2. Wang tiling conceptlibrary.utia.cas.cz/separaty/2019/MTR/zeman-0507081.pdfvol. 13/2017 Synthesizedmicrostructure-informedenrichmentsforXFEManalyses θˆ= 20 10 t x 10 t y ˆθ=

doi:10.14311/APP.2017.13.0029Acta Polytechnica CTU Proceedings 13:29–34, 2017 © Czech Technical University in Prague, 2017

available online at http://ojs.cvut.cz/ojs/index.php/app

SYNTHESIZED ENRICHMENT FUNCTIONS FOR EXTENDEDFINITE ELEMENT ANALYSES WITH FULLY RESOLVED

MICROSTRUCTURE

Martin Doškářa,∗, Jan Nováka, Jan Zemana,b

a Faculty of Civil Engineering, Czech Technical University in Prague, Thákurova 7/2077, 166 29 Prague 6, CzechRepublic

b The Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic,Pod Vodárenskou věží 4, 182 08 Prague 8, Czech Republic

∗ corresponding author: [email protected]

Abstract. Inspired by the first order numerical homogenization, we present a method for extractingcontinuous fluctuation fields from the Wang tile based compression of a material microstructure. Thefluctuation fields are then used as enrichment basis in Extended Finite Element Method (XFEM) toreduce number of unknowns in problems with fully resolved microstructural geometry synthesized bymeans of the tiling concept. In addition, the XFEM basis functions are taken as reduced modes ofa detailed discretization in order to circumvent the need for non-standard numerical quadratures. Themethodology is illustrated with a scalar steady-state problem.

Keywords: Wang tiling, microstructure synthesis, microstructure-informed enrichment functions,extended finite element method.

1. IntroductionAll materials exhibit heterogeneous microstructure atcertain level of detail. Besides the physical propertiesof microstructural constituents, the overall responseis also dictated by their spatial arrangement. Withthe advance of numerical methods and computationalpower, there is an emphasis on devising approachesthat take the microstructural characteristics into ac-count and propagate knowledge of material composi-tion into upper-scale models.

In our contribution, we focus on problems in whichthe underlying microstructural geometry is explicitlyknown. We consider problems where the character-istic length of the microstructure is comparable tothe dimension of an upper-scale model; hence, thenested numerical homogenization, such as the FE2method [1], is not applicable because the underlyingassumption of the separation of scales is violated [2,and references therein].

This work supplements our previous results regard-ing efficient modelling of materials with stochasticheterogeneous microstructure, e.g., [3, 4]. The mi-crostructural geometry of a macro-scale model—aswell as its finite element discretization—is synthesizedusing the Wang tiling concept, recalled in Section 2.Here, we present a method to extract dominant re-sponses of a compressed microstructure to prescribedmacroscopic loading. We reuse the extracted responsesto synthesize a microstructure-informed enrichmentbasis in order to accelerate the upper scale calculations.The numerical scheme, outlined in Section 4, combineseXtended Finite Element Framework (XFEM) [5, 6,and references therein], used for definition of local

basis functions, with the reduced order modellingapproach [7], allowing for utilization of standard fi-nite element procedures. Finally, Section 5 illustratesthe methodology with a two-dimensional problem ofsteady-state thermal conduction in an L-shaped do-main.

2. Wang tiling conceptMicrostructure of random heterogeneous materialsis usually represented by means of a periodic finite-size model—appearing in the literature under variousnames such as Statistically Optimal RepresentativeUnit Cell, Statistically Similar Representative VolumeElement, or Statistically Equivalent Periodic Unit Cell(SEPUC)—that resembles the material in terms ofvarious spatial statistics. This setting is particularlysuitable for the family of FE2 simulations,e.g., [1], inwhich it is often accompanied with periodic boundaryconditions imposed at the micro-scale level. How-ever, when used for microstructure synthesis, this rep-resentation introduces unrealistic, spurious periodiccorrelations.The formalism of Wang tiles represents a compro-

mise between the amount of induced correlation andcomplexity of a microstructure representation. In-stead of a single cell, microstructural information iscompressed into a set of equi-sized Wang tiles withpre-defined requirements on mutual compatibility. Inthe abstract setting, the compatibility constraints arerepresented by codes assigned to tile edges, illustratedwith distinct colours in Figure 1. A realization of thecompressed microstructure is then synthesized follow-ing a simple stochastic algorithm: an empty grid of

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M. Doškář, J. Novák, J. Zeman Acta Polytechnica CTU Proceedings

a desired size is sequentially filled with instances ofWang tiles such that the adjacent tile edges are com-patible. At each position, a tile is randomly chosenfrom a subset of tiles compliant with previously placedtiles; hence, the algorithm generates stochastic real-izations, providing that set of tiles is rich enough suchthat multiple tiles are admissible at each assemblystep; see Figure 1 for an illustration.

Figure 1. The compressed microstructural geome-try represented with the set of Wang tiles (top) andan illustration of a step in the tiling algorithm withpartially assembled microstructure (bottom).

Procedures designed to compress a given microstruc-ture into a SEPUC can be straightforwardly extendedto take into account generalized periodicity occurringin the tiling concept, e.g., the standard optimizationprocedure based on minimizing discrepancy in thetwo-point probability function was used in [3]. Wealso adapted a sample-based approach originated inComputer Graphics in order to address the high com-putational cost of the optimization-based design [4].Currently, a variety of material microstructures, rang-ing from particulate media to complex foam-like mi-crostructures, can be represented with the frameworkof Wang tiling. Individual realizations of the mi-crostructure are then generated in linear time withrespect to the required sample size, which makes thetiling concept appealing for applications where multi-ple microstructure samples are required, e.g., investi-gation of the RVE size [8].

3. Synthesized enrichments fieldsThe idea of compressing fluctuation fields using theWang tiling concept has been introduced in Novák etal. [9], where the microstructural geometry of individ-ual tiles was optimized with respect to (i) meetingprescribed spatial statistics and (ii) minimizing trac-tion discrepancies among congruent edges in an auxil-iary tiling. However, the resulting synthesized stressfields exhibited unavoidable discontinuities and theobjective function required careful tuning of weightparameters because the two requirements turned tobe hard to achieve simultaneously. In contrast, thepresent method generates continuous fields of theprimal unknowns for any existing microstructure com-pression.

We illustrate the methodology with the steady-stateheat conduction problem, governed by

∇ · (−K(x) · ∇θ(x)) = 0 , x ∈ Ω , (1)

where θ denotes the unknown temperature field, K isa local heat conductivity tensor and no heat sourcesare considered.

First, we assume the decomposition of θ into a fluc-tuation part θ̃, caused by the presence of hetero-geneities, and a macroscopic part controlled witha macroscopic temperature gradient H,

θ(x) = θ̃(x) + H · x , ∀x ∈ T(i),∀T(i) ∈ S, (2)

where T(i) denotes the domain of the i-th tile and S ={T(i), i = 1 . . . nT

}represent the set of nT available

tiles. Without loss of generality, each tile is assumedcentred, i.e., T(i) = [−α, α]2, where 2α is the size ofthe tile edge.

In order to enforce continuity of the extracted fields,all tiles are loaded with the prescribed macroscopicgradient H and solved simultaneously. Providing thatthe tile geometries are discretized with a compatibletriangulation—meaning that discretization of the cor-responding tile edges is the same across all tiles—thecontinuity is ensured by associating the correspondingedge unknowns during the localization of an elementcontribution into the conductivity matrix of the wholeset. The resulting system takes the standard form

KS θ̃ = F (H) , (3)

where KS = AT(i)∈S KT(i) is the conductivity matrixand F (H) is a H-dependent loading vector arisingfrom the weak form after plugging Eq. (2) into Eq. (1).KS has block-diagonal structure similar to primaldomain decomposition approaches.In addition, two types of constraints are imposed

on the set system. First, we eliminate zero-energymodes of KS by requirement

1|S|

∫Sθ̃(x) dx = 0 . (4)

The second constraint prevents the fluctuation field θ̃from compensating for the loading induced through H .

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Inspired in numerical homogenization, e.g. [2], weintroduce the second constraint enforcing

1|Ω|

∫Ω∇θ̃(x) dx = 0 (5)

in a chosen domain Ω loaded analogously to Eq. (2).Using Green’s first identity, Eq. (5) can be recast intoa boundary integral∫

∂Ωθ̃(x)n(x) dS = 0 , (6)

where n(x) denotes the outer normal of the boundary.In particular, we consider three mutually exclusivetypes of the second constraint:(K) The first type prescribes zero temperature fluctu-ations at tile boundaries,

θ̃|∂T(i) = 0 , ∀i = 1 . . . nT , (7)

where ∂T(i) stands for the boundary of the i-th tile.(P) In the second type, the requirement (6) is posedtile-wise, i.e.,∫

∂T(i)θ̃(x)n(x) dS = 0 , ∀i = 1 . . . nT . (8)

(S) The last type imposes Eq. (6) over the whole set,∫∂Sθ̃(x)n(x) dS = 0 , (9)

where ∂S ={∂T(i), i = 1 . . . nT

}.

Note that (K) corresponds directly to Kinematic Uni-form Boundary Conditions used in numerical homog-enization, while (P) and (S) mimic Periodic and Stat-ically Uniform Boundary Conditions, respectively.Instead of Eq. (3), the final fluctuations θ̃ follows

from KS CTI CTIICI 0 0CII 0 0

θ̃λIλII

=F(H)00

, (10)where CI and CII matrices represent the discretizedversions of Eq. (4) and one of Eqs. (7), (8), or (9), re-spectively, and λ• stands for the corresponding vectorof Lagrange multipliers.While CII straightforwardly eliminates edge un-

knowns for the constraint type (K), constructing con-tributions of individual tile separately and localizingthem into CII leads to an ill-posed problem for types(P) and (S) From Eq. (6), it clearly follows that if twoopposite sides of a tile are required to be compatible—hence they feature the same trace of θ̃—the corre-sponding component of ∇θ̃ average vanishes and sodoes the tile contribution to the relevant row in CII.In addition, the ill-posedness also stems from the rep-etition of edge code pairs within the tile set (swappingthe order of codes changes only the sign of the edgecontribution to Eq. (5)). In certain cases, this caneven result in zero matrix CII for the least restrictiveconstraint (S), in which tile contributions are summedtogether.

One way to treat the singularity of CII is to performSingular Value Decomposition on the constraint andtake into account only the non-zero singular values;however, such a procedure is sensitive to numericalprecision. Thanks to the simple structure of the con-straint, the regular part of the constraint can be di-rectly established as a projection of the original matrixCII. First, all nc combinations of edge codes pertinentto either horizontal or vertical edges are identified andenumerated. Next, an empty matrix P ∈ R2nT ×nc , for(P) type of the constraint, or P ∈ R2×nc , for (S) type,is created and populated with 1 (or −1) if the tilecontains an enumerated pair of codes (or a pair in theinverse order). Odd and even rows of the matrix cor-respond to vertical and horizontal edges, respectively.If P remains empty for (S), CII constraint is discardedentirely. Otherwise, the non-singular constraint ma-trix CII, replacing CII in Eq. (10), is obtained froma projection CII = PTCII.The outlined strategy allows to define one enrich-

ment field θ̃ for a prescribed H and the selected type ofthe second constraint. For the linear problem consid-ered in this work, two mutually orthonormal loadingcases H =

{1, 0}T and H = {0, 1}T cover all possible

load cases. Combined with three types of CII, we cangenerate up to six enrichment fields. Note that un-like the nested numerical homogenization, where thechoice of boundary conditions is based on their suit-ability in capturing effects of a surrounding medium,we combine different types of CII to control cardinalityof the set of compressed fluctuations.

Figure 2. A compressed fluctuation field obtained byprescribing (S) variant of the second constraint andsetting H = {1, 0}T.

4. Numerical strategyThe fields introduced above are employed as enrich-ments in a upper-scale model whose microstructuralgeometry is assembled by means of tiling from the setdepicted in Figure 1.

Assume a upper-scale problem governed by the samedifferential equation as in Eq. (1), valid in a upper-scale domain Ω and accompanied by boundary condi-tions

θ(x) = θ̂(x) x ∈ Γθ 6= ∅ , (11)(−K(x)∇θ(x)) · n(x) = q̂(x) x ∈ Γq , (12)

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M. Doškář, J. Novák, J. Zeman Acta Polytechnica CTU Proceedings

where θ̂ and q̂ are the given temperature and normalheat flux profiles, respectively, prescribed at parts Γθand Γq of the domain boundary ∂Ω. An approximatenumerical solution arises from the weak form of theproblem

Find θ ∈ V such that a(θ, ϑ) = b(ϑ), ∀ϑ ∈ V0 . (13)

and substitutes for the analytical solution that is usu-ally intractable. In Eq. (13), a(θ, ϑ) and b(ϑ) denotethe bilinear and linear form pertinent to Eq. (1); Vand V0 stand for the space of admissible temperaturefields.

Quality of the approximate solution is governed bysuitability of a finite-dimensional subspace Vc ⊂ V inwhich the solution is sought for. In particular, domaindiscretization must accurately resolve all geometricaldetails in the standard Finite Element (FE) setting.This requirement leads to very fine meshes in applica-tions where microstructural geometry of the domainis involved.

Besides local mesh refinements, the approximationspace can be enhanced by incorporating prior knowl-edge of local character of the solution. Concretely, inthe Extended Finite Element framework [5, 6], theapproximate solution θc ∈ Vc takes the form

θc(x) =nc∑i=1

Ni(x) θi +nc∑i=1

ne∑j=1

Ni(x)ψj(x) θji , (14)

where the first sum contains the standard finite el-ement shape functions Ni(x) and the second termadds ne enrichment functions ψj for each discretiza-tion node; θi denotes the regular Degree Of Freedom(DOF) associated with finite-element mesh node xi,θji are the additional DOFs.In traditional XFEM, only patches of the domain

are usually enriched with ψj , suited for one geomet-rical feature. Typically, the enrichment providesasymptotic solution near a crack tip or introducesstrong/weak discontinuities [6]. Here, we add globalenrichments that capture collective response of a ma-terial microstructure. The original shape functionsNi(x) model the macroscopic response of the do-main, while products Ni(x)ψj(x) cover the fluctua-tions caused by material heterogeneity. A similar ideawas introduced by Strouboulis et al. [10], who used nu-merical “handbook” functions for assemblies of closelypacked inclusions, or Plews and Duarte [11]. However,both approaches solve local boundary value problemsdefined on subdomains first and subsequently run theanalysis of the whole domain. In contrast, we pre-compute the fluctuations purely “off-line” at the levelof microstructure compression. i.e., without any in-formation of the domain geometry or loading, andconstruct the enrichments as an assembly of the tile-defined fluctuation fields.

In our approach, the upper scale discretizationis constructed irrespectively of the underlying mi-crostructural geometry of Ω. Hence, the approxima-tion basis functions in the form of Eq. (14) togetherwith fluctuations in material parameters due to thepresence of microstructural geometry preclude efficientuse of standard numerical procedures for evaluationof the bi-linear form in Eq. (13).

In order to circumvent this drawback, we constructanother, finer domain discretization space Vf , assem-bled from the tile discretization used in the off-linephase for extracting fluctuation fields. The shapefunctions Ni(x), defined at the coarser discretization,are projected onto the fine mesh; enrichments ψj(x)are already defined within the fine discretization byconstruction. Consequently, Eq. (14) can be under-stood as a definition of reduced modes for the finediscretization.

Let matrix Φ comprises individual reduced modes asits columns, computed as an element-wise product ofthe projectedNi(x) and ψj(x) following Eq. (14). Therange of Φ thus defines a subspace of Vf correspondingto Vc. Instead of solving the large, fine-discretizationsystem Kfθf = Ff , we restrict the fine-discretizationunknowns θf via

θf = Φ a , (15)

where a denotes the vector of unknown coefficientspertinent to the reduced modes. The final algebraicsystem then takes the form

ΦTKfΦa = ΦTFf . (16)

5. ResultsWe considered the material microstructure of mono-disperse elliptical inclusions, shown in Figure 1. Bothphases were assumed isotropic, with the inclusionmaterial being more conductive (K(x) = 100 I) thanthe matrix phase (K(x) = 1 I). Six fluctuation fieldsin total, discretized with linear triangular elements,were extracted, see Figure 2.

The microstructure was superimposed to an upper-scale problem represented by an L-shaped domainwith uniform temperature profiles prescribed at thebottom and right-hand side edges. The initial coarsediscretization of the domain is depicted in Figure 3.The linear Lagrange basis functions were assumedat the coarse level, whilst the fine space comprisedthe quadratic Lagrange approximation arising fromthe product of linear coarse shape functions and theenrichments.The solution θXFEM of Eq. (16) was compared

against Direct Numerical Simulation (DNS) θDNS,which was taken as the reference solution. A sequenceof five uniformly refined coarse discretizations, thefirst one shown in Figure 3, was investigated. Theproximity of the reference (DNS) and XFEM solutions

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θ̂ = 20

10 tx

10 ty

θ̂ = 010 tx

10 ty

Figure 3. A scheme of the problem definition. Theedge lengths tx and ty of the tiles were set as tx =ty = 6. All quantities are in consistent units.

was quantified via relative errors

�L2 =

∥∥θDNS − θXFEM∥∥L2

‖θDNS‖L2and (17)

�H1 =

∥∥θDNS − θXFEM∥∥H1

‖θDNS‖H1, (18)

where

‖u‖2L2 =∫

Ωu2(x) dx and (19)

‖u‖2H1 =∫

Ωu2(x) +∇u(x) · ∇u(x) dx . (20)

Mean and standard deviations of the errors computedfor five coarse discretizations and six different tilingrealizations are given in Figure 4.

10!4 10!3 10!2

XFEM/DNS DOFs

10!2

10!1

0 0

L2H1

Figure 4. Convergence of L2 and H1 errors withincreasing number of DOFs in the coarse discretiza-tion.

6. ConclusionsWe have presented a method for extracting continuousfluctuation fields from a microstructure compressed

by means of Wang tiles. We have also demonstratedutilization of these fields as enrichments in XFEM.In order to avoid non-standard quadratures, XFEMbasis functions were projected onto a discretizationarising from assembly of the tile discretization usedfor computing the fluctuation fields.

The proposed methodology was applied to a steady-state heat conduction problem defined in an L-shapeddomain with microstructural geometry provided bythe tiling concept. Only 1.6 % of the original degreesof freedom were sufficient to obtain 2 % relative errorcompared to DNS.

On the other hand, the reduction in DOFs acceler-ates only the solution of Eq. (16). For more significanttime savings, a low-rank additional approximation ofKf , e.g. using the idea of hyper-reduction [12], hasto be introduced and is the focus of our current work.We have also restricted ourself to the first-order de-composition, Eq. (2); higher order expansion of themacroscopic loading can be considered to further en-rich the approximation space.

AcknowledgementsM. Doškář and J. Novák acknowledge the endowmentof the Ministry of Industry and Trade of the CzechRepublic under the project No. FV10202. In addi-tion, M. Doškář was supported by the Grant Agencyof the Czech Technical University in Prague, project No.SGS17/042/OHK1/1T/11 “Numerical Methods for Mod-eling Uncertainties in Civil Engineering”.

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Acta Polytechnica CTU Proceedings 13:29–34, 20171 Introduction2 Wang tiling concept3 Synthesized enrichments fields4 Numerical strategy5 Results6 ConclusionsAcknowledgementsReferences