for the sampling process is manageable, the random-sampling approachappears to be the most appropriate sampling technique for the discussed problem setting.

Finally, the simulation of the whole complex viscoelastic behaviour with simultaneousapplication of all DDS is discussed. More precisely, time-efficient simulations applyingthe three deformation states ϕ1, u2 and u3 simultaneously within different time periods(representing the loading rates ϕ1, u2 and u3, lying within the limits given in (5.21)) shouldbe enabled by the performance of a random-sampling approach. Therefore, combinationsof uniformly distributed random values of ϕ1, u2 and u3 and log-normally distributedvalues of ϕ1, u2 and u3 are used to perform 50 pre-computations. The offline phaseis completed by a determination of the required reduction matrices using normalisedprojection errors PE(U ,V l) of at most 10−5 and PE(W ,Vk) of less than 10−9. Theperformance of the reduced system is investigated with 15 test simulations, which need intotal a computation time of 9 h 56min when using the full system. Due to the increasednumber of pre-computations, the computation time of the offline phase is with 138 h 26minsignificantly higher than in the previous examples. Moreover, the dimension of the reducedbases (consideration of 53 POD modes and 1144 DEIM points) are very large. As aconsequence, the numerical effort cannot be reduced as effective as before. Using thereduced system, the accumulated computation time of the 15 test simulations can onlybe reduced to 7 h 47min, which corresponds to a reduction to 78%. Nevertheless, thedetermined reduced systems gain satisfactory simulation results for the performed testsimulations, resulting in an averaged NRMS error of 2.406×10−3. As before, the resultinghomogenised moment and force coefficients M1, R2 and R3 are compared in order tofurther evaluate the accuracy of the reduced models, cf. Figure 5.43. Therefore, theintegral responses are plotted over that deformation state with the strongest influence.Once again, the coloured lines exclusively serve to identify the results related to the full-order simulation. It can be seen that the reduced model provides accurate results forall three quantities. Moreover, the additional influence of the other discrete deformationstates on the respective integral responses becomes obvious through the non-smooth formof the coloured lines.

Using lower projection errors PE(U ,V l) and PE(W ,Vk), the dimension of the reducedbases can be reduced, but the simulation results are too imprecise and some of the testsimulations are interrupted resulting from convergence problems. Even if a residual-basedGreedy procedure is performed instead of the random sampling, no further improvementis possible. In this case, the reduced-order simulations within the first iteration steps areassociated with convergence problems and thus are highly time-consuming. As a result,

5.5 Generalised approach for an application-driven model reduction 113

momentM

1[N

m]

forceR

2[N

]forceR

3[N

]

rotation ϕ1 [deg]



full system

full system

full system

reduced system

reduced system

reduced system

0

0

0

0

0

0

1

1

2

2

2 4 6

−1

−1

−2

−2

−2−4−6

5

10

−5

−10

100

200

600

1200

−100

−200

−600

−1200

−1800

Figure 5.43: Resulting moment M1 (top) and resulting forces R2 (middle) and R3 (bottom)plotted over that deformation state with the strongest influence and determined using either thefull or the POD-DEIM-reduced system (random sampling).

the computation time of the offline phase becomes even higher than for the randomsampling. Furthermore, it should be recalled that the presented results are derived fora simplified representation of a cylindrical IVD with homogeneous material properties.When a inhomogeneous wedge-shaped IVD is simulated instead, the number of DDSfurther grows and the sampling process requires an even greater effort.

5.5 Generalised approach for an application-driven

model reduction

After having reduced several porous-media models with varying complexity, a generalisedapproach for an application-driven reduction of a possibly coupled system of equations ispresented hereinafter to enable an adaptation of the discussed modifications to other mod-els. Therefore, the respective model needs to be characterised in a first step. Afterwards,the different tasks of the offline phase need to be treated, before reduced simulationswithin the online phase can be performed.


5.5.1 From a full-order system towards time-efficient simula-tions on the basis of a reduced model

In order to reduce a (coupled) system of equations within an offline phase performed inadvance and to run fast simulations based on the reduced model within a time-efficientonline phase, the subsequent steps need to be carried out:

Characterisation of the system of equations

For an efficient reduction of a general system of equations, one should first clarify theutilised type of equation system and the type of application for which the reduced modelwill be used for. Therefore, the full system of equations needs to be initially formulatedand classified into a certain group. More precisely, one should firstly identify if the systemis linear (time-invariant system matrices) or nonlinear, and, secondly, if it is a coupledsystem and how the block structure looks like. Moreover, the different primary variablesϑs need to identified and one should think about the different physical behaviour ofthe primary variables in time. Afterwards, those primary variables who have a similartemporal behaviour and whose values have the same order of magnitude can be combinedinto a group ςi. Additionally, consideration should be given to the envisaged range of theparameter domain depending on the type of application the reduced model will be used for.This means that one should know in advance if simulations of different loading scenariosor with changing material parameters are performed and which specific parameters arevaried within which limits.

Offline phase

Once the system of equations and the primary variables are characterised, the offlinephase can be realised. In a first step, all necessary pre-computations are performed usingthe initial full system. Therefore, depending on the envisaged range of the parameterdomain, different sampling schemes can be used to specify the parameter and/or theloading scenarios (and thus the parameter configurations µj) for the pre-computations,as discussed in detail in Subsection 4.2.4. For linear systems, the values of the vector ofunknowns are stored in all time steps and are written in one global snapshot matrix, whena decoupled system is treated, or in separated snapshot matrices allocated to the groups ςiof the primary variables, when reducing a coupled system. Moreover, the time-invariantsystem matrices are stored in the first time step of one (simulating different loadingscenarios) or multiple (simulations with varying material parameters) pre-computations.Afterwards, the required dimension of the reduced system, the POD reduction matrix,which consists of the separated POD reduction matrices when treating coupled systems,and the reduced system matrices are determined. If the system matrices are parameter-dependent and the material parameters are varied within the reduced simulations, thereduced system matrices need to be determined in the online phase, as discussed in Sub-section 5.1.3. Dealing with a nonlinear system, the system matrices do not have to bestored in the pre-computations. Instead, a second set of pre-computations is performedusing the POD-reduced system with the same parameter configurations as before. Duringthese simulations, the values of the nonlinear terms are stored in all Newton steps and are

5.5 Generalised approach for an application-driven model reduction 115

likewise written either in one global reduction matrix or, when treating a coupled system,in separated snapshot matrices allocated to ςi. This leads to significantly more preciseresults than storing the snapshots of the nonlinear terms in the pre-computations usingthe full system. Finally, the required number of magic points, the interpolation indicesand the DEIM reduction matrix are determined.

Online phase

After performing the time-consuming offline phase, a time-efficient online phase with sim-ulations based on the reduced system can be performed for different loading or parameterscenarios. In this regard, it should be considered that the reduced model generally canonly be expected to provide accurate results for simulations with parameters lying insidethe sampling parameter domain. Basically, it can be stated that the smaller the numberof varied parameters, the larger the trust in the simulation results. In order to certify thereduced model, an appropriate error bound can be used to calculate an error indicatorwithin the reduced-order simulation. However, such an error bound first has to be derivedin dependence of the underlying system of equations. While for specific systems, such assystems where the system matrix is time-invariant and possesses the full rank (cf. Am-sallem & Hetmaniuk [3], Haasdonk & Ohlberger [70]), appropriate error bounds alreadyexist, other systems require an individual formulation, as presented in Subsection 5.1.3for the biphasic standard problem with time-invariant but singular system matrix.

Finally, a comprehensive outline of the above stated steps is given in Figure 5.44.

characterisation of the system

· formulation of the initial full system· classification of the system:

linear/nonlinear, coupled/decoupled· identification of the primary variables· definition of the parameter configurations µj

offline phase

· pre-computations for µj

using the full system−→ uii=1,...,m, D, K

· determination ofl, Φu, Dµj

, Kµj

offline phase

· pre-computations for µj

using the full system−→ uii=1,...,m

· determination of l, Φu

· pre-computations for µj usingthe POD-reduced system−→ wii=1,...,w

· determination of k, Ψw, P

linear system nonlinear system

online phase

· determination of Dand K, if appropriate

· time-efficient simulationsbased on the POD-reduced system

online phase

· time-efficient simulations based onthe POD-DEIM-reduced system

Figure 5.44: Outline of the individual steps from a full-order system towards time-efficientsimulations within a reduced model.

Chapter 6:Summary and outlook

6.1 Summary

Time-efficient simulations of complex continuum-mechanical models, which neverthelessproduce sufficiently accurate simulation results for different parameter settings, are thebasis to enable practical applications, such as an accompanying use in clinical practice. Inthis monograph, an application-driven approach was presented, providing reduced mod-els capable of simulating specific porous materials in a time-efficient manner. Therefore,different projection-based model-reduction techniques were discussed and the most ap-propriate methods were customised to the specific systems of equations.

Taking into account the high structural complexity of porous materials, it was initiallynecessary to include all relevant physiological and anatomical properties of the multicom-ponent materials. An appropriate multiphasic and multicomponent modelling approachon the basis of the thermodynamically consistent Theory of Porous Media was thereforepresented and specified for three models of porous materials, namely a biphasic standardproblem of a saturated porous soil, a multiphasic and multicomponent description of hu-man brain tissue with application to drug-infusion processes and an extended biphasicmodel for the description of an inhomogeneous and anisotropic intervertebral disc. Re-garding the numerical solution of these models, the obtained balance equations were, ina next step, rewritten in their weak forms. This allowed a convenient numerical treat-ment of the governing equations on the basis of the finite-element method in space andthe finite-difference method in time. Therefore, mixed extended Taylor-Hood elementsand an implicit Euler time-integration scheme were used and led to a discrete system ofcoupled partial differential equations. In this regard, the numerical implementation wasrealised with the finite-element solver PANDAS.

As a next step, a brief overview of different model-reduction techniques was presentedin order to reinforce the suitability of two specific methods, namely the proper orthog-onal decomposition and, as a supplement for the occurrence of nonlinear terms withinthe system of equations, the discrete-empirical-interpolation method. In this regard, thegeneral idea and all necessary mathematical fundamentals of these two approaches werepresented. Furthermore, it was necessary to extend the classical POD method to a mod-ified POD approach which is based on a split of the snapshot matrix, yielding separatedPOD reduction matrices allocated to the different primary variables. In this manner,the different physical behaviour in time and the huge differences in the absolute valuesof the primary variables were taken into account. Moreover, the error between the fulland the reduced system was characterised. Finally, different sampling strategies to selectthe parameter and/or the loading scenarios for the pre-computations were pointed outin order to capture the physical behaviour of the model in the entire parameter/loadingdomain.

117

118 6 Summary and outlook

The last part of this monograph is concerned with the performance and reduction of thespecified porous-media models. Therefore, different numerical examples were presented,discussing various aspects of the reduction process. Starting with a biphasic standardproblem resulting in a linear system of equations, the suitability of a system reductionusing the (modified) POD method was demonstrated within different problem settings.Moreover, the formulation of an error indicator was presented in order to certify the sim-ulation results of the reduced model. Afterwards, the focus was placed on the applicationof the POD-DEIM approach when simulating a porous material undergoing large defor-mations. In this regard, the necessity of separated reduction matrices was demonstrated.Moreover, the suitability of an appropriate modified POD-DEIM approach was pointedout for different parameter settings. In a next step, a simplified as well as a general multi-component brain-tissue model was reduced and simulated. Regarding the numerical effortto solve the linearised system of equations in each iteration step, a significant reduction ofthe computing time could be demonstrated. Thereby, the computational effort was splitup into a costly offline phase and an inexpensive online phase. Since the offline phase iscomputed only once, the time saving especially grows when multiple reduced simulationsare performed. Moreover, the cheap online phase of the reduced simulations allows thejustified desire to realise even real-time simulations in daily clinical practice in the nearfuture. For both models, accurate results were obtained for the reduced simulations. Inaddition, a variation of the simulation parameters, i. e., the material parameters or theinfusion boundary conditions, was presented. Afterwards, the modified POD-DEIM ap-proach was applied to an intervertebral-disc model studying the simulation of differentdeformation states. In this manner, an accompanying use of the reduced finite-elementmodel in form of a co-simulation with a multi-body system was discussed. Therefore, theselection of specific snapshots was investigated since an appropriate choice strongly affectsthe quality of the reduced simulations. In this regard, it turned out that for the presentedapplication a random sampling, using uniformly and log-normally distributed random val-ues of the discrete deformation states, respectively, provides the most promising results.However, it was also shown that a (residual-based) Greedy procedure can be used when asystematic and automatic strategy with a more restricted number of pre-computations ispreferred. Finally, a generalised approach for the adaptation of the evolved modificationsof the reduction process to other models was presented.

In conclusion, it can be stated that the presented application-driven approach for thereduction of coupled systems of equations enables the performance of time-efficient sim-ulations of complex models. Moreover, the formulation and application of the reducedmodels provide the possibility to run a variety of simulations with changing material/sim-ulation parameters in a considerably reduced computation time. This opens the perspec-tive to realise even real-time simulations, like accompanied simulations during clinicalinterventions, for instance.

6.2 Outlook

The present work makes a considerable contribution to the theoretical and numerical ap-plication of model-order-reduction techniques to coupled systems of porous materials. In

6.2 Outlook 119

this regard, the presented approach provides the possibility to run a variety of simulationswith changing material/simulation parameters in considerably less time with the perspec-tive to realise even real-time simulations in daily routines. However, there are certainlyvarious open issues, which still need to be considered.

Due to the data-dependent nature of the applied model-reduction techniques, the reducedmodels generally cannot be expected to provide accurate results for simulations with pa-rameters lying outside the sampling parameter domain used to produce the snapshots.In this context, an adaptive framework that updates the POD basis may improve theaccuracy of the reduced model. Furthermore, the selected entries of the nonlinear terms,which are calculated within the reduced simulations, still depend on the full-dimensionalapproximation of the vector of unknowns, which therefore needs to be determined in eachiteration step. In this regard, a further reduction of the numerical effort is in general fea-sible. However, this would require some modifications within the source code of the usedfinite-element solver PANDAS. Another issue represents the validation of the developedreduced models with an appropriate error indicator, which ideally should be calculateddirectly within the reduced-order simulation. While for the biphasic standard problem,resulting in a linear DAE system, an error indicator was presented, an adequate errorindicator for the nonlinear DAE systems would also be of great interest. Moreover, thederived bounds overestimate the norms of the true errors, which hinder an accurate es-timation of the true error. Therefore, the presented approach offers future potentials foran improvement of the error bounds.

A further aspect is concerned with the required computation time of the pre-computationsperformed within the offline phase. In applications where the computation time of theoffline phase plays a role, parallel solution strategies could be contribute to the reductionof the offline time and also to a further reduction of the computational effort of the onlinephase. Additionally, the pre-computations could be performed simultaneously at severalcores to reduce the computation time of the offline phase.

Appendix A:Selected relations of tensor calculus

This part of the appendix provides a selected collection of important rules which allowfor a concise and convenient treatment of vector and tensor operations. For a morecomprehensive discussion, the interested reader is referred to the textbook of de Boer[14] and the vector- and tensor script of the Institute of Applied Mechanics (ContinuumMechanics) at the University of Stuttgart, which is available online (Ehlers [41]).

A.1 Tensor algebra

For the following considerations arbitrary placeholders are introduced. Let α, β ∈ R

be rational scalar quantities (zero-order tensors), a,b, c,d ∈ V3 be vectors (first-ordertensors) of the proper Euklidian 3-d vector space V3 and A,B,C,D ∈ V3 ⊗ V3 ten-sors (of second order) of the corresponding dyadic product space V3 ⊗ V3. Moreover,

n

A,n

B ∈ V3 ⊗ ...⊗ V3 (n times) are tensors of order n of the corresponding dyadic prod-uct space V3 ⊗ ...⊗ V3.

Collected rules for products of second-order tensors with scalars or vectors:

A (α a) = α(Aa) = (αA) a : associative law

(α + β)A = αA+ βA : distributive law

α (A+B) = αA+ αB : distributive law

A (a+ b) = Aa+Ab : distributive law

(A+B) a = Aa+Ba : distributive law

a = Ab : linear mapping

I a = a : linear mapping with identity element I

0 a = 0 : linear mapping with zero element 0

(A.1)

Collected rules for the scalar product of second-order tensors (inner product):

(αA) ·B = A · (αB) = α (A ·B) : associative law

A · (B+C) = A ·B+A ·C : distributive law

A ·B = B ·A : commutative law

A ·B = 0 ∀ A , if B ≡ 0

A ·A > 0 ∀ A 6= 0

a · (a⊗ b) = a ·Ab

(A.2)

121

122 Appendix A: Selected relations of tensor calculus

Collected rules for the tensor product of second-order tensors:

α (AB) = (αA)B = A (αB) : associate law

(AB) a = A (Ba) : associate law

(AB)C = A (BC) : associate law

A (B + C) = AB + AC : distributive law

(A + B)C = AC + BC : distributive law

IA = AI = A : linear mapping with identical element I

0A = A0 = 0 : linear mapping with zero element 0

(a⊗ b)(c⊗ d) = (b · c) a⊗ d(A.3)

Collected rules for the transposed second-order tensor:

(a⊗ b)T = (b⊗ a)

(αA)T = αAT

(AB)T = BTAT

a · (Bb) = (BTa) · bA · (BC) = (BTA) ·C(A + B)T = AT + BT

(A.4)

Computation rules for the trace operator, the determinant and the cofactor of a second-order tensor:

trA = A · IdetA = 1

6(A

@

@

@

@ A) ·A = 16(trA)3 − 1

2(trA) (AT ·A) + 1

3(AA)T ·A

cofA = 12A

@

@

@

@ A = 12(aik ano einj ekop) (ej ⊗ ep) =:

+ajp (ej ⊗ ep)

(A.5)

where cofA can be evaluated using relations (A.9) - (A.10) and index notation. Thus, the

coefficient matrix+ajp contains at each position ( · )jp the corresponding subdeterminant,

e. g.,+a11= a22 a33 − a23 a32 .

Collected rules for the inverse second-order tensor:

A−1 = (detA)−1 (cofA)T → A−1 exists if detA 6= 0

AA−1 = A−1A = I

(A−1)T = (AT )−1 =: AT−1

(AB)−1 = B−1A−1

(A.6)

A.1 Tensor algebra 123

Collected rules for the determinant and the cofactor of second-order tensors:

(cofA)T = cofAT

detAT = detA

det (AB) = detA detB

det (αA) = α3 detA

det I = 1

det(cofA) = (detA)2

detA−1 = (detA)−1

det(A+B) = detA + cofA ·B + A · cofB + detB

(A.7)

The third-order fundamental (Ricci) tensor and the axial vector:

a× b =3

E (a⊗ b) : where3

E is the permutation tensor, cf. (A.9)

A×B =3

E (ABT ) : with the specific case I×C =3

ECT = 2Ac

Ac = 1

2

3

ECT : whereAc is the axial vector of C

(A.8)

In index notation, the properties of the permutation tensor are given, viz.:

3

E = eijk (ei ⊗ ej ⊗ ek) with the “permutation symbol” eijk

eijk =

1 : even permutation

−1 : odd permutation

0 : double indexing

−→

e123 = e231 = e312 = 1

e321 = e213 = e132 = −1

all remaining eijk vanish

(A.9)

Collected rules for the outer (double cross) product of second-order tensors:

(a⊗ b)

@

@

@

@ (c⊗ d) = (a× c)⊗ (b× d)

(A

@

@

@

@ B) ·C = (B

@

@

@

@ C) ·A = (C

@

@

@

@ A) ·B (A.10)

Fourth-order fundamental tensors:

4

I := (I⊗ I)23T −→ (I⊗ I)

23T A = A : identical map

(I⊗ I)24T −→ (I⊗ I)

24T A = AT : transposing map

(I⊗ I) −→ (I⊗ I)A = (A · I) I : tracing map

(A.11)

Herein, the transpositions ( · )ik

T indicate an exchange of the i-th and k-th basis systemsincluded into the tensor basis of higher-order tensors.

124 Appendix A: Selected relations of tensor calculus

Properties of simple fourth-order tensors:

4

A = (A⊗B)23T = (BT ⊗AT )

14T

4

AT = [(A⊗B)23T ]T = (AT ⊗BT )

23T

4

A−1 = [(A⊗B)23T ]−1 = (A−1 ⊗B−1)

23T

4

B = (A⊗B)24T = [(A⊗B)

13T ]T

4

BT = [(A⊗B)24T ]T = (B⊗A)

24T

4

B−1 = [(A⊗B)24T ]−1 = (BT−1 ⊗AT−1)

24T

(A.12)

A.2 Tensor analysis

The product rule of derivatives of products of functions:

(a⊗ b)′ = a′ ⊗ b + a⊗ b′

(AB)′ = A′ B + AB′ (A.13)

Collected derivatives of tensors and their invariants:

∂A

∂A= (I⊗ I)

23T =

4

I

∂AT

∂A= (I⊗ I)

24T

∂A−1

∂A= − (A−1 ⊗AT−1)

23T

∂ trA

∂A= I

∂ detA

∂A= cofA = (detA)AT−1

∂ cofA

∂A= detA [(AT−1 ⊗AT−1) − (AT−1 ⊗AT−1)

24T ]

∂(A · I)I∂A

= (I⊗ I)

(A.14)

A.2 Tensor analysis 125

Selected computation rules for the gradient and the divergence operators:

grad (αβ) = α gradβ + β gradα

grad (αb) = b⊗ gradα + α gradb

grad (αB) = B⊗ gradα+ α gradB

div (αb) = b · gradα + α divb

div (a⊗ b) = a divb+ (grada)b

div (αB) = B gradα + α divB

div (Ab) = (divAT ) · b+AT · gradb

div (gradb)T = grad divb

(A.15)

Appendix B:Specific derivation of the overall systems ofequations in abstract formulation

A systematic derivation of the equilibrium equations formulated in Section 2.4 is providedin this appendix. Thereby, the operators of the operator equations are written down indetail and the associated nodal function vectors are formulated. Moreover, the specialcase of geometrically (and for the porous-soil model also materially) linear behaviour withthe corresponding simplifications is further discussed.

B.1 Overall system of a quasi-static biphasic model

of a porous material

Starting from the system of weak formulations (3.12) in form of an operator equation,the equilibrium equation (3.18) in global form can be obtained using, in a first step, thediscrete formulation (3.15) of the nodal unknowns and the related test functions, yielding

Ghu=

∫

Ωh

(

Dh(uh, uh, δuh) + Kh(uh, δuh))

dv −∫

Γh

Fh(δuh) da = 0 , (B.1)

with the operators

Dh = ρFR div

(n∑

j=1

(

N juS

(ujS)

′S

))

n∑

i=1

(N ip δp

i)

︸︷︷︸

Dhp

,

Kh =[

TSE −

( n∑

j=1

(N jp p

j))

I]

· grad( n∑

i=1

(N iuSδui

S))

−

− (nSρSR + nFρFR)b ·( n∑

i=1

(N iuSδui

S))

︸︷︷︸

KhuS

+

+ρFR kF

γFRI[

grad( n∑

j=1

(N jp p

j))

− ρFR b]

· grad( n∑

i=1

(N ip δp

i))

︸︷︷︸

Khp

,

Fh = t ·n∑

i=1

(N iuSδui

S)

︸︷︷︸

FhuS

− q

n∑

i=1

(N ip δp

i)

︸︷︷︸

Fhp

.

(B.2)

127

128 Appendix B: Specific derivation of the overall systems of equations

Therein, the Dirichlet boundary conditions uhS and ph are not considered, as they are

explicitly fulfilled during the assembling of the FE system. In the next step, the operatorequation is transferred from a scalar-valued form Gh

uto a vector-valued form GGGh

u. Herein,the function vector GGGh

u represents a system of N linearly independent equations, where theith entry Gh

u i is obtained by setting the discrete test function δui (either δpj or δujSd, with

d = 1, ..., D having D dimensions in space, at the corresponding nodal point Pj) to 1,while setting the remaining ones to δut = 0 for t = 1, ..., i− 1, i+ 1, ..., N . Following this,the nodal function vector GGGh j

u at node Pj (for the general three-dimensional FE model)can be written as

GGGh ju =

∫

Ωh

0 0 0 0

0 0 0 0

0 0 0 0

∂Dh jp

∂ukS1

∂Dh jp

∂ukS2

∂Dh jp

∂ukS3

0

ukS1

ukS2

ukS3

pk

+

Kh juS1

Kh juS2

Kh juS3

Kh jp

dv −∫

Γ h

Fh juS1

Fh juS2

Fh juS3

Fh jp

da

=

∫

Ωh

0 0

∂Dh jp

∂ukS

0

ukS

pk

+

KKKh juS

Kh jp

dv −∫

Γ h

FFFh juS

Fh jp

da = 0 ,

(B.3)with the entries

∂Dh jp

∂ukSd=

∂

∂(ukSd)′S

(n∑

k=1

[

ρFR(

divN kuS

· (ukS)

′S

)

N jp

])

=n∑

k=1

(ρFRNkuSd,d

N jp ) ,

Kh juSd

= T SEd1N

juSd,1

+ T SEd2N

juSd,2

+ T SEd3N

juSd,3

−n∑

k=1

(Nkp p

k)N juSd,d

− ρ bdNjuSd

,

Kh jp =

ρFR kF

γFR

[n∑

k=1

(pk gradNkp · gradN j

p )− ρFR (b1Njp,1 + b2N

jp,2 + b3N

jp,3)

]

,

Fh juSd

= tdNjuSd

,

Fh jp = − q N j

p .

(B.4)

Therein, the partial derivatives are identified by ( · ),d = ∂( · )/∂xd and T SE dl are the

coefficients of the extra stress tensor TSE of the solid and depend on the solid displacement

as well as the internal variables. Finally, the equilibrium equation (3.18) in global formcan be formulated by collecting the function vectors of all degrees of freedom in GGGh

u andusing the abstract discrete form (3.16) of the nodal unknowns of the FE mesh and therelated nodal test functions, yielding

GGGhu(t, u, u, q) =

[

0 0

D21 0

]

︸︷︷︸

D

[

uuuS

ppp

]

︸︷︷︸

u

+

[

k1(uuuS, ppp, q)

k2(ppp)

]

︸︷︷︸

k(u, q)

−[

f1

f2

]

︸︷︷︸

f

!= 0 ,

(B.5)

B.1 Overall system of a quasi-static biphasic model of a porous material 129

including the quantities

D21 = ρFR

∫

Ωh

NNNp 111(

div (NNNuSI))T

dv ,

k1(uuuS, ppp, q) =

∫

Ωh

(

grad (NNNuSI)TS

E − div (NNNuSI) (NNNp 111)

T ppp − NNNuSI ρb

)

dv ,

k2(ppp) =ρFR kF

γFR

∫

Ωh

[

grad (NNNp 111)(

grad (NNNp 111))T

ppp − grad (NNNp 111) ρFR b

]

dv ,

f1 =

∫

Γ huS

NNNuSI t da ,

f2 = −∫

Γ hp

NNNp 111 q da ,

(B.6)

where 111 = [1 ... 1]T ∈ Rn is the vector of all ones and I = [I ... I]T ∈ R

3n×3 is a matrixconsisting of n identity matrices I ∈ R

3×3. Furthermore, the assumption of materiallyincompressible constituents (under moderate pressure) is applied, yielding ραR =const.A reformulation of the equilibrium equation (B.5) yields

[

0 0

D21 0

]

︸︷︷︸

D

[

uuuS

ppp

]

︸︷︷︸

u

+

[

K11(uuuS, q) K12

0 K22

]

︸︷︷︸

K(uuuS, q)

[

uuuS

ppp

]

︸︷︷︸

u

=

[

b1(uuuS) + f1

b2 + f2

]

︸︷︷︸

fext(uuuS) = b(uuuS) + f

, (B.7)

with the quantities

K11(uuuS, q) =

∫

Ωh

grad (NNNuSI)∂TS

E(NNNuSuuuS, q)

∂ uuuSdv ,

K12 = −∫

Ωh

div (NNNuSI) (NNNp 111)

T dv ,

K22 =ρFR kF

γFR

∫

Ωh

grad (NNNp 111)(

grad (NNNp 111))T

dv ,

b1(uuuS) =

∫

Ωh

NNNuSI ρ(uuuS)b dv ,

b2 =(ρFR)2 kF

γFR

∫

Ωh

grad (NNNp 111)b dv ,

(B.8)


where Kij = ∂ki/∂uj are the particular blocks of the generalised stiffness matrix K.Proceeding from the special case of geometrically and materially linear behaviour, the

extra stress tensor TSE =

4

De εS = 2µSεS + λS(εS · I) I of the solid and the correspondinglinear strain tensor εS = 1

2(graduS + gradTuS) are related via a fourth-order elasticity

tensor4

De (elastic stiffness matrix), containing the macroscopic Lame constants µS andλS of the porous solid matrix. Thus, the quantity K11 can be expressed as

K11 =

∫

Ωh

grad (NNNuSI)

∂

∂ uuuS

[4

De1

2

[

grad(

(NNNuSI)TuuuS

)

+ gradT(

(NNNuSI)TuuuS

)]]

dv

=

∫

Ωh

1

2grad (NNNuS

I)4

De

[(

grad (NNNuSI)T)

23T

+[(

grad (NNNuSI)T)

23T ]

12T]

dv .

(B.9)

Furthermore, the dependency of the body-force vector b on the solid displacement uS

is negligible in that case, yielding an equilibrium equation (B.7) with (approximately)time-invariant stiffness matrix K.

B.2 Overall system of the dynamic porous-soil model

Starting from the system of weak formulations (3.20) and the supplementary weak for-mulation (3.21), the operators

D(u, u, δu) = (uS)′S · δuS

︸︷︷︸

D(vS)

+(

nSρSR (vS)′S + nFρFR (vS +wF )

′S

)

· δuS

︸︷︷︸

DuS

+

+ ρFR (vS +wF )′S · δwF

︸︷︷︸

DwF

,

(B.10)

K(u, δu) = − vS · δuS

︸︷︷︸

K(vS)

+ (TSE − p I) · grad δuS +

(

nFρFR grad (vS +wF )wF −

− (nS ρSR + nF ρFR)b)

· δuS

︸︷︷︸

KuS

+

+(nFγFR

kFwF + ρFR grad (vS +wF )wF − ρFR b

)

· δwF − p div δwF

︸︷︷︸

KwF

+

+ divvS δp − nFwF · grad δp︸︷︷︸

Kp

(B.11)

B.2 Overall system of the dynamic porous-soil model 131

and

F(δu) = t · δuS

︸︷︷︸

FuS

+ tF · δwF

︸︷︷︸

FwF

− q δp︸︷︷︸

Fp

.(B.12)

of an operator equation in the form (3.12), including the vector of unknownsu := [uS vS wF p ]T , can be found. Thus, the equilibrium equation (3.24) in globalform can be obtained using, in a first step, the discrete formulations

uS(x, t) ≈ uhS(x, t) =

n∑

j=1

N juS(x)uj

S(t) , δuS(x) ≈ δuhS(x) =

n∑

j=1

N juS(x) δuj

S ,

vS(x, t) ≈ vhS(x, t) =

n∑

j=1

N juS(x)vj

S(t) ,

wF (x, t) ≈ whF (x, t) =

n∑

j=1

N jwF

(x)wjF (t) , δwF (x) ≈ δwh

F (x) =n∑

j=1

N jwF

(x) δwjF ,

p(x, t) ≈ ph(x, t) =

n∑

j=1

N jp (x) p

j(t) , δp(x) ≈ δph(x) =

n∑

j=1

N jp (x) δp

j ,

(B.13)of the nodal unknowns and the related nodal test functions. As before, the Dirichletboundary conditions are not considered, as they are explicitly fulfilled during the assem-bling of the FE system. This yields

Ghu=

∫

Ωh

(


dv −∫

Γh

Fh(δuh) da = 0 , (B.14)

with the operators

Dh =n∑

j=1

(

N juS

(ujS)

′S

)

·( n∑

i=1

(N iuSδui

S))

︸︷︷︸

Dh(vS)

+

+[

ρn∑

j=1

(

N juS

(vjS)

′S

)

+ nFρFRn∑

j=1

(

N jwF

(wjF )

′S

)]

·( n∑

i=1

(N iuSδui

S))

︸︷︷︸

DhuS

+

+ ρFR

n∑

j=1

(

N juS

(vjS)

′S + N j

wF(wj

F )′S

)

·( n∑

i=1

(N iwF

δwiF ))

︸︷︷︸

DhwF

,

(B.15)


Kh = −n∑

j=1

(N juS

vjS) ·

( n∑

i=1

(N iuSδui

S))

︸︷︷︸

Kh(vS)

+

+[

TSE −

( n∑

j=1

(N jp p

j))

I]

· grad( n∑

i=1

(N iuSδui

S))

+

+[

nFρFR grad( n∑

k=1

(N kuS

vkS +N k

wFwk

F )) n∑

j=1

(N jwF

wjF )− ρb

]

·( n∑

i=1

(N iuSδui

S))

︸︷︷︸

KhuS

+

+[nFγFR

kF

n∑

j=1

(N jwF

wjF ) + ρFR grad

( n∑

k=1

(N kuS

vkS +N k

wFwk

F )) n∑

j=1

(N jwF

wjF )−

− ρFR b]

·( n∑

i=1

(N iwF

δwiF ))

−n∑

j=1

(N jp p

j) div( n∑

i=1

(N iwF

δwiF ))

︸︷︷︸

KhwF

+

+ div( n∑

j=1

(N juS

vjS)) n∑

i=1

(N ip δp

i) − nF

n∑

j=1

(N jwF

wjF ) · grad

( n∑

i=1

(N ip δp

i))

︸︷︷︸

Khp

(B.16)and

Fh = t ·n∑

i=1

(N iuSδui

S)

︸︷︷︸

FhuS

+ tF ·n∑

i=1

(N iwF

δwiF )

︸︷︷︸

FhwF

− qn∑

i=1

(N ip δp

i)

︸︷︷︸

Fhp

.(B.17)

Transferring the scalar-valued form Ghuof the operator equation to a vector-valued form

GGGhu, the function vector GGGh

u represents a system ofN linearly independent equations, wherethe nodal function vector GGGh j

u at node Pj can be written as

GGGh ju =

∫

Ωh

∂Dh j

(vS )

∂ukS

0 0 0

0∂Dh j

uS

∂vkS

∂Dh juS

∂wkF

0

0∂Dh j

wF

∂vkS

∂Dh jwF

∂wkF

0

0 0 0 0

ukS

vkS

wkF

pk

+

KKKh j(vS)

KKKh juS

KKKh jwF

Kh jp

dv −∫

Γ h

0

FFFh juS

FFFh jwF

Fh jp

da = 0 ,

(B.18)


with the entries

∂Dh j(vSd)

∂(ukSd)′S

=

n∑

k=1

(NkuSd

N juSd

) ,

∂Dh juSd

∂(vkSd)′S

= ρ

n∑

k=1

(NkuSd

N juSd

) ,∂Dh j

uSd

∂(wkFd)

′S

= nFρFR

n∑

k=1

(NkwFd

N juSd

) ,

∂Dh jwFd

∂(vkSd)′S

= ρFR

n∑

k=1

(NkuSd

N jwFd

) ,∂Dh j

wFd

∂(wkFd)

′S

= ρFR

n∑

k=1

(NkwFd

N jwFd

) ,

Kh j(vSd)

= −n∑

k=1

(vkSdNkuSd

N juSd

) ,

Kh juSd

= T SEd1N

juSd,1

+ T SEd2N

juSd,2

+ T SEd3N

juSd,3

−n∑

k=1

(Nkp p

k)N juSd,d

+

+[(

nFρFR

n∑

i=1

n∑

k=1

(viSd gradNiuSd

+ wiFd gradN

iwFd

)·(NkwF

wkF ))

− ρ bd

]

N juSd

,

Kh jwFd

= −n∑

k=1

(Nkp p

k)N jwFd,d

+[nFγFR

kF

n∑

k=1

(wkFdN

kwFd

) +

+(

ρFR

n∑

i=1

n∑

k=1

(viSd gradNiuSd

+ wiFd gradN

iwFd

)·(NkwF

wkF ))

− ρFR bd

]

N jwFd

,

Kh jp =

n∑

k=1

(vkS1NkuS1,1

+ vkS2NkuS2,2

+ vkS3NkuS3,3

)N jp − nF

n∑

k=1

(NkwF

wkF ) · gradN j

p ,

Fh juSd

= tdNjuSd

, Fh jwFd

= tFd NjwFd

, Fh jp = − q N j

p .(B.19)

However, since the last equation of the nodal function vector (B.18), and thus analogouslythe last equation of the corresponding equilibrium equation in global form, does notdepend on the pore pressure, such a formulation leads to difficulties within the timeintegration (DAE with higher differential index than 1). For this reason, the term nFwF

in the operator term Kp from equation (B.11) is replaced by

nFwF = − kF

γFR

[

grad p − ρFR[

b−(

(vS + wF )′S + grad (vS + wF )wF

)]]

, (B.20)

using the momentum balance (2.49)2 of the fluid. Thus, additional terms

∂Dh jp

∂(vkSd)′S

=kFρFR

γFR

n∑

k=1

(NkuSd

N jp,d) ,

∂Dh jp

∂(wkFd)

′S

=kFρFR

γFR

n∑

k=1

(NkwFd

N jp,d) (B.21)


enter in the nodal function vector (B.18) and the term Kh jp is changed to

Kh jp =

n∑

k=1

(vkS1NkuS1,1

+ vkS2NkuS2,2

+ vkS3NkuS3,3

)N jp +

+kF

γFR

( n∑

k=1

(pk gradNkp )− ρFR b

)

· gradN jp +

+kFρFR

γFR

n∑

i=1

n∑

k=1

(

(viS1 gradNiuS1

+ wiF1 gradN

iwF1

)N jp,1+

+ (viS2 gradNiuS2

+ wiF2 gradN

iwF2

)N jp,2+

+ (viS3 gradNiuS3

+ wiF3 gradN

iwF3

)N jp,3

)

·(NkwF

wkF ) .

(B.22)

Finally, the equilibrium equation (3.24) in global form can be formulated by collectingthe function vectors of all degrees of freedom in GGGh

u, yielding

GGGhu(t, u, u, q) =

I 0 0 0

0 D22 D23 0

0 D32 D33 0

0 D42 D43 0

︸︷︷︸

D

uuuS

vvvS

wwwF

ppp

︸︷︷︸

u

+

−vvvS

k2(u, q)

k3(u, q)

k4(u, q)

︸︷︷︸

k(u, q)

−

0

f2(t)

f3(t)

f4(t)

︸︷︷︸

f

!= 0 ,

(B.23)including the quantities

D22 = ρ

∫

Ωh

NNNuSI (NNNuS

I)T dv , D23 = nFρFR

∫

Ωh

NNNuSI (NNNwF

I)T dv ,

D32 = ρFR

∫

Ωh

NNNwFI (NNNuS

I)T dv , D33 = ρFR

∫

Ωh

NNNwFI (NNNwF

I)T dv ,

D42 =kFρFR

γFR

∫

Ωh

grad(NNNp 111) (NNNuSI)T dv , D43 =

kFρFR

γFR

∫

Ωh

grad(NNNp 111) (NNNwFI)T dv

(B.24)


of the generalised system matrix D, the quantities

k2 =

∫

Ωh

[

grad(NNNuSI)TS

E − div(NNNuSI) (NNNp 111)

Tppp + (NNNuSI) ρb+

+ (NNNuSI)nFρFR

[(

grad(NNNuSI))

13T

vvvS +(

grad(NNNwFI))

13T

wwwF

]T

(NNNwFI)TwwwF

]

dv ,

k3 =

∫

Ωh

[

div(NNNwFI) (NNNp 111)

Tppp + (NNNwFI)nFγFR

kF(NNNwF

I)TwwwF + (NNNwFI) ρFR b+

+ (NNNwFI) ρFR

[(

grad(NNNuSI))

13T

vvvS +(

grad(NNNwFI))

13T

wwwF

]T

(NNNwFI)TwwwF

]

dv ,

k4 =kF

γFR

∫

Ωh

[

γFR

kF(NNNp 111)

(

div(NNNuSI))T

vvvS + grad(NNNp 111)(

grad(NNNp 111))T

ppp− grad(NNNp 111) ρFR b+

+grad(NNNp 111) ρFR[(

grad(NNNwFI))

13T

wwwF

]T

(NNNwFI)TwwwF

]

dv

(B.25)of the generalised stiffness vector k, and the quantities

f2 =

∫

Γ huS

NNNuSI t da , f3 =

∫

Γ hwF

NNNwFI tF da , f4 = −

∫

Γ hp

NNNp 111 q da (B.26)

of the generalised force vector f . Assuming materially incompressible constituents (un-der moderate pressure), yields ραR =const. Furthermore, proceeding from the specialcase of geometrically and materially linear behaviour, the extra stress tensor can be for-

mulated as TSE = 1

2

4

De (graduS + gradTuS), the nonlinear convective terms (gradvS)wF

and (gradwF )wF can be omitted and the dependencies of the densities ρ and ρα on thesolid displacement uuuS are negligible. Thus, a reformulation of the equilibrium equation(B.23) yields

I 0 0 0

0 D22 D23 0

0 D32 D33 0

0 D42 D43 0

︸︷︷︸

D

uuuS

vvvS

wwwF

ppp

︸︷︷︸

u

+

0 −I 0 0

K21 0 0 K24

0 0 K33 K34

0 K42 0 K44

︸︷︷︸

K

uuuS

vvvS

wwwF

ppp

︸︷︷︸

u

=

0

b2 + f2

b3 + f3

b4 + f4

︸︷︷︸

fext = b+ f

,

(B.27)



K21 =

∫

Ωh

1

2grad (NNNuS

I)4

De

[(

grad (NNNuSI)T)

23T

+[(

grad (NNNuSI)T)

23T ]

12T]

dv ,

K24 = −∫

Ωh


T dv ,

K33 =nFγFR

kF

∫

Ωh

NNNwFI (NNNwF

I)T dv , K34 =

∫

Ωh

div(NNNwFI) (NNNp 111)

T dv ,

K42 =

∫

Ωh

NNNp 111(

div(NNNuSI))T

dv , K44 =kF

γFR

∫

Ωh

grad (NNNp 111)(

grad (NNNp 111))T

dv ,

b2 = ρ

∫

Ωh

NNNuSI b dv , b3 = ρFR

∫

Ωh

NNNwFI b dv , b4 = −k

FρFR

γFR

∫

Ωh

grad (NNNp 111)b dv ,

(B.28)yielding an equilibrium equation with (approximately) time-invariant system matrices Dand K.

B.3 Overall system of the simplified drug-infusion

model for brain tissue

Starting from the system of weak formulations (3.31), the equilibrium equation (3.32) inglobal form can be obtained using, in a first step, the discrete formulation of the nodalunknowns and the related nodal test functions, yielding the operator equation

Ghu=

∫

Ωh

(


dv −∫

Γh

Fh(δuh) da = 0 , (B.29)

with the operator Dh = DhpBR +Dh

pIR +DhcDm

with entries

DhpBR = nB

0S div[ n∑

j=1

(

N juS(uj

S)′S

)] n∑

i=1

(N ipBR δp

BR i) ,

DhpIR = (1− nB

0S) div[ n∑

j=1

(

N juS(uj

S)′S

)] n∑

i=1

(N ipIR δp

IR i) ,

DhcDm

= nI

n∑

j=1

(

N jcDm(cD j

m )′S

) n∑

i=1

(N icDmδcD i

m ) +

+

n∑

k=1

(NkcDmcDkm ) div

[ n∑

j=1

(

N juS(uj

S)′S

)] n∑

i=1

(N icDmδcD i

m ) ,

(B.30)

B.3 Overall system of the simplified drug-infusion model for brain tissue 137

the operator Kh = KhuS

+KhpBR +Kh

pIR +KhcDm

with entries

KhuS

=(

TSE − 1

1− nS

n∑

j=1

(nB0S N

jpBR p

BR j + nIN jpIR

pIR j) I)

·

· grad( n∑

i=1

(N iuSδui

S))

,

KhpBR =

KSB

µBRgrad

( n∑

j=1

(N jpBR p

BR j))

· grad( n∑

i=1

(N ipBR δp

BR i))

,

KhpIR =

KSI

µIRgrad

( n∑

j=1

(N jpIR

pIR j))

· grad( n∑

i=1

(N ipIR δp

IR i))

,

KhcDm

= −n∑

k=1

(NkcDmcDkm ) div

[KSB

µBRgrad

( n∑

j=1

(N jpBR p

BR j))] n∑

i=1

(N icDmδcD i

m ) +

+DDgrad( n∑

j=1

(N jcDmcD jm ))

· grad( n∑

i=1

(N icDmδcD i

m ))

+

+n∑

k=1

(NkcDmcDkm )

KSI

µIRgrad

( n∑

j=1

(N jpIR

pIR j))

· grad( n∑

i=1

(N icDmδcD i

m ))

,

(B.31)

and the operator Fh = FhuS

+ FhpBR + Fh

pIR + FhcDm

with entries

FhuS

= t ·n∑

i=1

(N iuSδui

S) , FhpBR = − vB

n∑

i=1

(N ipBR δp

BR i) ,

FhpIR = − vI

n∑

i=1

(N ipIR δp

IR i) , FhcDm

= − Dn∑

i=1

(N icDmδcD i

m ) .

(B.32)

Afterwards, the operator equation is transferred from a scalar-valued form Ghuto a vector-

valued form GGGhu, containing the nodal function vectors

GGGh ju

=

∫

Ωh

0 0 0 0

∂Dh jpBR

∂ ukS

0 0 0

∂Dh jpIR

∂ ukS

0 0 0

∂Dh jcDm

∂ ukS

0 0∂Dh j

cDm

∂ cDkm

ukS

pBRk

pIR k

cDkm

+

KKKh juS

Kh jpBR

Kh jpIR

Kh jcDm

dv −∫

Γ h

FFFh juS

Fh jpBR

Fh jpIR

Fh jcDm

da = 0 (B.33)


at node Pj (for the general tree-dimensional FE model with D = 3) with the entries

∂Dh jpBR

∂ ukSd=

n∑

k=1

(nB0S N

kuSd,d

N jpBR) ,

∂Dh jpIR

∂ ukSd=

n∑

k=1

(

(1− nB0S)N

kuSd,d

N jpIR

)

,

∂Dh jcDm

∂ ukSd=

n∑

k=1

( n∑

i=1

(N icDmcD im )Nk

uSd,dN j

cDm

)

,

∂Dh jcDm

∂ cDkm

=

n∑

k=1

(nI NkcDmN j

cDm) ,

Kh juSd

= T SE d1N

juSd,1

+ T SE d2N

juSd,2

+ T SE d3N

juSd,3

−

− 1

1− nS

n∑

k=1

(nB0S N

kpBR p

BRk + nINkpIR p

IR k)N juSd,d

,

Kh jpBR =

KSB

µBR·( n∑

k=1

(pBRk gradN jpBR ⊗ gradNk

pBR))

,

Kh jpIR

=KSI

µIR·( n∑

k=1

(pIR k gradN jpIR

⊗ gradNkpIR))

,

Kh jcDm

= − 1

µBR

n∑

i=1

(N icDmcD im )

[[

grad (KSB)T I ·( n∑

k=1

(pBRk gradNkpBR)

)]

+

+ (KSB)T ·[ n∑

k=1

(

pBRk grad (gradNkpBR)

)]]

N jcDm

+

+DD ·( n∑

k=1

(cDkm gradN j

cDm⊗ gradNk

cDm))

+

+

n∑

i=1

(N icDmcD im )

KSI

µIR·( n∑

k=1

(pIR k gradN jcDm

⊗ gradNkpIR))

,

Fh juSd

= tdNjuSd

, Fh jpBR = − vBN j

pBR ,

Fh jpIR

= − vIN jpIR

, Fh jcDm

= − DN jcDm.

(B.34)

Finally, the equilibrium equation (3.32) in global form can be formulated by collectingthe function vectors of all degrees of freedom in GGGh

u and using the abstract discrete form

B.3 Overall system of the simplified drug-infusion model for brain tissue 139

(3.29) of the nodal unknowns of the FE mesh, yielding

GGGhu =

0 0 0 0

D21 0 0 0

D31 0 0 0

D41(cccDm) 0 0 D44(uuuS)

︸︷︷︸

D(u)

uuuS

pppBR

pppIR

cccDm

︸︷︷︸

u

+

k1(uuuS,pppBR,pppIR)

k2(pppBR)

k3(pppIR)

k4(pppBR,pppIR,cccDm)

︸︷︷︸

k(u)

−

f1

f2

f3

f4

︸︷︷︸

f

!= 0 .

(B.35)A reformulation of the equilibrium equation, using the generalised stiffness matrix K withthe particular blocks Kij = ∂ki/∂uj , yields

0 0 0 0

D21 0 0 0

D31 0 0 0

D41(cccDm) 0 0 D44(uuuS)

︸︷︷︸

D(uuuS, cccDm)

uuuS

pppBR

pppIR

cccDm

︸︷︷︸

u

+

+

K11(uuuS) K12(uuuS) K13(uuuS) 0

0 K22 0 0

0 0 K33 0

0 K42(cccDm) K43(ccc

Dm) K44

︸︷︷︸

K(uuuS, cccDm)

uuuS

pppBR

pppIR

cccDm

︸︷︷︸

u

=

f1

f2

f3

f4

︸︷︷︸

f

,

(B.36)


D21 = nB0S

∫

Ωh

NNNpBR 111(

div (NNNuSI))T

dv ,

D31 = (1− nB0S)

∫

Ωh

NNNpIR 111(

div (NNNuSI))T

dv ,

D41 =

∫

Ωh

[

NNNcDm111(

div (NNNuSI))T]

(NNNcDm111)TcccDm dv ,

D44 = nI

∫

Ωh

NNNcDm111 (NNNcDm

111)Tdv

(B.37)


of the system matrix, the quantities

K11 =

∫

Ωh

1

2grad (NNNuS

I)4

De

[(

grad (NNNuSI)T)

23T

+[(

grad (NNNuSI)T)

23T ]

12T]

dv ,

K12 = − nB0S

1− nS

∫

Ωh

div (NNNuSI) (NNNpBR 111)T dv ,

K13 = − nI

1− nS

∫

Ωh

div (NNNuSI) (NNNpIR 111)

T dv ,

K22 =1

µBR

∫

Ωh

grad (NNNpBR 111)KSB(

grad (NNNpBR 111))T

dv ,

K33 =1

µIR

∫

Ωh

grad (NNNpIR 111)KSI(

grad (NNNpIR 111))T

dv ,

K42 = − 1

µBR

∫

Ωh

[

(NNNcDm111)⊗

[

grad (NNNpBR 111) div (KSB)T +

+grad(

grad (NNNpBR 111))

(KSB)T]]


K43 =1

µIR

∫

Ωh

grad (NNNcDm111)KSI

(

grad (NNNpIR 111))T


K44 =

∫

Ωh

grad (NNNcDm111)DD

(

grad (NNNcDm111))T

dv

(B.38)

of the stiffness matrix and the quantities

f1 =

∫

Γ huS

NNNuSI t da , f2 = −

∫

Γ h

pBR

NNNpBR 111 vB da ,

f3 = −∫

Γ h

pIR

NNNpIR 111 vI da , f4 = −

∫

Γ h

cDm

NNNcDm111 D da

(B.39)

of the force vector, where 111 = [1 ... 1]T ∈ Rn is the vector of all ones and I = [I ... I]T ∈

R3n×3 is a matrix consisting of n identity matrices I ∈ R

3×3. Proceeding from the specialcase of geometrically linear material behaviour and a constant blood volume fraction, thedependencies of the system matrix D and the stiffness matrix K on the solid displacementuuuS (approximately) vanish.

Appendix C:Specific derivation of the reduced systems inabstract formulation

This part of the appendix provides a systematic derivation of specific reduced systemsof equations. Proceeding from the respective equilibrium equations derived in Section3.3, the reduced systems are determined using the model-reduction techniques presentedin Chapter 4. Thereby, a distinction is made between the application of the classicalmethods and their modified variants.

C.1 Reduced formulation of the quasi-static porous-

soil model with linear system of equations

Proceeding from the special case of geometrically and materially linear behaviour in com-bination with negligible body forces (b = 0), the equilibrium equation (3.19) can beformulated in form of a linear system. Moreover the generalised vector of unknowns con-tains the nodal unknowns of the two primary variables, namely the solid displacementuS and the fluid-pore pressure p. Thus, the reduced system (4.11) on the basis of theclassical POD method can be formulated in abstract form as

ΦTu

[

0 0

D21 0

]

Φu ured + ΦTu

[

K11 K12

0 K22

]

Φu ured = ΦTu

[

f1

f2

]

, (C.1)

finally resulting in

ΦT2 D21 Φ1

︸︷︷︸

D

ured + (ΦT1 K11Φ1 +ΦT

1 K12Φ2 +ΦT2 K22Φ2

︸︷︷︸

K

)ured = ΦT1 f1 +ΦT

2 f2︸︷︷︸

f

.(C.2)

Therein, the reduction matrix Φu = [Φ1 Φ2]T is a full matrix which is for demonstration

purposes separated into two parts. Thereby Φ1 contains the entries which are allocatedto the nodal values of the solid displacement, whereas the entries of Φ2 are allocated tothe nodal values of the fluid-pore pressure. Inserting the relations (B.6), (B.8) and (B.9)

141

142 Appendix C: Specific derivation of the reduced systems in abstract formulation

into the abstract formulation (C.2), the individual quantities can be expressed as

D = ρFR ΦT2

[ ∫

Ωh

NNNp 111(

div (NNNuSI))T

dv]

Φ1 ,

K = ΦT1

(∫

Ωh

1

2grad (NNNuS

I)4

De

[(

grad(NNNuSI)T)

23T

+[(

grad(NNNuSI)T)

23T ]

12T]

dv

)

Φ1−

−ΦT1

(∫

Ωh


T dv)

Φ2+

+ρFR kF

γFRΦT

2

[ ∫

Ωh

grad (NNNp 111)(

grad (NNNp 111))T

dv]

Φ2 ,

f = ΦT1

∫

Γ huS

NNNuSI t da − ΦT

2

∫

Γ hp

NNNp 111 q da .

(C.3)

Using the modified POD method instead of the classical version, as presented in Subsec-tion 4.2.2, two separate snapshot matrices UuS

and Up, and thus two reduction matricesΦuS

and Φp, respectively, are determined. The reduction matrices allocated to the pri-mary variables are afterwards summarised in accordance to (4.13) in the reduction matrixΦu = blkdiag[ΦuS

, Φp]. Following this, the reduced system (4.11) can be formulated inabstract form as

[

0 0

ΦTp D21ΦuS

0

] [

uuuS red

pppred

]

+

+

[

ΦTuSK11ΦuS

ΦTuSK12Φp

0 ΦTp K22Φp

] [

uuuS red

pppred

]

=

[

ΦTuSf1

ΦTp f2

]

,

(C.4)


[0 0

D21 0

]

︸︷︷︸

D

[

uuuS red

pppred

]

︸︷︷︸

ured

+

[

K11 K12

0 K22

]

︸︷︷︸

K

[

uuuS red

pppred

]

︸︷︷︸

ured

=

[

f1

f2

]

︸︷︷︸

f

. (C.5)

C.2 Reduced system of the dynamic porous-soil model 143

Therein, the individual quantities can be expressed as

D21 = ρFR ΦTp

(∫

Ωh

NNNp 111(

div (NNNuSI))T

dv)

ΦuS,

K11 = ΦTuS

(∫

Ωh

1

2grad (NNNuS

I)4

De

[

(grad (NNNuSI)T )

23T + [(grad (NNNuS

I)T )23T ]

12T]

dv)

ΦuS,

K12 = −ΦTuS

( ∫

Ωh


T dv)

Φp ,

K22 =ρFR kF

γFRΦT

p

(∫

Ωh

grad (NNNp 111)(

grad (NNNp 111))T

dv)

Φp ,

f1 = ΦTuS

( ∫

Γ huS

NNNuSI t da

)

,

f2 = −ΦTp

( ∫

Γ hp

NNNp 111 q da)

.

(C.6)

Comparing the reduced systems (C.2) and (C.5), it becomes obvious that only if themodified POD method is used, the block structure of the coupled equation system canbe preserved while considering the different temporal (physical) behaviour of the primaryvariables.

C.2 Reduced system of the dynamic porous-soil

model

Proceeding from the special case of geometrically and materially linear behaviour in com-bination with negligible body forces (b = 0), the equilibrium equation in (3.24) can beformulated in form of a linear system, cf. Appendix B.2. In contrast to the quasi-staticporous-soil model, the generalised vector of unknowns contains, in addition to the nodalunknowns of the solid displacement uS and the fluid-pore pressure p, the nodal unknownsof the solid velocity vS and the fluid seepage velocity wF . Thus, the reduced system(4.11) on the basis of the classical POD method can be formulated in abstract form as

ΦTu

I 0 0 0

0 D22 D23 0

0 D32 D33 0

0 D42 D43 0

Φu ured + ΦTu

0 −I 0 0

K21 0 0 K24

0 0 K33 K34

0 K42 0 K44

Φuured = ΦTu

0

f2

f3

f4

,

(C.7)



D ured + K ured =:(

ΦT1Φ1 +ΦT

2D22Φ2 +ΦT2D23Φ3 +ΦT

3D32Φ2+

+ΦT3D33Φ3 +ΦT

4D42Φ2 +ΦT4D43Φ3

)

ured+

+(

−ΦT1 Φ2 +ΦT

2K21Φ1 +ΦT2K24Φ4 +ΦT

3K33Φ3+

+ΦT3K34Φ4 +ΦT

4K42Φ2 +ΦT4K44Φ4

)

ured

= ΦT2 f2 + ΦT

3 f3 + ΦT4 f4 := f .

(C.8)

Therein, the reduction matrix Φu = [Φ1 Φ2 Φ3 Φ4]T is a full matrix, which is for demon-

stration purposes separated into four parts. Thereby Φs contains the entries which areallocated to the nodal values of the primary variables ϑs. Thus, the approximation u ofthe vector of unknowns u can be formulated in a separated form, yielding

u = [uuuS vvvS wwwF ppp]T ≈ [Φ1 ured Φ2 ured Φ3 ured Φ4 ured]T = Φuured = u . (C.9)

As in the previous section, the block structure of the system is not preserved using theclassic POD method. Instead, the reduced system matrix D (and analogously the re-duced stiffness matrix K and the reduced force vector f , respectively) results from asummation of the reduced matrices ΦT

s DstΦt (or the reduced matrices ΦTs KstΦt and

the reduced vectors ΦTs fs, respectively), which are allocated to the nodal values of the

primary variables ϑs and ϑt.

Using the modified POD method instead of the classical version, as presented in Sub-section 4.2.2, four separate reduction matrices Φϑs

are determined and afterwards sum-marised in accordance to (4.13) in the reduction matrixΦu = blkdiag[ΦuS

, ΦvS, ΦwF

, Φp].Following this, the reduced system (4.11) can be formulated in abstract form as

I 0 0 0

0 ΦTvSD22ΦvS

ΦTvSD23 ΦwF

0

0 ΦTwFD32ΦvS

ΦTwFD33 ΦwF

0

0 ΦTp D42 ΦvS

ΦTp D43 ΦwF

0

uuuS red

vvvS red

wwwF red

pppred

+

+

0 −ΦTuS

ΦvS0 0

ΦTvSK21ΦuS

0 0 ΦTvSK24Φp

0 0 ΦTwFK33ΦwF

ΦTwFK34Φp

0 ΦTp K42ΦvS

0 ΦTp K44Φp

uuuS red

vvvS red

wwwF red

pppred

=

0

ΦTvSf2

ΦTwFf3

ΦTp f4

,

(C.10)

C.3 Reduced system of a nonlinear biphasic model of a porous material 145

finally resulting in a reduced system

I 0 0 0

0 D22 D23 0

0 D32 D33 0

0 D42 D43 0

︸︷︷︸

D

uuuS red

vvvS red

wwwF red

pppred

︸︷︷︸

ured

+

0 K12 0 0

K21 0 0 K24

0 0 K33 K34

0 K42 0 K44

︸︷︷︸

K

uuuS red

vvvS red

wwwF red

pppred

︸︷︷︸

ured

=

0

f2

f3

f4

︸︷︷︸

f

, (C.11)

where the block structure of the strongly coupled system of equations is preserved. More-over, the reduced vector of unknowns ured can be separated in four vectors uϑsred, whichare allocated to the four primary variables ϑs.

C.3 Reduced system of a nonlinear biphasic model

of a porous material

In the first instance, the POD method is used to reduce a general (nonlinear) system ofequations of a quasi-static biphasic model with two primary variables, namely the soliddisplacement uS and the fluid-pore pressure p (or rather the hydraulic pressure P), asgiven in (3.18) for the simulation of a porous soil undergoing large deformations or in(3.37) for the simulation of an intervertebral disc. For both materials, the assumption ofmaterially incompressible constituents (under moderate pressure) is applied. Thus, thereduced system (4.10) on the basis of the classical POD method can be formulated inabstract form as

ΦTu

[

0 0

D21 0

]

Φu ured + ΦTu

[

k1(Φu ured)

k2(Φu ured)

]

= ΦTu

[

f1

f2

]

. (C.12)

In contrast, using the modified PODmethod, as presented in Subsection 4.2.2, the reducedsystem can be given as

[

0 0

ΦTp D21 Φp 0

][

uuuS red

pppred

]

+

[

ΦTuSk1(Φuured)

ΦTp k2(Φu ured)

]

=

[

ΦTuSf1

ΦTp f2

]

. (C.13)

Therein, two separate reduction matrices ΦuSand Φp (or rather ΦP for the IVD model)

are determined and afterwards summarised in accordance to (4.13) in the reduction matrixΦu = blkdiag[ΦuS

, Φp] in order to preserve the block structure of the coupled equationsystem while considering the different temporal (physical) behaviour of the primary vari-ables. Moreover, in order to account for the nonlinearities, the DEIM approach is appliedaccording to (4.31) to the reduced system (C.12) or (C.13). Thus, using the classicalPOD-DEIM approach, the reduced system in abstract form finally results in

ζ

[

0 0

D21 sel 0

]

Φu

︸︷︷︸

D

ured + ζ

[

k1 sel(Φuured)

k2 sel(Φuured)

]

︸︷︷︸

k(Φuured)

= ΦTu

[

f1

f2

]

︸︷︷︸

f

, (C.14)


with the coefficient matrix

ζ = ΦTu Ψw (P TΨw)

−1P T . (C.15)

If, on the other hand, the modified approach is used, the snapshot matrix W , whichcontains the nonlinear snapshots wi(ui)i=1,...,w of all Newton steps, is also separatedinto smaller snapshot matrices WuS

and Wp (or rather WP). Following this, reductionmatrices ΨuS

and Ψp (or rather ΨP) are computed and summarised in the reductionmatrix Ψw = blkdiag [ΨuS

, Ψp]. The matrix P contains the information on the selectedmagic (or DEIM) points and can be allocated to matrices PuS

and Pp (or rather PP),containing the information on the selected DOF corresponding to the respective primaryvariable. Inserting these quantities in the POD-DEIM-reduced system (4.31), the follow-ing formulation of the global equilibrium equation can be found:

[

0 0

ζpD21 selΦuS0

]

︸︷︷︸

D

[

uuuS red

pppred

]

︸︷︷︸

ured

+

[

ζuSk1 sel(Φuured)

ζp k2 sel(Φuured)

]

︸︷︷︸

k(Φuured)

=

[

ΦTuSf1

ΦTp f2

]

︸︷︷︸

f

. (C.16)

Therein, the coefficient matrices are introduced as

ζuS= ΦT

uSΨuS

(P TuS

ΨuS)−1P T

uS,

ζp = ΦTp Ψp (P

Tp Ψp)

−1P Tp .

(C.17)

Even though the system matrix D can assumed to be (approximately) constant, thesnapshots wi of the composed nonlinear term w(u) = D u + k(u) are sampled. Thisis done due to the underlying implementation within the coupled FE solver PANDAS.However, the reduced system matrix D can be computed in the offline phase and canafterwards be used in the online phase to enable a significantly faster determination ofthe residual tangent J .

C.4 Reduced system of the different drug-infusion

models for brain tissue

C.4.1 Simplified brain-tissue model with linear equation system

Proceeding from the special case of geometrically and materially linear behaviour in com-bination with negligible body forces (b = 0) and a constant blood volume fraction, theequilibrium equation (3.33), containing as primary variables the solid displacement fielduS, the effective pore-liquid pressures pIR and pBR and the molar concentration cDm of thetherapeutic agent, can be formulated in form of a linear system assuming time-invariantsystem matrices D and K when simulating a diffusion-dominated spreading. Applyingthe modified POD approach on this coupled system, separate snapshot matrices UuS

,UpBR , UpIR and UcDm

are used to determine the subspace matrices ΦuS, ΦpBR, ΦpIR and

C.4 Reduced system of the different drug-infusion models for brain tissue 147

ΦcDm, which are summarised in the reduction matrix Φu = blkdiag[ΦuS

, ΦpBR, ΦpIR, ΦcDm].

Thus, the reduced system can be formulated in abstract form as

0 0 0 0

ΦTpBRD21ΦuS

0 0 0

ΦTpIRD31ΦuS

0 0 0

ΦTcDmD41ΦuS

0 0 ΦTcDmD44ΦcDm

uuuS red

pppBRred

pppIRred

cccDm red

+

+

ΦTuSK11ΦuS

ΦTuSK12ΦpBR ΦT

uSK13ΦpIR 0

0 ΦTpBRK22ΦpBR 0 0

0 0 ΦTpIRK33ΦpIR 0

0 ΦTcDmK42ΦpBR ΦT

cDmK43ΦpIR ΦT

cDmK44ΦcDm

uuuS red

pppBRred

pppIRred

cccDm red

=

ΦTuSf1

ΦTpBRf2

ΦTpIRf3

ΦTcDmf4

,

(C.18)finally resulting in

0 0 0 0

D21 0 0 0

D31 0 0 0

D41 0 0 D44

︸︷︷︸

D

uuuS red

pppBRred

pppIRred

cccDm red

︸︷︷︸

ured

+

K11 K12 K13 0

0 K22 0 0

0 0 K33 0

0 K42 K43 K44

︸︷︷︸

K

uuuS red

pppBRred

pppIRred

cccDm red

︸︷︷︸

ured

=

f1

f2

f3

f4

︸︷︷︸

f

.(C.19)

C.4.2 General brain-tissue model with nonlinear equationsystem

Reducing the global equilibrium equation (3.30), which describes the general brain-tissuemodel, it is crucial for an efficient reduction to account for nonlinearities. Therefore, thePOD-DEIM approach is applied on the nonlinear and strongly coupled system of equa-tions, leading to the reduced system (4.31). Again, the different physical time behaviour ofthe primary variables needs to be considered. Consequently, separated snapshot matricesUuS

, UpBR, UpIR and UcDm, containing the nodal values of the FE-grid for the different pri-

mary variables at each time step, are generated to compute separated reduction matricesΦuS

, ΦpBR , ΦpIR and ΦcDm, which are summarised in accordance to (4.13) in the reduc-

tion matrix Φu. Moreover, the snapshot matrix W , containing the nonlinear snapshotswii=1,...,w of all Newton steps, needs to be separated into smaller snapshot matricesWuS

,WpBR ,WpIR andWcDmfor each primary variable. Following this, reduction matrices

ΨuS, ΨpBR , ΨpIR and ΨcDm

are computed and summarised in the reduction matrix Ψw.The matrix P contains the information on the selected DOF and can be allocated tothe matrices PuS

, PpBR , PpIR and PcDm, containing the information on the selected DOF

corresponding to the respective primary variable. Inserting these quantities in the POD-DEIM-reduced system (4.31), the following abstract formulation of the global equilibrium


equation can be found:

0 0 0 0

D21 D22 D23 0

D31 D32 D33 0

D41 0 0 D44

︸︷︷︸

D(Φuured)

uS red

pBRred

pIRred

cDm red

︸︷︷︸

ured

+

ζuSk1 sel(Φuured)

ζpBR k2 sel(Φuured)

ζpIR k3 sel(Φuured)

ζcDm k4 sel(Φuured)

︸︷︷︸

k(Φuured)

=

ΦTuSf1

ΦTpBR f2

ΦTpIR f3

ΦTcDmf4

︸︷︷︸

f

, (C.20)

with the coefficient matrices

ζuS= ΦT

uSΨuS

(P TuS

ΨuS)−1P T

uS,

ζpBR = ΦTpBR ΨpBR (P T

pBR ΨpBR)−1P TpBR ,

ζpIR = ΦTpIR ΨpIR (P T

pIR ΨpIR)−1P T

pIR ,

ζcDm = ΦTcDm

ΨcDm(P T

cDmΨcDm

)−1P TcDm.

(C.21)

Moreover, the entries of the reduced system matrix D take the form:

D21 = ζpBRD21 sel ΦuS, D22 = ζpBRD22 sel ΦpBR ,

D23 = ζpBRD23 sel ΦpIR , D31 = ζpIRD31 selΦuS,

D32 = ζpIRD32 selΦpBR , D33 = ζpIRD33 selΦpIR ,

D41 = ζcDmD41 sel ΦuS, D44 = ζcDmD44 sel ΦcDm

.

(C.22)

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List of Figures

2.1 Motion of a biphasic material . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 REV of a porous microstructure and homogenisation . . . . . . . . . . . . 17


2.4 Axial cut through an IVD and schematic illustration . . . . . . . . . . . . 23


3.1 Spatial discretisation of a continuous domain . . . . . . . . . . . . . . . . . 26

3.2 Extended Taylor-Hood elements . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Overview of projection-based model-order-reduction methods . . . . . . . . 43

4.2 Visualistation of the selected magic points . . . . . . . . . . . . . . . . . . 59

5.1 Example 1: geometry, loading and boundary conditions and FE mesh . . . 65

5.2 Example 1: normalised error (POD-reduced system) . . . . . . . . . . . . . 66

5.3 Example 1: normalised error (POD-reduced system, high calculation accu-racy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Example 1: normalised error (modified POD-reduced system) . . . . . . . 68

5.5 Example 1: normalised error (modified POD-reduced system, high calcu-lation accuracy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.6 Example 1: normalised error (classical/modified POD-reduced system) . . 69

5.7 Example 1: solid displacement and pore pressure at point A using thefull/reduced system (classical POD method) . . . . . . . . . . . . . . . . . 70

5.8 Example 1: solid displacement and pore pressure at point A using thefull/reduced system (modified POD method) . . . . . . . . . . . . . . . . . 70


5.10 Example 2: values of the primary variables at point A using the full/POD-reduced system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.11 Example 2: values of the primary variables and deformed grid during thesimulation using the full/reduced system . . . . . . . . . . . . . . . . . . . 73

5.12 Example 3: geometry, loading and boundary conditions and FE mesh . . . 74


5.14 Example 3: solid displacement in the full/reduced system (test simulations) 77

5.15 Example 3: time behaviour of the relative 2-norms of the estimated andthe true errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.16 Example 3: time behaviour of the G (semi)norms of the estimated and thetrue errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

161

162 List of Figures

5.17 Example 4: geometry, loading and boundary conditions . . . . . . . . . . . 82

5.18 Example 4: solid displacement and pore pressure at point A using thefull/reduced system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.19 Example 4: normalised error and computation time to solve the equationsystem (classical/modified POD-reduced system) . . . . . . . . . . . . . . 84

5.20 Example 4: normalised error and computation time to solve the equationsystem (classical/modified POD-DEIM-reduced system) . . . . . . . . . . . 85

5.21 Example 4: values of the primary variables and deformed grid during thesimulation using the full/reduced system . . . . . . . . . . . . . . . . . . . 85

5.22 Example 4: solid displacement and pore pressure at point A for differentelastic material constants using the full/reduced system (pre-computationswith 1 parameter set) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.23 Example 4: solid displacement and pore pressure at point A for differentelastic material constants using the full/reduced system (pre-computationswith 3 different parameter sets) . . . . . . . . . . . . . . . . . . . . . . . . 87

5.24 Example 5: rectangular geometry and mesh of the initial-boundary-valueproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.25 Example 5: normalised concentration of the therapeutic agent at point 1,obtained from the full/reduced system . . . . . . . . . . . . . . . . . . . . 91

5.26 Example 5: normalised error between the full and the reduced simulation . 91

5.27 Example 5: therapeutical spreading at the end of an infusion process . . . 92

5.28 Example 5: molar concentration at points 1 and 2 for a variation in theapplied initial values, obtained from the full/reduced system . . . . . . . . 93

5.29 Example 5: effective pressure at points 1 and 2 for a variation in the appliedsolution influx, obtained from the full/reduced system . . . . . . . . . . . . 93

5.30 Example 5: molar concentration at points 1 and 2 for a variation in theapplied solution influx, obtained from the full/reduced system . . . . . . . 94

5.31 Example 6: rectangular geometry and mesh of the IBVP and evaluatedpoints within the domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.32 Example 6: evaluated DEIM points within the domain . . . . . . . . . . . 96

5.33 Example 6: values of the primary variables at selected points obtained fromthe full/reduced system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.34 Example 6: therapeutical spreading and stress at the end of an infusionprocess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.35 Example 6: values of different primary variables at selected points 1-3 fora variation in the material parameters . . . . . . . . . . . . . . . . . . . . 98

5.36 Example 7: geometry and mesh of the initial-boundary-value problem . . . 100

5.37 Example 7: parameterisation of the Dirichlet boundary conditions . . . . 101

5.38 Example 7: viscoelastic response of the IVD model and corresponding limits103

List of Figures 163

5.39 Example 7: stress and deformed grid at the end of different deformationprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.40 Example 7: integral behaviour of the cylindrical IVD . . . . . . . . . . . . 105

5.41 Example 7: integral behaviour of the cylindrical IVD for simultaneouslyapplied deformation states . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.42 Example 7: integral behaviour of the cylindrical IVD for an application ofu2 within accordingly adjusted time periods . . . . . . . . . . . . . . . . . 111

5.43 Example 7: integral behaviour of the cylindrical IVD for a simultaneousapplication of ϕ1, u2 and u3 within accordingly adjusted time periods . . . 113

5.44 Generalised approach for an application-driven model reduction . . . . . . 115

List of Tables

5.1 Example 1: material parameters of the biphasic porous-soil model . . . . . 65

5.2 Example 2: computing time for the simulation in the full and in the reducedsystem with 25 POD modes . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 Example 5: material parameters of the multi-component brain-tissue model 90

5.4 Example 5: computing time for the simulation in the full and in the reducedsystem with 25 POD modes . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5 Example 6: computing time for the simulation in the full and in the POD-DEIM-reduced system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6 Example 6: computing time for different numbers of performed simulations 99

5.7 Example 7: limiting deformation states . . . . . . . . . . . . . . . . . . . . 102

5.8 Example 7: computing time for the performed test simulations . . . . . . . 106

5.9 Example 7: characteristic values of the different sampling strategies whensimulating the purely elastic behaviour of an IVD . . . . . . . . . . . . . . 107

5.10 Example 7: factors influencing the total computing time of the offline phase 108

5.11 Example 7: characteristic values of the different sampling strategies whensimulating the viscoelastic behaviour of an IVD with a seperate applicationof u2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

165

Curriculum Vitae

Personal data:

Name: Davina Fink, nee Otto

Date of birth: September 23, 1987

Place of birth: Kassel, Germany

Nationality: German

Parents: Iris and Joachim Otto

Siblings: Jana Tomasevic and Ronja Otto

Marital status: married to Jury Fink

Children: Jonathan and Maximilian Fink

Education:

09/1993 – 07/1998 elementary school “Grundschule Gudensberg”,

09/1998 – 07/2001 secondary school “Gymnasium Ursulinenschule Fritzlar”,

09/2001 – 06/2007 secondary school “Ludwig-Uhland-Gymnasium”,

Kirchheim unter Teck, Germany

06/2007 degree: “Allgemeine Hochschulreife” (high school diploma)

10/2007 – 04/2012 studies in civil engineering at the

University of Stuttgart, Germany

major subjects: “Modellierungs- und Simulationsmethoden”,

“Konstruktiver Ingenieurbau”

11/2008 – 04/2012 student teaching assistant at the

Institute of Applied Mechanics (Civil Engineering) at the


04/2012 degree: “Diplom-Ingenieur (Dipl.-Ing.) Bauingenieurwesen”

Professional occupation:

05/2012 – 02/2019 teaching assistant and research associate at the

Institute of Applied Mechanics (Civil Engineering) at the


Presently published contributions in this report series

II-1 Gernot Eipper: Theorie und Numerik finiter elastischer Deformationen in fluid-gesattigten porosen Festkorpern, 1998.

II-2 Wolfram Volk: Untersuchung des Lokalisierungsverhaltens mikropolarer poroser Me-dien mit Hilfe der Cosserat-Theorie, 1999.

II-3 Peter Ellsiepen: Zeit- und ortsadaptive Verfahren angewandt auf Mehrphasenproblemeporoser Medien, 1999.

II-4 Stefan Diebels: Mikropolare Zweiphasenmodelle: Formulierung auf der Basis derTheorie Poroser Medien, 2000.

II-5 Dirk Mahnkopf: Lokalisierung fluidgesattigter poroser Festkorper bei finiten elasto-plastischen Deformationen, 2000.

II-6 Heiner Mullerschon: Spannungs-Verformungsverhalten granularer Materialien amBeispiel von Berliner Sand, 2000.

II-7 Stefan Diebels (Ed.): Zur Beschreibung komplexen Materialverhaltens: Beitrageanlaßlich des 50. Geburtstages von Herrn Prof. Dr.-Ing. Wolfgang Ehlers, 2001.

II-8 Jack Widjajakusuma: Quantitative Prediction of Effective Material Parameters ofHeterogeneous Materials, 2002.

II-9 Alexander Droste: Beschreibung und Anwendung eines elastisch-plastischen Ma-terialmodells mit Schadigung fur hochporose Metallschaume, 2002.

II-10 Peter Blome: Ein Mehrphasen-Stoffmodell fur Boden mit Ubergang auf Interface-Gesetze, 2003.

II-11 Martin Ammann: Parallel Finite Element Simulations of Localization Phenomenain Porous Media, 2005.

II-12 Bernd Markert: Porous Media Viscoelasticity with Application to Polymeric Foams,2005.

II-13 Saeed Reza Ghadiani: A Multiphasic Continuum Mechanical Model for Design In-vestigations of an Effusion-Cooled Rocket Thrust Chamber, 2005.

II-14 Wolfgang Ehlers & Bernd Markert (Eds.): Proceedings of the 1st GAMM Seminaron Continuum Biomechanics, 2005.

II-15 Bernd Scholz: Application of a Micropolar Model to the Localization Phenomena inGranular Materials: General Model, Sensitivity Analysis and Parameter Optimiza-tion, 2007.

II-16 Wolfgang Ehlers & Nils Karajan (Eds.): Proceedings of the 2nd GAMM Seminar onContinuum Biomechanics, 2007.

II-17 Tobias Graf: Multiphasic Flow Processes in Deformable Porous Media under Con-sideration of Fluid Phase Transitions, 2008.

II-18 Ayhan Acarturk: Simulation of Charged Hydrated Porous Materials, 2009.

II-19 Nils Karajan: An Extended Biphasic Description of the Inhomogeneous and AnisotropicIntervertebral Disc, 2009.

II-20 Bernd Markert: Weak or Strong – On Coupled Problems In Continuum Mechanics,2010.

II-21 Wolfgang Ehlers & Bernd Markert (Eds.): Proceedings of the 3rd GAMM Seminaron Continuum Biomechanics, 2012.

II-22 Wolfgang Ehlers: Porose Medien – ein kontinuumsmechanisches Modell auf derBasis der Mischungstheorie, 2012. Nachdruck der Habilitationsschrift aus dem Jahr1989 (Forschungsberichte aus dem Fachbereich Bauwesen der Universitat-GH-Essen47, Essen 1989).

II-23 Hans-Uwe Rempler: Damage in multi-phasic Materials Computed with the ExtendedFinite-Element Method, 2012.

II-24 Irina Komarova: Carbon-Dioxide Storage in the Subsurface: A Fully Coupled Anal-ysis of Transport Phenomena and Solid Deformation, 2012.

II-25 Yousef Heider: Saturated Porous Media Dynamics with Application to EarthquakeEngineering, 2012.

II-26 Okan Avci: Coupled Deformation and Flow Processes of Partial Saturated Soil:Experiments, Model Validation and Numerical Investigations, 2013.

II-27 Arndt Wagner: Extended Modelling of the Multiphasic Human Brain Tissue withApplication to Drug-Infusion Processes, 2014.

II-28 Joffrey Mabuma: Multi-Field Modelling and Simulation of the Human Hip Joint,2014.

II-29 Robert Krause: Growth, Modelling and Remodelling of Biological Tissue, 2014.

II-30 Seyedmohammad Zinatbakhsh: Coupled Problems in the Mechanics of Multi-Physicsand Multi-Phase Materials, 2015.

II-31 David Koch: Thermomechanical Modelling of Non-isothermal Porous Materials withApplication to Enhanced Geothermal Systems, 2016.

II-32 Maik Schenke: Parallel Simulation of Volume-coupled Multi-field Problems with Spe-cial Application to Soil Dynamics, 2017.

II-33 Steffen Mauthe: Variational Multiphysics Modeling of Diffusion in Elastic Solidsand Hydraulic Fracturing in Porous Media, 2017.

II-34 Kai Haberle: Fluid-Phase Transitions in a Multiphasic Model of CO2 Sequestra-tion into Deep Aquifers: A fully coupled analysis of transport phenomena and soliddeformation, 2017.

II-35 Chenyi Luo: A Phase-field Model Embedded in the Theory of Porous Media withApplication to Hydraulic Fracturing, 2019.

II-36 Sami Bidier: From Particle Mechanics to Micromophic Continua, 2019.

II-37 Davina Fink: Model Reduction applied to Finite-Element Techniques for the Solutionof Porous-Media Problems, 2019.

dissertation_davina_fink.pdf - Universität Stuttgart

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