Numerical Multiscale Solution Strategy for Fracturing Heterogeneous Materials

Lukasz Kaczmarczyk1, Chris J. Pearce1, Nenad Bicanic1 and Eduardo de Souza Neto2

1Department of Civil Engineering, University of Glasgow 2Civil and Computational Engineering Centre, Swansea University

Technical Report TR/CELK0901

Submitted to: Computer Methods in Applied Mechanics & Engineering, June 2009

Abstract

This paper presents a numerical multiscale modelling strategy for simulating frac-turing in concrete where the fine-scale heterogeneities arefully resolved. The fine-scaleis modelled using a hybrid-Trefftz stress formulation for modelling propagating cohe-sive cracks. The very large system of algebraic equations that emerges from detailedresolution of the fine-scale structure requires an efficientiterative solver with a precon-ditioner that is appropriate for fracturing heterogeneousmaterials. This paper proposesa two-grid strategy for construction of the preconditionerthat utilizes scale transitiontechniques derived for computational homogenization and represents an adaptation ofthe work of Miehe and Bayreuther (IJNME, 2007) and its extension to fracturing het-erogeneous materials. For the coarse scale, this paper investigates both classicalC 0-continuous displacement-based finite elements as well asC

1-continuous elements. Thepreconditioned GMRES Krylov iterative solver with dynamicconvergence tolerance isintegrated with a constrained Newton method with local arc-length control and linesearches. The convergence properties and performance of the parallel implementationof the proposed solution strategy is illustrated on two numerical examples.

Keywords: fracturing, multiscale, concrete, hybrid-Trefftz finite elements

0.1 Introduction

This paper presents a numerical multiscale modelling strategy for fracturing heteroge-neous materials. Physical multiscale analysis aims to predict the macroscopic consti-tutive behaviour of materials with heterogeneous microstructures. Such techniques notonly determine macroscopic ”effective” continuum material properties but also pro-vide understanding of the relationship between microstructural phenomena and theoverall macroscopic behaviour. Computational approaches(e.g. Suquet [8], Miehe[27], Kouznetsova [5]) typically utilize nested multi-level finite element analyses withdiscretisation at both the microscale and macroscale - so-called computational homog-enization. A fundamental restriction of these techniques is a clear separation of scales,such that the characteristic length of a representative volume element (RVE) is suffi-ciently small compared to the macrostructural characteristic length.

Clear separation of scales permits the assumption of uniformity of the macro-scopic strain field across the microstructure, as adopted infirst-order homogenizationschemes. In cases where the existence of an RVE necessitatesa less well definedseparation of scales, the assumption of uniform strains maybe inappropriate in somesituations, e.g. strain localization, boundary layers, etc. Second-order schemes havebeen proposed to overcome such short-comings [5, 16, 12], whereby the macroscopicmaterial behaviour is described using a higher-order continuum theory (e.g. strain gra-dient, Cosserat, micropolar). In such cases the material response at a macroscopic pointalso depends on the response in the neighbourhood of that point, thereby introducing amaterial length scale into the macroscopic constitutive model, and enables geometricalsize effects to be captured.

Material softening or fracturing lead to an evolving microstructure which makesit impossible to definea priori the size of the RVE. Once strains start to localise orfractures coalesce, material instability occurs, scale separation is no longer possible,the RVE becomes undefined and it is not possible to use scale-transition homogeniza-tion techniques. In Belytschko et al. [7], a modelling approach to overcome materialinstability in a multiscale setting has been presented in which discontinuities are re-moved from the fine-scale model and introduced directly intothe coarse-scale model.Gitman [10] presents the coupled volume approach where the concept of an RVE asso-ciated with a coarse-scale material point is abandoned; instead a coarse-scale elementis uniquely linked with a fine-scale cell. Markovic and Ibrahimbegovic [35] reporteda similar strategy. Miehe and Bayreuther [2] presented unifying computational proce-dures for the analysis of heterogeneous materials in the extremes of scale separationand in particular a multi-grid solution strategy (referredto asnumerical multiscale) forsituations without scale separation. This approach was inspired by the formulations,and in particular the scale transition techniques, of computational homogenization.

Here Miehe and Bayreuther’s numerical multiscale solutionstrategy is extendedfor the case of softening or fracturing materials. An efficient two-grid (fine and coarsemesh) preconditioner for Krylov iterative solvers is constructed and derivation of thehomogenization-based projection operators is fully described. Although the approachpresented has application to a broad range of materials, thefocus of attention is con-crete material at the level of observation below the macroscale (so-called meso-level(1-10cm)), identifying individual aggregates embedded ina matrix, with a weak in-terfacial transition zone. The analysis of fracturing heterogeneous materials requiresa robust solution strategy for tracing the unstable equilibrium path. Thus the precon-ditioned iterative solver is embedded in a constrained Newton method with local arc-length control and line searches.

1

In this paper, the cohesive crack methodology is utilized together with Hybrid-Treffz stress elements [13] for the fine mesh that fully resolves the heterogenous struc-ture and this is briefly discussed in Section 2. Next, the overall solution strategy forfracturing heterogeneous materials is discussed, before describing in detail the pro-posed two-grid preconditioner and in particular the construction of the homogenization-based projection operators in Section 5. Section 6 demonstrates the performance of theproposed model with some numerical examples and investigates the use of bothC 0-continuous andC 1-continuous elements for the coarse mesh.

0.2 Fine-scale framework for fracturing heterogeneousmaterials

The analysis of fracturing heterogeneous materials necessitates full resolution of thefine-scale structure, that evolves during mechanical loading. This requires a robustmodel for cohesive cracking, where multiple cracks, crack branching and crack coales-cence are the norm and where different constitutive models are required for the variousphases of matrix, inclusions and interface. A hybrid-Trefftz stress (HTS) formulationis adopted, the detailed description of which can be found in[3, 1] and in [13] for theextension to heterogeneous quasi-brittle materials.

Displacement discontinuities are restricted to element interfaces and the materialresponse within each element is assumed to be hyperelastic.Such an approach isdeemed realistic [18], and can be justified by the observation of fracturing phenomena,for which localization occurs and material unloads in the vicinity of the crack. Un-structured fine-scale meshes are adopted in order to reduce the influence of the meshon fracture propagation. Furthermore, for heterogenous materials, the finite elementmesh size is significantly constrained by the size and spacing of the inclusions. It hasbeen shown that for the type of problem considered here, the results are mesh objective[13].

The HTS finite element formulation is characterised by the approximation of stresseswithin the domain of the element and by the fact that the stiffness can be expressed viaa boundary, rather than domain, integral. Thus, compared totheir classical FEM coun-terpart, HTS elements exhibit faster convergence of the stress fields. Furthermore, thedisplacements are approximated on element boundaries and the displacement basis isdefined independently on each element interface. Consequently, the overall bandwidthof the stiffness matrix is very small and computationally efficient to solve.

0.2.1 Discretization and boundary value problem for cohesive crackmethodology

Consider the domainΩe of the solid body occupied by a quasi-brittle material thathas a boundaryΓ = Γσ ∪ Γu and crack interfacesΓint, whereΓσ ∩Γu = Ø andΓ∩Γint = Ø. Given boundary displacementsu on Γu and boundary tractionst on Γσ, thestatic mechanical problem is mathematically defined as finding σσσ, εεε andu given thegoverning relations:

Ω

Γu

Γσ

Γint

Γint

N

n

2

Figure 1: Body with cracks

LTσσσ = 0, in Ωe (1)

σσσ = Cεεε, in Ωe (2)

Lu = εεε, in Ωe (3)

u = u, on Γu (4)

Nσσσ = t, on Γσ (5)

nσσσ = Dg on Γint (6)

whereL is a differential operator,N is the unit normal to the domain boundary,Cis the stiffness matrix for the bulk material andD(g,κκκ) is the interface stiffness, ex-pressed as a function of the displacement jumpg and history variableκκκ. All equationsare expressed in Voigt’s notation. For simplicity, body forces are omitted from theformulation.

The bulk response of material is described by stress and displacement fields whichare approximated by means of a Hybrid-Trefftz approximation. The stress field withinan element is approximated directly as:

σσσ = Svv (7)

whereSv is a matrix of field approximation functions andv is the unknown vector ofgeneralised stress degrees of freedom. In Equation (7), thestress approximation fieldis chosen so as to automatically satisfy equilibrium:

LTSv = 0. (8)

In order to retain the independence of the descriptions of kinematics and statics, an ad-ditional and independent approximation of displacementsuΓ on the traction boundaryΓσ of the element is introduced:

uΓ = UΓq. (9)

Thus, the primary unknowns are the stress degrees of freedomv within the ele-ment and the displacement degrees of freedom on the element boundaryq. However,the stress degrees of freedom are eliminated from the globalsystem of equations byapplication of static condensation at the element level.

The cohesive crack model on the element interfaces adopted here assumes that allinelastic deformation in the vicinity of a crack is concentrated onto a line and expressedin terms of tractions and displacements [28]. In order not toover-complicate the for-mulation, a straightforward material model for cohesive failure in two-dimensions isadopted. The model response depends on three material parameters: tensile strengthft, fracture energyGf andα which assigns different weights to the shear and normalopening displacements. As such, a discontinuity (i.e. displacement jump) is intro-duced when the effective tractiont on an element face exceeds the tensile strength ofthe material; the subsequent traction transferred across the interface is dependent onthe magnitude of the displacement jump. A bilinear softening law is adopted, Figure 2.

Full details of the Hybrid-Trefftz stress element formulation for modelling cohesivecracking in heterogeneous materials is described in [13]. The current paper focuses onthe solution strategy for this class of problem.

3

σ1

κ1 κ2 κ

ft

t

Figure 2: Bilinear softening law.

0.3 Solution strategy for fracturing heterogeneous ma-terials

The analysis of fracturing heterogeneous materials represents a significant computa-tional challenge. First, tracing the unstable equilibriumpath, including overcomingcritical stability points, requires a robust solution scheme. Second, the full resolutionof the heterogeneous fine-scale structure results in a largesystem of algebraic equationsthat needs to be solved efficiently. This section will initially discuss a Newton’s methodwith local arc-length control and line searches for tracingthe nonlinear response beforegoing on to present a numerical multi-scale preconditionerfor Krylov solvers that isspecifically designed for this application.

0.3.1 Constrained Newton-multigrid algorithm

For each load steps, the equilibrium equation can be expressed as

r(q,λ) = fext(λ)− fint(q) = 0 (10)

whereq are the displacement degrees of freedom,λ is the load parameter,fext andfint

are the external and internal forces, andr is the vector out-of-balance forces.In order to trace the dissipative load-displacement path, an arc-length scheme is

adopted, detailed later, which is based on the introductionof a control function∆b foreach load increment, which only depends on the displacementsq [6]. The correspond-ing constraint equation is written as:

rλ = ∆b(q)−∆l = 0 (11)

where∆l represents the increment length for the current load step, the magnitudeof which is computed via an automatic incremental proceduredescribed later.

The equilibrium and constraint equations together form a nonlinear system of equa-tions that is solved using Newton’s method in combination with an efficient iterativesolver [2] for the associated linear system of equations. Linearization of (10) resultsin:

4

Kδq−pδλ = r (12)

whereK =−∂qr is the tangent stiffness matrix,p = ∂λfext is the vector of externallyapplied reference loads.

Since it is assumed that function∆b(q) remains unchanged during each load step[6], ∆b in Equation (11) can be expressed as

∆b = bT∆q with bTδq = 0 (13)

where∆q is the displacement increment for steps andδq is the iterative change indisplacement for iterationm.

For automated load control, the increment length must be adapted as the analysisproceeds in order to take account of the changing degree of convergence. A simple ap-proach is adopted here based on the ratio of the actual numberof iterations required forconvergence of the previous increment Is and the desired optimum number of iterationsId as follows [6, 20]:

∆ls =

(Id

Is

)0.5

∆ls−1 (14)

Algorithm 1 Constrained Newton method with line-search1: Initialize ∆q = 0, ∆λ = 0 and set iterationm= 02: if m> 0 and|r| < β∗ tolr1 ∗ |r0| then3: Update set of active cracks4: end if5: AssembleK(q+ ∆q) andr(q+ ∆q,λ + ∆λ)6: Update preconditioner7: if m = 0 then8: Set local control arc-length vectorb and determinerλ(q,λ, l)9: Call Linear Solver(Km;b, rλ;Kc

m,Pm) ⇒ δq,δλ;10: else if |rm| > tolr1 ∗ |r0| then11: Call Linear Solver(rm,Km;b, rλ = 0;Kc

m,Pm) ⇒ δq,δλ12: Call Line Search(δq,∆q,q) ⇒ η13: Update∆q = ∆q+ ηδq, ∆λ = ∆λ + ηδλ14: else15: Updateq = q+ ∆q, λ = λ + ∆λ and history variables16: if Set of active cracks unchangedthen17: Next increment,go to 118: end if19: end if20: m= m+121: go to 2

Details of a single Newton increment of the solution processare given in Algo-rithm (1). This procedure is relatively standard and does not require detailed discus-sion, however some of the salient points are described here.The parameter tolr1 rep-resents the convergence tolerance for the Newton iterations (typically×10−5). Line 6refers to the update of the preconditioner which will be discussed in the next section.Where cracks initiate at the interface between two elements, as defined by the consti-tutive law, the degrees of freedom (DOF’s) are duplicated. Due to the likelihood of

5

cracks developing in the weak interfacial transition zone,duplicate DOF’s are includedfrom the beginning of the analysis and displacement continuity enforced. It can benoted that for stability of the algorithm, the set of active cracks is only updated onceequilibrium is achieved (within a loose tolerance defined byβ). Overall convergence isachieved once both equilibrium is satisfied and the set of active cracks is not changing.

0.3.2 Local arc-length control

The concept of local arc-length control with line searches was presented in [6] for de-lamination analysis. In this paper this approach is adaptedto the analysis of fracturingheterogeneous materials. The basic idea oflocal arc-length control is to restrict atten-tion to those DOF’s associated with the active process zone and thereby control thedissipative evolution of the structure.

The arc-length vectorb in (13) is a function of the displacement jumps at the inte-gration points along element interfaces. These are collected together inδδδ as:

δδδT = [g1,g2, . . . ,gN] (15)

wheregi is the displacement jump in the normal direction to the face,at integrationpoint i. Only active cracks are included, i.e.∆gi > 0 in the previous increment. Acritical value,gc

i , is associated with each component ofδδδ, and is chosen to be thedisplacement jump associated with the change in slope of theadopted interface bilinearsoftening curve (see Figure 2). The magnitude of this value will vary for differentphases of the composite material.

(δδδc)T = [gc1,g

c2, . . . ,g

cN]. (16)

The constraint equation is now written as a function of displacement jumps:

bT∆q =N

∑i=1

visgn(δs−1i )(δi − δs−1

i ) = ∆l with vi =

1δc

i

∑Ni=1

1δc

i

. (17)

For the first increments = 1, all points are assumed undamaged and a dummyconstraint equation is created by settingδs−1

i = 0.Although local arc-length control has proven to be generally effective, the proce-

dure is still found to be unstable in some specific situations, typically related to over-coming sharp limit points and non-smooth nonlinearities [6]. Line searches have beenshown to provide an effective remedy [6] and is adopted here.A detailed descriptionof line search techniques can be found in [6], amongst others.

0.3.3 GMRES and preconditioner for linearized problem

The linearized problem (12) and Equation (11) can be expressed together in matrixform as: [

K −pbT 0

][qλ

]=

[ff λ

](18)

whereδ has been omitted fromδq andδλ for simplicity. In condensed form, theabove equation can be rewritten as:

Kq = f (19)

6

0.4 Preconditioner and projection operators

Solution to this system of equations is found using a Generalized Minimum Resid-ual Method (GMRES) solver that is based on an oblique projection process onto theKrylov subspace and minimization of the residual over all vectors in the Krylov sub-space. Although the GMRES method and other Krylov subspace methods are wellfounded theoretically, they suffer from slow convergence for problems of the type dis-cussed in this paper. Since the rate of convergence of a Krylov projection method isstrongly dependent on its spectrum, preconditioning can accelerate the convergencerate by altering that spectrum.

Preconditioning can be applied to the system described by Equation (19) [21, 24]as follows:

(M−1L KM−1

R )(MRq) = M−1L f (20)

whereML and MR indicate ”left” and ”right” preconditioning matrices respec-tively. In this paper, attention is restricted to left preconditioning, i.e. MR = I &M = ML , although other possibilities can be investigated in future work. The straight-forward application of GMRES to the linear system (20) with left preconditioning ispresented in Algorithm (2) [24].

Algorithm 2 GMRES with left preconditioner

1: Call Preconditioner⇒ r0 = M−1(f− Kq0)2: β = |r0| andv0 = r0/β3: for j = 0 tom do4: Call Preconditioner⇒ w = M−1Kv j

5: for i = 0 to j −1 do6: hi, j = w · vi

7: w = w− hi, j vi

8: end for9: Computeh j+1,i = |w| andv j+1 = w/h j+1,i

10: end for11: DefineVT

m = [v0, . . . , vm], Hm = hi, j12: Computeym = argminy|V

Tmr0−Hmy| andq = q0 + Vmy

13: if |ri | < tolr2|f| then14: stop15: else16: Setq0 = q17: go to 118: end if

Algorithm (2) utilises the Gram-Schmidt process for vectorothogonolization im-plemented in PETSc [21, 22, 23]. All residual norms that are computed by the algo-rithm correspond to the preconditioned residuals. However, for checking the conver-gence of the presented approach, the original (unpreconditioned) residuals have to becomputed explicitly.

An important feature of this solution strategy is that the convergence tolerance tolr2

is dynamic and ensures that the GMRES algorithm does not converge to an unnecessar-ily small tolerance while the Newton iterations are still far from equilibrium, therebyrepresenting an important speed-up with respect to direct solvers.

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0.4.1 Two-grid preconditioner

Iterative solvers exhibit a smoothing property, whereby the short-wave modes of thesolution error are eliminated effectively during the iteration process. Conversely, thelong-wave modes are damped very slowly. However, these long-wave modes appearrelatively higher in frequency when projected onto coarsermeshes. Thus, a multi-gridpreconditioning strategy utilises coarser meshes for fastsmoothing of the long-wavemodes of the error. Typically, a hierarchy of coarse meshes would be used, althoughhere we restrict ourselves to just two meshes - fine and coarse.

In order to smooth the long wave component of the error, the fine mesh residual isrestricted onto a coarser mesh using a projection matrixR as:

rc = R(f−Kqf) (21)

where the superscript f has been added to indicate the fine mesh and c indicates thecoarse mesh. Next, the following system of equations associated with the coarse meshare solved in order to determine the coarse mesh solution:

qc = (Kc)−1rc (22)

The construction of the stiffness matrixKc on the course mesh will be discussedin the next section. Typically, since the size of the coarse mesh system of equationsis significantly smaller than that of the fine mesh, the coarsemesh solution can becomputed using a direct solver. The coarse mesh correction is then prolongated backto the fine mesh using the projectionP as:

qf = P(qc + λqλ) (23)

where the load factor is determined as:

λ = ( f λ −bTPqc)/bTqλ (24)

andqλ stands for:qλ = P(Kc)−1PTf (25)

These steps are presented in Algorithm (3). The algorithm also includes pre- (Spre)and post-smoothing (Spost) in order to reduce the contribution of the fine mesh resid-ual. The purpose of this relaxation process is not to reduce the overall solution error,but smooth it out so that it can be well approximated by functions on the coarse mesh.This is achieved by removing high frequency errors, which exhibit local variations inthe solution, with a straightforward relaxation method such as Gauss-Seidel [25].

A standard method to parallelize Gauss-Seidel on an unstructured mesh is to colourthe matrix graph so that adjacent nodes have different colours. Having done so, allnodes of a given color within one convergence iteration can be executed in parallel.For classical FEM, the main problem is the potentially largenumber of colors leadingto large inter-process communication. However, for the HTSelements, displacementdegrees of freedom are associated with element faces, whichreduces the number ofneighbours to 3 for a 2D mesh of triangles and 4 for a 3d mesh of tetrahedrals. Anappropriate algorithm for parallelized Gauss-Seidel withcolors can be found in [25, 24,26]. The coloured version of Gauss-Seidel is implemented bythe authors and used forall numerical analyses presented in this paper. Alternatives for classical displacement-based FEM can be found in [26].

8

Algorithm 3 Two-grid preconditioner

1: Call Spre(qf , f,K)2: Restrict residualrc = R(f−Kqf)3: Determineqc = (Kc)−1rc andqλ = P(Kc)−1PTf4: Determine load factorλ = ( f λ −bTPqc)/bTqλ

5: Prolongate coarse displacementsqf = P(qc + λqλ)6: Call Spost(qf , f + λp,K)

Before Algorithm (3) is initiated the coarse mesh tangent matrix Kc, qλ and theprojection matrices are computed. The tangent matrix is factorized in order to make theiterations more efficient. In subsequent sections we focus on the key issue of derivingthe coarse-to-fine projection operatorP, fine-to-coarse projection operatorR and coarsestiffness matrixKc.

0.5 Construction of homogenization based transfer pro-jection operators

The two-grid preconditioner utilises a coarse mesh in addition to the fine mesh. A patchof fine mesh elements, which fully resolves the fine-scale heterogeneities, is associatedwith a single coarse mesh element, see Figure 3. The two-gridpreconditioner, used inAlgorithm (3), can be expressed in explicit form as:

ML = P(Kc)−1R (26)

The construction of the homogenization-based coarse mesh stiffness matrixKc isdescribed later. Attention is currently focused on derivation of the projection operatorsbased on a decomposition of the fine-scale displacements into long-waveandshort-wavecontributions by the additive split:

qf = qf

︸︷︷︸ + qf

︸︷︷︸long-wave short-wave

(27)

The long-wave contribution of the fine-scale displacementsare associated with the ho-mogeneous contribution of the coarse mesh approximation and determined from in-terpolation of the coarse-grid displacements. The short-wave displacements representfluctuations due to the fine-scale heterogeneous structure of the patch, e.g. inclusionsand deformation induced cracks. Long- and short-wave contributions to the fine meshapproximation of displacements are obtained by prolongation of the coarse mesh dis-placementsqc

qf = Pqc and qf = Pqc. (28)

As a consequence, the prolongation operator in (26) is decomposed into long-wavePand short-wave fluctuationP components as:

P = P+ P (29)

This decomposition is also described in Figure 3. With the coarse-to-fine mesh projec-tion operator at hand, the fine-to-coarse mesh projection operatorR in Algorithm 3, isdefined as

R ≡ PT. (30)

9

X

Nodal displacements &element shape functions

qf = Pqc, uf = uc

Boundary conditions &Solve BVP

qf = Paveqc, uf + wf = uc

qf = Phomqc

uf = εεεx

εεε

εεε

Figure 3: Long-wave and short-wave projection operators.

10

In constructing the projection operators, small displacements and small strains are as-sumed.

0.5.1 Long-wave component of projection operator

The long-wave coarse-to-fine mesh prolongation operatorP relates the displacementfield in a coarse mesh element to the displacement field in the corresponding patch offine mesh elements and is computed from interpolation of the displacement field onthe coarse mesh. With 2D classical displacement type finite elements for both fine andcoarse meshes,P is evaluated as [2]:

P =

Nc1 (x1) 0 · · · Nc

nc (x1) 00 Nc

1 (x1) · · · 0 Ncnc (x1)

.... . .

...Nc

1 (xnf ) 0 · · · Ncnc (xnf ) 0

0 Nc1 (xnf ) · · · 0 Nc

nc (xnf )

(31)

whereNci (xi) is a coarse mesh element shape function computed at positionxi on the

fine mesh,nc is the number of nodes for a coarse mesh element andnf is the number ofnodes comprising the correspondingfine mesh patch. However, in this particular imple-mentation two different discretization methods are used, i.e. displacement type finiteelements and hybrid-Trefftz stress elements for the coarseand fine mesh problems, re-spectively. For such problem, the generalized projection operator, relating coarse meshnodal degrees of freedom with those on the faces of the corresponding fine mesh patchis given by

P = A(

Nc(

xfi

))(32)

where matrix functionA (. . . ) depends on the fine mesh face approximation functionsadopted and relates the interpolated coarse mesh nodal displacements with the finemesh face displacements of the fine mesh patch such that the following equalities arefulfilled for all position vectorsxf

i

uf(xi) = UΓ(xi)qf = UΓ(xi)Pqc = Nca(xi)qc = uc(xi) (33)

In a classical geometric multi-grid strategy, this long-wave projection operator asso-ciated with the homogeneous part of the deformation represents the only componentof the total projection operator, i.e.P = P. However, following the approach in [2],the long-wave projection operator is augmented by a short-wave component associatedwith the the fine-scale heterogeneities.

0.5.2 Short-wave component of projection operator

The short-wave component of the projection operator,P, reflects the influence of theheterogeneous nature of the fine mesh patch of elements corresponding to a singlecoarse-scale element. Following [2], this contribution can be formulated using thescale transition techniques for computational homogenization.

A truncated Taylor series expansion of the displacement vector about the geometriccentre of the coarse mesh element, using Voigt notation, yields:

u(x) = u0 + Xεεε+ w(x) (34)

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whereu0 is the displacement vector at the geometric centre of the coarse mesh element,εεε is the strain state at element centre,w(x) is the displacement fluctuation over the finemesh patch andX is a matrix of relative position coordinates:

X =12

[2x 0 y0 2y x

](35)

Rearranging (34) yields an expression for the displacementfluctuation:

w(x) = u(x)−Xεεε−u0 (36)

Relating the displacement fluctuation to the coarse elementdegrees of freedomqc

yields, after HTS discretization (9), the short-wave component of the projection op-erator is introduced as

w(x) = UΓPqc (37)

Similarly,u(x) = UΓPaveqc + u0 and Xεεε = UΓPhomqc (38)

Substitution of (37) and (38) into (36) yields

UΓPqc = UΓ(Pave− Phom)qc (39)

where the short-wave component of the projection operator is:

P = Pave− Phom (40)

The homogeneous component of this projection operator,Phom, associated with coarsemesh deformations, simply takes the form:

Phom= A (XiBc) (41)

whereXi is evaluated at the fine mesh nodes and the approximation projection functionA is the same function as used in (32). Note thatA is the identity function if classicalLagrange shape functions are used for interpolation of displacements on the fine mesh.Bc is the coarse mesh strain-displacement matrix.

The fluctuation component of the short-wave projection operator, Pave, is deter-mined by solution of aboundary value problem(BVP), whereby deformation of thefine mesh patch is enforced, via appropriate boundary conditions, according to a givencoarse element average strain. The boundary conditions forthe fine mesh patch mustsatisfy:

Z

∂Ωc

n⊗wdΓ = 0 (42)

wheren is the normal vector to the coarse element boundary domain∂Ωc. It can beshown, that such boundary conditions represent a minimal condition that fulfill the Hill-Mandel theorem, e.g. see [12, 5, 2]. Given the HTS discretization of displacements (9)and the expression for the displacement fluctuation (36), the constraint equation (42) isexpressed as:

Cqf −DBcqc = 0 (43)

whereC is a constraint matrix given by

C =

Z

∂Ωc

HUTΓUΓdΓ, (44)

12

andD asD =

Z

∂Ωc

HUTΓXdΓ (45)

The terms of matrixH reflects the specific nature of the boundary conditions used,i.e. linear deformation or uniform traction. It is worth noting that since the proposedformulation is not restricted to regular meshes at either the coarse or fine scales andgiven that unstructured triangular meshes are typically preferred, periodic boundaryconditions are not considered as an option. The influence of the type of boundarycondition will be investigated with numerical examples in the next section. Each termin matrix H can be interpreted as an admissible distribution of nodal traction forces onthe boundary of the elements patch. Details on the construction of H can be found in[12].

The solution of the discretized BVP for the fine-scale patch of elements can beexpressed as a constrained quadratic problem. A common method to solve such aproblem is to introduce Lagrange multipliers. However, such an approach increasesthe number of unknowns and alters the character of the systemmatrix (to an indefinitesaddle point problem). Moreover, the numerical solution ofEuler’s conditions for thestationary point of the Lagrangian is rather inefficient andtherefore not suitable forsolving computationally complex multiscale problems where the constrained quadraticproblem has to be solved for every integration point. Alternative numerical techniquesinclude the penalty method or Uzawa method [36]. However, inorder to express theshort wave projection operator in closed-form, the approach taken in [12] is brieflypresented, where the work of Ainsworth [29] is applied. The fine-scale BVP is can beexpressed as:

Kfqf + CTλλλ = 0 (46)

Cqf = DBcqc (47)

whereλλλ are the Lagrange multipliers. Following [29], the following matrices are de-fined:

Q = I−CT(CCT)−1C, R = CT(CCT)−1 (48)

Matrix Q projects the fine mesh stiffness matrixKf onto a sub-space that is orthogonalto that of the constraints expressed byCTC. Solution forqf is then given as:

qf = K−1f (49)

andλλλ = −RTKfqf (50)

whereK = CTC+ QTKfQ (51)

andf, expressed in terms of the coarse element displacement degrees of freedom, is:

f = gBcqc, g =(

CTD−QTKfRD)

(52)

Thus, Equations (49) & (52) provide an expression forPave as:

qf = Paveqc with Pave= K−1gBc (53)

Recall thatPhom is subtracted fromPave to giveP, which is then added toP to yieldthe projection operatorP = P+(Pave− Phom).

13

0.5.3 Construction of coarse element stiffness matrix

The coarse scale element stiffness is computed from averaging of the evolving fine-scale stiffness and represents a coupled-volume approach [10]. For computation ofthe coarse element stifnessKc, it is convenient to re-write (53) in terms of the coarseelement strain, rather than degrees of freedom, as:

qf = Paveε εεε with Pave

ε = K−1g (54)

To determine the coarse element stress-strain relationship, the average stress can beexpressed in terms of the Lagrange multipliers [12] as

σσσ =1V

DTλλλ = −1V

DTRTKfqf (55)

where the final expression has been arrived at by substitution of λλλ from Equation (50)andV is the coarse element volume. Inserting (54) into the above,the coarse elementstress-strain relationship is obtained:

σσσ = Cεεε where C = −1V

DTRTKfPaveε (56)

0.5.4 Update of projection operators

At the beginning of an analysis, all coarse mesh element stiffness matrices and pro-jection operators are computed from the derivations above.For all coarse elementsfor which the response is elastic, the material stiffness matrix and projection operatorsremain unchanged and are stored in distributed memory of parallel processors. Whenlocalization at the fine-scale takes place, it is observed that material non-linearities onlyoccur in the vicinity of the crack bands; taking advantage ofthis knowledge, only thosecoarse element stiffness matrices and projection operators associated with cracking inthe respective fine-scale patches are updated. A set of coarse elements to be updatedare determined by the following criteria:

1. softening takes place at the integration points of the finemesh patch in the pre-vious Newton iteration;

2. one or more integration points of the fine mesh patch changeregime from soft-ening to unloading in the previous Newton iteration.

This strategy makes computations of the preconditioner very efficient, since, for eachload-step, updates are only made for a limited number of coarse mesh elements. Wealso note that computations for the fine mesh patches do not need inter-processor com-putations, which is efficient for parallel implementation.

0.5.5 Generalization to second-order continuum

For problems involving softening and fracturing of materials, there is potential forimprovement of the preconditioner by enhancing the coarse mesh displacement fieldapproximation space and thereby introducing higher-orderdeformation modes in thecomputation ofPave. A similar strategy of taking account of higher order deformationmodes is applied in [11].

Therefore, a natural extension of the first-order two-grid preconditioner, describedin this section, can be achieved by taking into account second-order strain measures in

14

the spirit of the second-order homogenization method [5, 12]. Such an enhancementimproves numerical efficiency of the preconditioner, whichis manifested by faster con-vergence of the Newton method with Krylov iterative solver.This will be demonstratedwith numerical examples in the next section.

A second-order scheme is realized by developing the fine-grid displacements witha Taylor series expansion and truncated after the second-order term

u(x) = u0 + x ·εεε+12

x⊗x : ηηη+ w(x) (57)

whereηηη = ∂xxu is a second-order strain measure. Detailed description of second-order computational homogenization can be found in [12], here we limit ourselves tothe salient equations required for extension of the first-order scheme developed in theprevious sections.

The displacement fluctuation in (57) expressed in terms of the coarse mesh degreesof freedom have the form

w(x) = u(x)−XBcqc−ZGcqc−u0 (58)

where

Z =14

[2x2 0 2y2 0 xy 00 2y2 0 2x2 0 xy

](59)

andGc is the coarse element second-order equivalent ofBc. The homogeneous part ofthe projection operator has the form

Phom= A (XiBc + ZiGc) (60)

whereXi andZi are evaluated at the fine mesh nodes.The boundary conditions for the fine mesh patch, in addition to Equation (42), are

given byZ

∂Ωc

n⊗x⊗wdΓ = 0. (61)

This equation enforces deformation of the fine mesh patch according to the averagecoarse mesh second-order strainηηη. Following discretization of the fine mesh displace-ments, the boundary conditions (42) and (61) are expressed in a modified constraintequation (compare with (43)):

Cqf −DBcqc + EGcqc = 0 (62)

whereE =

Z

∂Ωc

HUTΓZdΓ (63)

With this enhanced constraint equation (62) the second-order fluctuation part of theshort-wave contribution to the coarse-to-fine mesh operator Pave can be constructed ina similar manner to the first-order scheme in section (??).

In order to keep the coarse mesh problem as simple and efficient as possible forpractical implementation, the formulation is restricted to the antisymmetric part ofηηη.Thus, in 2D, there are two bending modes in addition to the simple first-order defor-mation modes, as depicted in Figure 4.

Significantly, this second-order approach leads to a strain-gradient continuum, re-quiring C 1-continuous displacements. A common approach for such problems is to

15

Figure 4: Deformation modes: first-order modes associated with second-order straintensorεεε (upper row) and second-order modes associated with antisymmetric part ofthird-order strain tensorηηη (lower row).

utilize mixed finite elements withC 0-continuity and relaxed approximation for the dis-placement gradients. However, for the two-grid solution strategy presented here, suchan approximation is not suitable and requires a smooth approximation of displace-ments. Appendix 0.7 presents a formulation for a simple and efficient C 1-continuousfinite elements.

The next section will investigate and compare performance of both the first andsecond-order schemes with two numerical examples.

0.6 Numerical examples

Two numerical examples are presented to demonstrate the performance of the proposednumerical homogenization two-grid solution strategy. Readers are directed to [2] for astudy of the efficiency of first-order multi-grid strategy (i.e. more than two hierarchicalgrids) for elastic and hardening plasticity materials and their superiority over the localSchur complement method.

In the first example, numerical convergence and efficiency ofthe first and second-order schemes are investigated with focus on elastic heterogeneous materials. In thesecond example, performance of the solution strategy is investigated for fracturing het-erogenous materials.

0.6.1 Elastic heterogeneous specimen with a hole

The problem under consideration is shown in Figure 5, which is based on an examplefrom [2], investigates the performance of a plate with a central hole and comprising anelastic heterogeneous material. The plate has dimensions 20m x 20m with a hole of ra-dius 2m. The specimen is subjected to a uniform tensile traction of 1N/m on the left andright hand side. Due to symmetry, only a quarter of the plate is discretized using trian-gular hybrid-Trefftz stress elements. As is shown in Figure5, three different fine-scaleheterogeneities are considered in order to investigate theinfluence of the microstructureon the performance of the two-grid preconditioner, Figure 5. These are: randomly dis-tributed heterogeneities, laminates and checker board pattern. The distribution of the

16

Figure 5: Plate with a hole and heterogeneous microstructures (randomly distributedheterogeneity, laminate and checker board).

heterogeneities are obtained by imposition of a specific density function at the center ofeach fine mesh element. Each fine mesh consists of 449,464 displacement degrees offreedom and 896,520 stress degrees of freedom. The materialfor all constituents is as-sumed to be elastic. The stiff elastic inclusion has an elastic modulusEhard= 214.9MPaand Poisson’s ratioν = 0.34. The soft elastic matrix hasEsoft = Ehard/100 andν = 0.34.

For each microstructure, four analyses are considered (Table 1) in order to investi-gate the difference between first and second-order schemes and the influence of the typeof boundary conditions utilised in computing the projection operator. The coarse meshcomprises constant strain triangular (CST) elements with 1959 nodes for the first-orderscheme andC 1-continuous triangles (see Appendix 0.7) for the second-order scheme.

Table 1: Details of plate analyses.

Analysis Boundary conditions Coarse mesh type (number dof’s)

CST/TRAC uniform traction C0-continuous (3918)

CST/DISP linear displacements C0-continuous (3918)

GRADIENT/TRAC linear traction C1-continuous (11754)

GRADIENT/DISP quadratic displacements C1-continuous (11754)

The stress distributionσx is shown in Figure 6 for each microstructure. Despitea strong heterogeneous distribution of stiffness, all the analyses show good conver-gence, see Figure 7, although the second-order scheme is fastest. Moreover, the useof displacement boundary conditions leads to better convergence compared to tractionboundary conditions. When using the traction boundary conditions, displacement fluc-tuations do not vanish locally on the boundary of the fine meshpatches and this resultsin a mismatch of fluctuations on common edges. Averaging of this difference leadsto residual forces and the resulting displacement errors are the cause of slower conver-gence. On the contrary, with displacement boundary conditions, a zero fluctuation fieldis enforced ‘a priori‘ on the patch boundary.

All analyses have been run on a parallel computer, up to a maximum of 128 proces-sors. Speed-up is shown in Figure 8 and it can be observed thatefficiency deteriorateswith increasing number of processors due to the increasing proportion of time spenton non-parallelized aspects of the algorithm. In the current implementation the coarsemesh stiffness matrix factorization and solution on each cycle is computed on a singleprocessor and is the main reason for loss of efficiency on a large numbers of processors.Moreover, this shortcoming explains the differences in parallel efficiency between thefirst and second-order schemes, where the latter comprises alarger number of coarse

17

mesh degrees of freedom, see Table 1.The longer computational times noted for the traction boundary conditions are not

only the result of a larger number of iterations. In addition, the imposition of trac-tion boundary conditions result in multi-point constraints which cause larger fill-in offactorized element stiffness matrices, resulting in greater computational effort.

Figure 6: Stress distributionσx for different heterogeneous microstructures.

0 50 100 150 200

0 50 100 150 200 1

0.1

0.01

0.001

1e−04

1e−05

1e−06

1e−07

0.1

0.001

1e−04

1e−05

1e−06

0.01

1

random

check board

laminate

CST/DISP

GRADIENT/DISPCST/TRAC

GRADIENT/TRAC

Number of Iterations

Number of Iterations

Absolute residual (not preconditioned)

Figure 7: Convergence rates for different heterogeneous microstructures.

0.6.2 Tensile fracture of dog-bone specimens

The fracture of dog-bone concrete specimens subject to tensile loading is analysed inorder to demonstrate the performance of the proposed numerical homogenization strat-egy for relatively large-scale problems and fracturing heterogeneous materials. Detailsabout geometry, boundary conditions, aggregate size distribution, material model andparameters can be found in [4]. In Figure 9, a simplified geometry for the specimens ispresented. Three sizes of dog-bone samples are considered,D = 50mm,D = 100mm

18

40 80 120 0 0

20

40

60 random

Speed Up

Number of Processors

CST/DISPGRADIENT/DISP

CST/TRACGRADIENT/TRAC

250

200

150

100

50

randomcheck boardlaminate

CST/DISP GRADIENT/DISP CST/TRAC GRADIENT/TRAC

Cumulative time [seconds]

Figure 8: Speed-up performance for randomly distributed heterogeneities and cumula-tive computational time for all heterogeneous microstructures.

Mesostructure Coarse mesh50mm

100mm

200mm

0.25D

0.2

D

DD

Ls = 1.2D

Figure 9: Dog-bone geometry (left), mesostructure for different specimen sizes (mid-dle) and corresponding coarse meshes (right).

andD = 200mm. Each sample size has a meso-structure generated withthe same ag-gregates size distribution. Specific details of the aggregate structure can be found inAppendix B of [4] and in [31].

The mesostructure for all problems is fully resolved, whereby aggregates, matrixand the interfacial transition zone are discretely modelled. The characteristic size ofthe fine mesh for all sample sizes are the same and controlled by the aggregate size andspacing. Such an approach has been shown to lead to mesh independent results [13].The circular shape of aggregates is discretized by a piecewise linear approximation,with the length of the segment equal to 1/10 of the radius of the smallest aggregate.The characteristic size of the coarse mesh is chosen arbitrarily as the distance betweenaggregate centres. Table 2 shows the number of degrees of freedom for both the fineand coarse meshes. For the aggregate, the elastic modulus isEA = 98MPa and for thematrix EM = 35MPa. Poisson’s ratioν = 0.2 for all phases. The steel loading platesare also modelled as an elastic material withES = 420MPa andν = 0.3. The cohesivecrack material parameters for matrix, aggregate and interfacial transition zone havethe following values : f M

t = 12MPa,GMf = 190N/m, f A

t = 6MPa, GAf = 190N/m,

f ITZt = 2.0MPa andGITZ

f = 30N/m, respectively. For all interfacesα = 1.0 is assumed.For the smallest sampleD = 50mm attention is focused on convergence for four

19

Table 2: Number of degrees of freedom for each sample size (number of staticallycondensed stress degree of freedom in brackets).

Sample size Coarse mesh Fine mesh

50mm 3342/1114 71496(+114324)100mm 9084 248080(+390204)200mm 28296 868936(+1364160)

0.5

2.5

2.0

0.0

1.5

1.0

0

1120 1100 1080 1060 1040 1020 1000

0.1

1e−04

6k

5k

4k

3k

2k

1k

0

0.01

1e−05

0.001

10 60 50 40 30 20

CST/DISP

GRADIENT/DISP

CST/TRAC

GRADIENT/TRAC

Number of iterations

Number of iterations Nominal Stress [MPa]

Relative Error

Elongation of Ls [µm]

notc

onve

rged

disp. magnification x103

disp. magnification x102

resNewton resGMRES

Figure 10: Model performance for 50mm specimen. Nominal stress-displacement re-sponse and cumulative number of iterations (top left). Deformed fine mesh at peakand post-peak (top right). Illustration of convergence of GRADIENT/TRAC analysis,demonstrating the influence of the dynamic tolerance adopted for the GMRES algo-rithm (bottom).

types of projection operators, as in the previous example, i.e. CST/DISP, CST/TRAC,GRADIENT/DISP and GRADIENT/TRAC, see Figure 10 and Table 2.In all casesthe preconditioned GMRES solver was used in combination with local arc-length con-trol and line searches as outlined in Section 0.3. The analysis using uniform tractionboundary conditions to compute the short-wave projection operator together with ele-ments (i.e. CST/TRAC) does not converge. The analyses with linear and quadratic dis-placement boundary conditions (i.e. CST/DISP and GRADIENT/DISP) exhibit goodconvergence. TheC 1-continuous elements with linear distribution of tractions (GRA-DIENT/TRAC) shows the fastest convergence. The relative number of iterations and

20

relative cumulative computational time are shown in Table 3. The GRADIENT/TRACanalysis, with 8 processors, is 10% faster than GRADIENT/DIS, whereas the cumula-tive number of iterations is 40% smaller. However, implementation of the second-orderscheme is not optimized and the authors believe that there ispotential to significantlyimprove efficiency.

Table 3: Relative number of iterations and relative cumulative computational time.Analysis Relative number iterations Relative cumulative time

GRADIENT/TRAC 1.0 1.0CST/TRAC not converged -

GRADIENT/DISP 1.3 1.2CST/DISP 1.4 1.1

In contrast to the results from the previous section (elastic materials), the analyseswith displacement boundary conditions (DISP) converged slower than for the second-order scheme with traction boundary conditions (GRADIENT/TRAC). The reasonsfor this difference is that where a crack crosses the fine meshpatch boundary, dis-placement fluctuation continuity is not enforced by the traction boundary conditions.Furthermore, the coarse mesh problem gives too stiff a response as a result of the resid-uals restricted from the fine mesh in the case of displacementboundary conditions.

10 20 30 40 50 60 0

3.0

2.5

2.0

1.5

1.0

0.5

0.0

50mm100mm200mm

Nominal Stress [MPa]

Elongation of Ls [µm]

Figure 11: Nominal stress vs displacement plots forD = 50mm, 100mm and 200mmdog bone specimens. Experimental peak stress and standard deviation also show.

Four random mesostructures were created for each of the D=50mm, D=100mm andD=200mm specimens. Figure 11 shows the nominal stress-displacement response forall 12 analyses. These results highlight a degree of scatterin the response due to therandomness of the fine-scale structure. Moreover, these numerical results support the

21

50mm

100mm

200mm

Figure 12: Fracture patterns forD = 50mm, 100mm and 200mm dog bone specimens.

experimental observation of van Vliet [4], that for a range of small sample sizes (i.e.50-200mm) no clear size effect for nominal strength is observed (i.e. reducing withsize). It can also be seen that the response becomes more brittle with increasing size.Figure 12 shows the crack patterns that developed, illustrating that the shape of thedeveloping crack is clearly influenced by the spatial arrangement of aggregates.

0.7 Final remarks

This paper presents a modelling strategy for simulating thebehaviour of fracturing het-erogeneous materials such as concrete where the fine-scale heterogeneities are fully re-solved. The fine-scale is modelled using hybrid-Trefftz stress elements, where cracksare restricted to element interfaces. This represents an efficient framework for mod-elling propagating cohesive cracking. The very large nonlinear system of algebraicequations that emerges from this fine scale resolution requires an efficient iterativesolver with a preconditioner that is appropriate for fracturing heterogeneous materials.A significant extension to the work of Miehe and Bayreuther (Miehe 2007) has beenproposed that constructs a preconditioner using a two-gridstrategy that utilizes thescale transition techniques derived for computational homogenization. Both displace-ment and traction boundary conditions are considered in construction of the projectionoperators for both first and second-order schemes. The proposed two-grid strategy hasbeen demonstrated for both elastic and fracturing heterogeneous materials, illustratingboth the convergence properties of the proposed scheme and efficiency of the parallelimplementation. The use of the preconditioned GMRES Kryloviterative method withdynamic tolerance in combination with a constrained Newtonmethod with local arc-length control and line searches represents a robust and efficient simulation framework.

22

The extension of the solution strategy to 3D is relatively straightforward and will bethe subject of a future paper.

Acknowledgements

The authors gratefully acknowledge the support of the UK Engineering and PhysicalSciences Research Council (Grant Ref: EP/D500273).

The authors also wish to thank Dr. Lee Margetts of The University of Manchesterfor access to HECToR (UK’s national supercomputing service) through the UK Engi-neering and Physical Sciences Research Council HECToR capability challenge grant(Grant Ref: EP/F055595/1).

Appendix A: Coarse-scale problem

For the strain-gradient continuum, it is assumed that the local variation of internal workat the coarse-scale can be expressed as

δW = σσσ : εεε+τττ...ηηη, (A-1)

where the first-order stressσσσ and second order stressτττ are work conjugate to the firstorder strainεεε and second-order strainηηη respectively. Following [32] the boundaryvalue problem for second-order continuum (leading to a fourth-order partial differentialequation) is expressed by the following equilibrium equation:

∇ · σσσ+ f = 0 (A-2)

where the real stress is defined as:

σσσ = σσσ+ ∇ ·τττ (A-3)

dofs

dofs

dofs

Ai

A j

Ak

p Ei

E j

Ek

A ji

A jk

L j

n

Figure 13:C 1-continuous element

When undertaking conventional displacement based finite element analysis of second-order (i.e. strain gradient) continua, the interpolation of displacements should exhibitat leastC 1-continuity of displacements to ensure convergence. An alternative is theuse of shape functions withC 0-continuity [33] with relaxed displacement gradients in-troduced as additional degrees of freedom, together with anadditional kinematic con-straint between displacement gradients. For practical computations involving second-order continua in two dimensions, Shu et al. [33] recommend the use of the nine-node

23

isoparametric quadrilateral element (QU34L4). Such an element is stable and foundto work well for nearly incompressible solids. However, this is not guaranteed for tri-angular (2D) and tetrahedral (3D) elements. Here an alternative approach is adoptedbased on an efficient triangular plate element withC 1-continuity [34]. This element ismodified for strain gradient continua and can easily be extended to 3D.

After [34], a triangular domainΩe is defined by three nodes - Figure 13. Thedisplacement functionu(Ωe) is required to vary at least quadratically along each edge,yielding linear or higher order polynomials for the first derivatives of displacement.Displacements and gradient of displacements along an edge cannot be influenced bydisplacements of the opposite node, in order to establish a conformalC 1-continuousfinite element.

To determine the displacements and displacement gradientsat point p, three fic-titious linesLi , L j andLk are defined that intersect atp and lie parallel to the edgesEi , E j andEk. Focussing on lineL j , to compute the displacements and displacementgradients at pointsA ji andA jk, first the nodal displacement gradient degrees of free-dom are mapped to the normal and tangential slopes of edgesEi andEk. Next a linearLagrange interpolation of the normal gradient∂nu, cubic Hermite interpolation of dis-placementsu and quadratic Hermite interpolation of the transverse gradient ∂su areapplied. The values determined atA ji andA jk are then interpolated in the same wayalong the imaginary lineL j . Finally, the weighted average of the values on each of theintersecting imaginary linesLi , L j andLk at pointp are determined from the normal-ized coordinates of pointp. It is worth noting that generalization of this procedure to3D is straightforward by first interpolating the nodal degrees of freedom on each faceof the tetrahedral element.

The stress-strain relationship can be expressed as (restricted here to linear elasticitywithout loss of generality)

σσσ = C1εεε+ C2ηηη, (A-4)

τττ = C3εεε+ C4ηηη (A-5)

where material stiffness operatorsC1-C4 are computed as a second-order extension [5]to the first-order formulation presented in Section 0.5.

In order to calculate the coarse-scale stiffness matrix fortheC 1-continuous finiteelement, an appropriate integration scheme should be adopted. However, determiningthe strain gradient at the integration points is a non-trivial task. Instead, the element’saverage strain and strain gradient are computed and used with a single point integra-tion rule. The element average strain and strain gradient can be defined by boundaryintegrals as

εεεel =12

Z

∂Ωe

x⊗u+ u⊗xdΓ, (A-6)

andηηηel =

Z

∂Ωe

x⊗ ∂xudΓ. (A-7)

which are determined using numerical integration.However, the resulting stiffness matrix is rank deficient due to the single point inte-

gration rule adopted and this leads to pathological zero energy deformation modes. Toovercome this problem, a stabilization matrix is proposed that is energetically orthog-onal toKc.

24

The strain field in the element is given by

εεε =12

[∇u+(∇u)T

]. (A-8)

The homogeneous part of this strain can be determined in terms of the element averagestrain and strain gradient

εεεhom= εεεel + x ·ηηηel (A-9)

Thus, the “stabilizing” strain can be defined as

εεεstab= εεε−εεεhom = (B−Bel)qc = Bstabqc (A-10)

whereB andBel are matrices relating the strains to nodal degrees of freedom for thestrain and average strain respectively. A-10 is then used toconstruct the stabilizationmatrix as

Kstab=

Z

Ωe

BstabC1BstabdΩ. (A-11)

Thus, the coarse-scale system of equations are written as

(Kc + ρKstab)qc = fc (A-12)

whereρ = 1 is a stabilization parameter used in all simulations.

25

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