Module Linear Structural Computational Mechanics for Wind Energy Systems i Lecture Notes Linear Computational Structural Mechanics for Wind Energy Systems Prof. Dr.-Ing. habil. Detlef Kuhl Online M.Sc. Wind Energy Systems University of Kassel and Fraunhofer IWES www.uni-kassel.de/wes
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Module Linear Structural Computational Mechanics for Wind Energy Systems i
Lecture Notes
Linear Computational Structural Mechanics for Wind Energy Systems
Prof. Dr.-Ing. habil. Detlef Kuhl
Unive
Online M.Sc. Wind Energy Systems University of Kassel and Fraunhofer IWES
Table 0.1: How this Module fits into the Online M.Sc. Wind Energy Systems
Table 0.1 shows the present module Linear Computational Structural Mechanics for Wind Energy
Systems embedded in the specialist studies Simulation and Structural Technology for Wind Ene r-
gy Systems and the master’s course Wind Energy Systems. The present lecture is based on the
knowledge of the modules of Fundamental Studies of Mathematics and Engineering. In particu-
lar, very good knowledge of Mathematics for Wind Energy Systems, Design of Mechanical and
Electrical Components of Wind Energy Systems and Practice of Software Tools for Wind Ene rgy
Systems is essential for the successful graduation of the present module. Since in the present
module almost all continuum and structural mechanical problems, previously presented in mod-
ule Solid Mechanics for Wind Energy Systems, are solved numerically, it is quite important to
understand the topics of this fundamental module. The present module is extended to the nu-
Module Linear Structural Computational Mechanics for Wind Energy Systems 5
merical analysis of non-linear static and dynamic problems in module Non-Linear Computational
Structural Mechanics for Wind Energy Systems (NCSM) and to the valuation of strength, failure,
low and high cycle fatigue in lecture Strength Durability and Reliability for Wind Energy Systems.
The present module can be combined with the fluid mechanics modules of specialist studies
Simulation and Structural Technology for Wind Energy Systems in order to obtain the knowledge
to overcome traditional borders between solid and fluid mechanics with study of both and finally
with the analysis of fluid structure interaction. Furthermore, it can be combined with the tech-
nology modules of the specialist study in order to use numerical analysis of towers, foundations
and rotor blades to improve or optimize these components of wind turbines.
Learning Schedule (Beispiel) The simulation of wind turbines under real operating conditions enforces the consideration of
time dependent loads and inertial forces. These simulations are performed by applying time
integration schemes. Since these schemes are requiring a large numerical effort and significantly
influencing the quality of the prognosis of the dynamic behavior of structures, it is worth to care-
fully develop these methods in Chapters 6 to 8 and to enrich the basic time integrations schemes
by error measures and adaptive time stepping procedures. Methodologically oriented we will
review continuum mechanics and we will discuss the dynamic characteristic and analytical sol u-
tion of structural dynamics.
Basics Static
analysis
Spatial
discreti-
zation
Dynamic
analysis
Tem-
poral
discreti-
zation
Chapter 1, page 1: Introduction to Linear Com-
putational Structural Mechanics
Chapter 2, page 25: Finite Element Method for
One Dimensional Continua
Chapter 3, page 35: Advanced Topics and Spa-
tial Truss Structures
Chapter 4, page 7: Generalized Finite Element
Method for n-Dimensional Continua
Chapter 5, page 13: Dynamic Characteristics
and Analytical Solution of Dynamics
Chapter 6, page 17: Central Difference Method
Chapter 7, page 21: Newmark Time Integration
Schemes
Chapter 8, page 25: Galerkin Time Integration
Schemes
Figure 1: Learning Schedule of Linear Computational Structural Mechanics for Wind Energy Systems
Afterwards, as main tasks of the present lecture, methods for the numerical solution of statics
and dynamics are presented. In particular, the spatial and temporal discretization methods are
6 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
thought and intensively studied by means of analytical analyses and representative and illustra-
tive examples. The simulation of wind turbines under real operation condition enforces the con-
sideration of time dependent loads and inertial forces. These simulations are performed by ap-
plying time integration schemes. Since these schemes are requiring a large numerical effort and
significantly influencing the quality of the prognosis of the dynamic behavior of structures, it is
worth to carefully develop these methods in Chapters 6 to 8 and to enrich the basic time inte-
grations schemes by error measures and adaptive time stepping procedures. Methodologically
oriented we will review continuum mechanics and we will discuss the dynamic characteristic and
analytical solution of structural dynamics. Afterwards, as main tasks of the present lecture,
methods for the numerical solution of statics and dynamics are presented. In particular, the spa-
tial and temporal discretization methods are thought and intensively studied by means of analyt-
ical analyses and representative and illustrative examples.
Module Linear Structural Computational Mechanics for Wind Energy Systems 7
1 Finite Element Method for One
Dimensional Continua and Truss
Elements
Abstract In the present chapter ... Kurze Zusammenfassung des Kapitels.
Jedes Kapitel beginnt mit einem Deckblatt, auf welchem die Kapitelüberschrift, eine kurze
Zusammenfassung des Kapitels sowie die Key Words zu finden sind.
Key Words linear elasticity, finite element method, history of mechanics
8 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
1.1 Learning Goals
reviewing linear continuum mechanics
knowing different, also non-linear, models of continuum mechanics
having a idea of numerical methods applied for the solution of continuum mechanical
models
having fun with the histories of the finite element method and the time integration
schemes
1.2 Required Prior Knowledge (Empfehlung) Welche Voraussetzungen müssen erfüllt sein, um dieses Kapitel zu verstehen.
1.3 Section 2 Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d-
unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo
duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
1.3.1 Section 3 Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d-
unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo
duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
Example for citation:
"Fluid ows at and below the earth's surface are the cause and the cure for problems of water
and soil pollution" (Wendland & Efendiev, 2003, S. 37).
Section 4 The section 4 will not be consecutively numbered.
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d-
unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo
duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua1. At vero eos et
1 This is a footnote.
Module Linear Structural Computational Mechanics for Wind Energy Systems 9
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
Example for citation:
"Fluid ows at and below the earth's surface are the cause and the cure for problems of water
and soil pollution" (Wendland & Efendiev, 2003, S. 37).
1.3.2 Formula
(𝒙 + 𝒂)𝒏 = ∑ (𝒏𝒌
)𝒙𝒌𝒂𝒏−𝒌𝒏
𝒌=𝟎 (1.1)
(𝟏 + 𝒙)𝒏 = 𝟏 +𝒏𝒙
𝟏!+
𝒏(𝒏−𝟏)𝒙𝟐
𝟐!+ ⋯ (1.2)
𝒙 =−𝒃±√𝒃𝟐−𝟒𝒂𝒄
𝟐𝒂 (1.3)
1.3.3 Essenz (Empfehlung)
Chapter Checks 1. (Question/Task 1 of the paragraph 1.1)
2. (Question/Task 1 of the paragraph 1.1)
3. (Question/Task 1 of the paragraph 1.1)
Special texts like examples, excursions or tips are framed in a box: At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet.
Enumeration 1. level - Enumeration 2. level
* Enumeration 3. level
Memotechnic verse: Field shaded in gray to give short (!) memos or advices.
10 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Figure 1.1: Wind power Plant
Nomenclature
Symbol Equivalent Uni Explanation
T s Time
𝚯 Κ temperature
𝚾𝟏 m position
𝝊𝟏 m displacement
Table 1.1: Nomenclature for one dimensional linear continuum mechanics and linear truss me-
chanics
References Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Applications.
Berlin: Springer.
Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multield Problems. Berlin:
Springer.
Verschiedene Darstellungsweisen möglich!
[1] M. Ameen. Computational Elasticity. Theory of Elasticity and Finite and Boundary Element
[2] T. L. Anderson. Fracture Mechnics. Fundamentals and Applications. Taylor & Francis Group,
Broken, 3. edition, 2005.
Module Linear Structural Computational Mechanics for Wind Energy Systems 11
[3] Archimedes. De planorum aequilibriis. 285-212 v.Chr.
[4] J. Argyris. Dynamics of Structure. Elsevier, Amsterdam, 1991.
[5] V. I. Arnold. Lectures on Partial Differential Equations. Springer & Phasis, Berlin & Moscow,
2004.
[6] G. Galilei. Discorsi e dimostrazioni matematiche intorno a due nuove scienze. Leiden, 1638.
Homework (Möglichkeit)
Hausaufgaben können auch in Moodle oder in anderer Form den
Studierenden zur Verfügung gestellt werden.
Figure 1.2: Tension of a truss: Geometry and loading cases
In the present homework your own finite element program for the static analysis of one dime n-
sional continua should be extended in order to allow for the application of the p finite element
method. Therefore, higher order (𝜌 = 1; 2; 3; 4; 5; 6), one dimensional continuum elements
should be applied together with the Gauss-Legendre integration. The correct implementation of
the finite element and finite element procedure on the structural level should be verifie d by
means of above sketched model problems. These examples are described by a truss loaded by
load cases i, ii and iii. They should be analyzed using ΝΕ = 1; 2; 4; 8; 16; 32 p finite elements for
the discretization of the truss. For these reasons the following working stages are proposed:
Develop a finite element routine for calculation of the element stiffness 'tensors' 𝑘𝑒𝑖𝑗
and the consistent load 'tensors' 𝑟𝑒𝑖 for all load cases using the Gauss-Legendre integra-
tion with GAUSS point coordinates and weights as given in the file gauss.f provided in the
Moodle course.
Chose the number of GAUSS points 𝑁𝐺 such that the stiffness tensors and the load ten-
sors for load cases i and ii are exactly integrated. The load tensors according to load case
iii cannot integrated exactly. For these integrations please use a integration rule
with 𝑁𝐺 = 𝑝 + | 5.
Develop finite element procedure for analyses with 𝑁𝐸 = 1; 2; 4; 8; 16; 32 finite el-
ements of polynomial degrees 𝑝 = 1; 2; 3; 4; 5; 6 and check your solutions for all
load cases.
Perform all forthcoming tasks only for load case iii, but for all implemented polynomial
degrees p.
12 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Extend your p finite element program by a post-processing procedure, calculating the
approximations of the displacement 𝑢1, stress 𝜎11 and residuum 𝜎11,1 + 𝑝𝑏1
Calculate the local (at position X1) and global (of the hole system) displacement errors
with respect to the analytical solution.
Plot diagrams of the displacements, stresses, the residuum and the local displacement
error.
Your homework submission should include
a brief report documenting your results in form of diagrams
your program code
Module Linear Structural Computational Mechanics for Wind Energy Systems 13
2 Finite Element Method for One Di-
mensional Continua and Truss Ele-
ments
Abstract In the present chapter ... Kurze Zusammenfassung des Kapitels
Key Words linear elasticity, finite element method, history of mechanics
14 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
2.1 Introduction
2.1.1 Learning goals
reviewing linear continuum mechanics
knowing different, also non-linear, models of continuum mechanics
having a idea of numerical methods applied for the solution of continuum mechanical
models
having fun with the histories of the finite element method and the time integration
schemes
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d-
unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo
duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
2.1.2 Section 3 Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invi d-
unt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo
duo dolores et ea rebum. Stet clita kasd gubergren, no sea taki mata sanctus est Lorem ipsum
dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod
tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est
Lorem ipsum dolor sit amet.
Example for citation:
"Fluid ows at and below the earth's surface are the cause and the cure for problems of water
and soil pollution" (Wendland & Efendiev, 2003, S. 37).
2.1.3 Sections 3
Memotechnic verse: Field shaded in gray to give short (!) memos or advices.
Special texts like examples, excursions or tips could be framed in a box: At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet.
Enumeration 1. level - Enumeration 2. level
* Enumeration 3. level
Module Linear Structural Computational Mechanics for Wind Energy Systems 15
2.1.4 Formula
(𝒙 + 𝒂)𝒏 = ∑ (𝒏𝒌
)𝒙𝒌𝒂𝒏−𝒌𝒏
𝒌=𝟎 (2.1)
(𝟏 + 𝒙)𝒏 = 𝟏 +𝒏𝒙
𝟏!+
𝒏(𝒏−𝟏)𝒙𝟐
𝟐!+ ⋯ (2.2)
𝒙 =−𝒃±√𝒃𝟐−𝟒𝒂𝒄
𝟐𝒂 (2.3)
Chapter Checks 1. (Question/Task 1 of the paragraph 1.1)
2. (Question/Task 1 of the paragraph 1.1)
3. (Question/Task 1 of the paragraph 1.1)
Figure 2.1: Wind power Plant
2.2 Essenz Example for citation:
"Fluid ows at and below the earth's surface are the cause and the cure for problems of water
and soil pollution" (Wendland & Efendiev, 2003, S. 37).
References Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Appli-
cations. Berlin: Springer.
Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multield Problems.
Berlin: Springer.
16 D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel
Homework (Möglichkeit)
Hausaufgaben können auch in Moodle oder in anderer Form den
Studierenden zur Verfügung gestellt werden.
Module Linear Structural Computational Mechanics for Wind Energy Systems 17
Bibliography (Möglichkeit)
Beinhaltet die gesamte Literatur im Text. Die Literatur sollte jedoch in
jedem Kapitel aufgelistet sein. Die Gesamtdarstellung stellt ein Zusatz dar.
Capital 1
Ehlers, W. & Bluhm, J. (2002). Porous Media. Theory, Experiments and Numerical Applications .
Berlin: Springer.
Wendland, W. L. & Efendiev, M. (2003). Analysis and Simulation of Multifield Problems. Berlin:
Springer.
Verschiedene Darstellungsmöglichkeiten. Diese müssen einheitlich im Dokument sein!
Capital 1
[1] M. Ameen. Computational Elasticity. Theory of Elasticity and Finite and Boundary Element
[2] T. L. Anderson. Fracture Mechnics. Fundamentals and Applications. Taylor & Francis Group,
Broken, 3. edition, 2005.
[3] Archimedes. De planorum aequilibriis. 285-212 v.Chr.
Module Linear Structural Computational Mechanics for Wind Energy Systems 18
Appendix
Glossary (Möglichkeit)
Der Glossary stellt ein Zusatz dar.
Actuator
Actuator is a device to convert an electrical control signal to a physical action. Actuators may be
used for flow-control valves, pumps, positioning drives, motors, switches, relays and meters.
Floating-Point Operations Per Second (FLOPS)
Floating-Point Operations Per Second (FLOPS) is a measurement of performance of capability
assigned to a floating-point processor. It is usually noted as MFLOPS or Million FLOPS.
Local Area Network
A Local Area Network is a group of interconnected devices that share common processing and
file management resources, usually within a specific physical area. An example would be an of-
fice computer network.
Resolution
Resolution is a measure of accuracy or dynamic range of an A/D or D/A converter.
D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel 19
Index
(Möglichkeit)
Der Index stellt ein Zusatz dar.
Actuator 8
Formula 5
Internet adress 7
Questions/Tasks 2, 5
D. Kuhl, Online M.Sc. Wind Energy Systems, University of Kassel 20
Nomenclature (Möglichkeit)
Beinhaltet die gesamte Nomenklatur aller Kapitel. Diese sollte jedoch in
jedem Kapitel aufgelistet sein. Die Gesamtdarstellung stellt ein Zusatz dar.
𝝂 Poisson ratio
𝝂 Poisson ratio
𝝈𝟏𝟏 normal stress / normal stress component in direction 𝑒1
𝝈𝟏𝟏 normal stress / normal stress component in direction 𝑒1
𝝈𝟏𝟏 normal stress / normal stress component in direction 𝑒1
𝜺𝟏𝟏 normal stress / normal strain component in direction 𝑒1
𝜺𝟏𝟏 normal stress / normal strain component in direction 𝑒1
Module Linear Structural Computational Mechanics for Wind Energy Systems 21
Online M.Sc. Wind Energy Systems www.uni-kassel.de/wes University of Kassel and Fraunhofer IWES
Lecture Notes
Linear Computational Structural Mechanics for Wind Energy Systems Prof. Dr.-Ing. habil. Detlef Kuhl These lecture notes are designed to assist students of the online master’s study wind energy systems with their learning process in linear finite ele-ment methods and linear structural dynamics of wind energy systems. For this reason it includes various elements: the theoretical development, ap-plication of methods in selected examples and program flowcharts, as well as coding instructions supporting the homework and the final case study of the course.