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Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360www.elsevier.com/locate/cma

Adaptive wavelet-enriched hierarchical finite element model forpolycrystalline microstructures

Yan Azdoud1, Somnath Ghosh∗,2

Department of Civil, Mechanical and Materials Science & Engineering, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218,United States

Received 31 December 2016; received in revised form 17 April 2017; accepted 19 April 2017Available online 26 April 2017

Highlights

• Overcome shortcomings of FE and FFT methods for modeling polycrystalline microstructures.• Adaptive, wavelet enriched hierarchical FE method for computational efficiency and accuracy.• Lifted second-generation wavelets for hierarchical interpolation functions.• Significant error and convergence rate enhancement compared to uniformly refined FEM.

Abstract

This paper proposes an adaptive, wavelet-enriched hierarchical finite element model (FEM) to enhance computational efficiencyand solution accuracy for heterogeneous domains of anisotropic elastic materials. It is motivated by the need to overcomeshortcomings of conventional methods like the standard FEM or fast Fourier transformation (FFT) based method for analyzingdeformation in large polycrystalline microstructures. The multi-resolution wavelet functions are advantageous for selection of anoptimal set of functions that can adaptively enrich the solution space with a prescribed level of accuracy. The proposed methodintroduces a discretization space that conforms to the profile of the solution sought. The method engages a second generationfamily of wavelets with a lifting scheme to generate hierarchical interpolation functions. An iterative algorithm efficiently calculatesestimates of the solution from the previous iterate using a modified Jacobi method. The adaptive FE method performs significantlybetter than uniformly refined FEM in validation tests with focus on the convergence rate and error mitigation. A polycrystallinemicrostructure with elastically anisotropic grains is simulated by the adaptive wavelet-enriched hierarchical method, showing highconvergence rates.c⃝ 2017 Elsevier B.V. All rights reserved.

Keywords: Second-generation wavelets; Hierarchical finite elements; Adaptive method; Solution estimate; Anisotropic elasticity

∗ Corresponding author. Fax: +1 410 516 7473.E-mail address: sghosh20@jhu.edu (S. Ghosh).

1 Post-doctoral Researcher.2 M.G. Callas Professor.

http://dx.doi.org/10.1016/j.cma.2017.04.0180045-7825/ c⃝ 2017 Elsevier B.V. All rights reserved.

338 Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360

1. Introduction

Image-based modeling of polycrystalline microstructures, often using finite element models, is increasingly usedfor predicting microstructure-property relationships of metallic materials [1–5]. These models involve generationof virtual microstructures capturing details of microstructural features, e.g. grain morphology, crystallographicorientations etc., followed by discretization into finite element mesh. Full field analysis, representing short and longrange grain interactions in the microstructure, is subsequently conducted to predict various deformation and failuremechanisms. Representation of the complex 3D microstructures, along with deformation and failure phenomena oftenrequires very high resolution in the 3D computational domain with high density finite element meshes. Coupled withthe complexities of constitutive models, the high resolution model requirements result in very high computationalefforts, often challenging current computational capabilities. This has prompted the development of alternativecomputational methods for simulating deformation in polycrystal aggregates.

The elasto-visco-plastic self-consistent method has been developed in [6,7] for modeling polycrystalline mi-crostructures, where each grain is treated as an ellipsoidal inclusion embedded in a homogeneous medium representingthe averaged behavior of all other grains. A fast Fourier transform (FFT) based method has been introduced forperiodic, linear and nonlinear heterogeneous materials in [8,9]. This method has gained popularity because of itssimplicity of implementation and use. It is based on a point-wise fast Fourier transformed representation and solutionof the equilibrium equations with strain compatibility constraints for periodic microstructures. The FFT method hasbeen extended to full-field crystal plasticity simulations using a Green’s function method in [10–12]. For the samespatial resolution, the FFT method has been shown to be very efficient in comparison with standard FEM for someproblems. However for problems involving localization, methods involving periodic global interpolations can sufferfrom low convergence rates and large truncation errors.

The present paper develops an adaptive hierarchical enrichment method to augment the computational efficiencyand accuracy of FE analyses of polycrystalline microstructures. Elastic material properties are considered in thispaper as a precursor to crystal plasticity. Adaptive finite element techniques have evolved over the years to optimizethe overall computational effort and provide a measure of reliability through error indicators. Various methods, suchas the h-method of mesh refinement [13–15], p-method of basis enrichment [16,17], and their combination in theh-p-method [18–20] have been proposed. The h-p-method of adaptation has proved to be very effective for a wideclass of problems with localization. More recently, the generalized FEM or GFEM type methods with global–localenrichment have been developed in e.g. [21,22] to enrich the global solution space using solutions of the local problem.All of these methods define error indicator functions, whose intensity is used to trigger mesh refinement or enrichmentof interpolation functions [23,24]. In many of the above methods, the mesh enhancement strategy does not guaranteeconformity of the new discretized/enriched space to the profile of the solution. Furthermore, old shape functions arenot preserved in the new enriched solution space, which adds to the difficulty of mapping internal variables.

1.1. Synopsis of the wavelet enhanced hierarchical method

The proposed adaptive method overcomes these shortcomings by developing a wavelet-based enrichment strategy.Wavelet methods can project any given field onto a set of scaling and multi-resolution wavelet bases. Thescaling functions span an approximation space for the field at a given scale, while wavelet functions augment theapproximated field by estimating residuals at higher scales. The multi-resolution property is particularly advantageousfor approximating a field using a minimal set of wavelet basis functions. The wavelet decomposition of an estimate ofthe solution allows for the selection of an optimal set of functions (in the sense of the wavelet compression method)that can then adaptively enrich the solution space with a prescribed level of accuracy. This property is depicted inthe error plot of Fig. 1. It is a significant advantage over other conventional methods, for which the enrichment is notrelated to the conformity to the solution profile. An elastic simulation of a heterogeneous beam under compression,detailed in Section 5.2, is performed to generate Fig. 1. The error eu(x) =

|u(x)−u f (x)||u f (x)|max

, which corresponds to the relativeerror in the absolute value of the displacement, is plotted in Fig. 1. Here u f (x) is a reference displacement solution andu(x) are the coarse mesh and wavelet-enhanced solutions with two-scale and four-scale augmentations respectively.Suitable estimates of the solution may be selected such that their profile is compatible with the enriched solutionssought.

An important criterion in the selection of the family of wavelet functions is its compatibility with the finite elementframework. In the Galerkin FE approach, the set of wavelet and scaling functions must constitute an admissible set

Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360 339

Fig. 1. Comparison of percentage error eu(x) =|u(x)−u f (x)||u f (x)|max

× 100(%) for the wavelet enhanced displacement solutions of an elastic heterogeneousbeam under compression, discussed in Section 5.2.

of shape functions, when associated with standard element shape functions. Galerkin-Wavelet methods have beendeveloped to use wavelet functions shape functions in [25,26]. Although successful with 1D and 2D implementations,these methods have faced difficulties with numerical quadrature, irregular mesh generation and definition nearinterfaces. Their use in modeling heterogeneous microstructures with non-linear material properties is consequentlya challenge. Alternatively, the second generation family of wavelets [27] is used in this paper for developing theadaptive enrichment formulation. These wavelets can be procedurally generated from hierarchical shape functionsusing the so-called lifting scheme. Given scaling functions at a coarse and fine scale, the lifting scheme defines theset of wavelet functions that complement the coarser set of interpolation functions to uniquely project any functiondecomposed on the finer set. Complex irregular meshes are trivially constructed with these wavelets, which makethem ideal candidates for enrichment functions in the wavelet-enriched hierarchical finite element method developedin this paper.

This paper begins with a comparison study of the FFT and FE Methods for an elastic polycrystalline microstructurein Section 2. The formulation of the adaptive wavelet-enriched hierarchical FE method is developed in Section 3, thatis followed by implementation strategy for 3D problems in Section 4. Validation studies of the adaptive method withrespect to convergence rates are conducted in Section 5, and results of simulations of a polycrystalline microstructureare presented in 6. The paper concludes with a summary in Section 7.

2. A comparison study of the FFT and FE methods for an elastic polycrystalline microstructure

This section simulates a polycrystalline microstructure of an anisotropic elastic material, shown in Fig. 2(a), by theFFT [28,29] and a conventional FE method for comparing their accuracy and efficiency. Similar comparison studieshave been conducted in [30–32] for crystalline elasto-viscoplastic models, but did not focus on conforming meshesfor the FE method. The computational domain is a 6.4 × 6.4 × 6.4 µm cubic microstructural volume consists of 25grains of average grain size 2.5 µm. Each grain is assumed to have a hexagonal close packed or hcp lattice structure.The crystallographic lattice misorientation between grains is depicted in the contour plots of Fig. 2(a). The elasticproperties are for a transversely isotropic material with stiffness components (expressed in a material coordinatesystem) as: C11 = 170 GPa, C12 = 98 GPa, C13 = 86 GPa, C11 = 170 GPa, C33 = 204 GPa, C55 = 51 GPa.The cubic computational domain is subjected to a uniaxial macroscopic strain ⟨ε11⟩ = 0.05 with periodic boundaryconditions. The FFT method is generally limited to periodic microstructures and periodic boundary conditions. TheFFT model is simulated using the CraFT:1.0.12 code [33] based on the formulations in [29,34,35]. For convergencein FFT simulations, five different voxel grids ranging from 83 to 1283 cells are generated as shown in Fig. 2(b). Thefinite element simulations are conducted using the ABAQUS/Standard code [36], using a conforming mesh of 4-nodedlinear tetrahedral elements, ranging from 11 016 elements (2308 nodes) to 5 640 192 elements (962 281 nodes) shownin Fig. 2(c).

340 Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360

Fig. 2. (a) Computational model of a polycrystalline microstructure with 25 hcp grains showing lattice misorientation distribution, modeled byFFT and FEM methods, (b) the FFT grid with 1283 grid points obtained by rasterization on a Cartesian voxel grid, and (c) the conforming FE meshwith 5 640 192 tetrahedral elements.

Fig. 3. Comparison of: (a) CPU time and (b) convergence rate, for simulations using the FFT CraFT:1.0.12 code [33] and the FEMABAQUS/Standard code [36]. The percentage of the relative L2 norm of the error in stress eσ is used.

The CPU time for simulations by the FFT and FE methods are plotted as a function of the number of DoFs inFig. 3(a). For this linear problem, the FFT simulations have a one to four orders of magnitude efficiency advantage overthe FEM simulations. This is the direct consequence of the number of operations in the solution. For a problem withN degrees of freedom, the FFT method requires O(N log(N )) operations, while FEM with a direct solver generallyrequires O(N 3) operations. For an iterative solver, this can be improved to O(N 2 log(N )) operations.

Fig. 3(b) plots the solution convergence rate as a function of degrees of freedom, for the relative L2 error norm ofstress on a log–log graph. The ordinate in this plot corresponds to the percentage error with respect to the referencesolution, given in terms of the L2 norm of stress on the domain of interest Ω as:

eσ =∥σ (x) − σ f (x)∥2

∥σ f (x)∥2 × 100(%), where ∥σ (x)∥2=

∫x∈Ω

σ (x) · σ (x)dΩ . (1)

The reference solutions σ f (x) for the FFT and FEM models correspond to results with the finest grid and themaximum number of elements, as shown in Fig. 2(b) and (c) respectively. The FEM solution has a faster convergencerate of O(N−0.52) in comparison with that of the FFT method O(N−0.32). The lower convergence rate for the FFTsimulations is due to stress discontinuities in the heterogeneous polycrystalline microstructure. The approximationspace in the FFT method is a finite set of trigonometric polynomials, for which truncation introduces Gibbs

Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360 341

Fig. 4. Contour plots of the relative error field elocal (x) for (a) FFT with 262 144 sampling points, and (b) the FEM with 123 349 nodes results.

instabilities. Projections on a uniform grid cause some inherent inaccuracies in regions of discontinuities or largegradients.

The convergence rate of FEM simulations is lower than the theoretical rate of O(N−1) for well posed problems,due to the fact that the periodic boundary condition constraints depend on the resolution. The convergence rate of FFTsimulations is much less than the theoretical exponential rate for well posed problems, because of Gibbs instabilitiesemanating from series truncation as well as stress discontinuities in the microstructure. A theoretical convergence rateof O(N−1/2) has been demonstrated for solution fields with discontinuous derivatives in [37]. This outcome can besomewhat alleviated with proper Gibbs regularization schemes discussed in [37,38].

Fig. 4 depicts contour plots of the local relative value of the error in stress, given by:

elocal(x) =

√[σ (x) − σ f (x)

]2[σ f (x)

]2 (2)

both for the FFT and FE methods. Isosurfaces, corresponding to 2% relative error in the microstructure, are shown.For FEM, the error is higher near the domain boundary, since periodic boundary conditions are altered with themore refined mesh. In the interior, the error is clustered near multiple grains intersections, where elements areshared by more than two grains. The FFT result shows a larger relative error that is clustered with striations onthe grain boundaries. This is due to the error induced through non-conformity of the uniform sampling grid to themicrostructure, which introduces aliasing. The FFT simulations result in a sub-optimal convergence rate in the stressof ∼O(N−1/3).

Finally the loading direction stress σ11 by the two methods are plotted in Fig. 5 along a line in the x2 direction. Thisline cuts through three grains having large misorientation at x2 = 2.1 µm, 4.2 µm and 6.3 µm respectively. The plotsare results for coarse and fine discretization resolutions, in each of the FE and the FFT methods. While both the FFTand FEM stress results are close to the converged values, high frequency Gibbs instabilities are seen in the FFT resultsnear the grain boundaries. The differences in the FFT and FEM results are due to non-conformity to the microstructureand interfacial Gibbs instability for the FFT method and over-constraining periodic boundary conditions for FEM.Note that while the local error reduces with a finer FFT grid, the amplitude of the Gibbs instabilities at the interfacesdoes not. This can have serious consequences in non-linear problems where localization is an important factor.

This paper is aimed at overcoming the shortcomings encountered with both these methods through the use ofwavelet basis enhanced adaptive hierarchical enrichment of the solution fields.

3. The adaptive solution enrichment method

Most adaptive strategies e.g. in [39–41] are comprised of identification and evaluation of discretization errorindicators on the mesh, which is followed by mesh refinement (h-adaptation) and/or basis function enrichment

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Fig. 5. Comparison of the stress component σ11(x2) (MPa) along a x2 line with coordinates x1 = 2.2 µm and x3 = 3.2 µm by (a) a coarseFFT mesh ( 643 elements), (b) a fine FFT mesh ( 1283 elements) and (c) a coarse FEM mesh (75 880 elements), with a fine FEM mesh (607 040elements) respectively.

(p-adaptation) to improve the solution quality. Rather than building on an error indicator, the proposed adaptivewavelet-enrichment method focuses on generating a suite of basis functions that directly conform to the profile of thesolution estimate.

Let Ω ⊂ R3 denote an open bounded domain with Lipschitz boundary Γ = Γu⋃

Γt , where Γu and Γt correspondto Dirichlet and Neumann boundaries respectively. The weak form of the boundary value problem seeks the solutionu ∈ V (Ω ) such that:

a(u, v) = b(v) ∀v ∈ Vo(Ω ) (3)

where a(·, ·) is a symmetric bilinear form on V (Ω ) × V (Ω ), b is a linear functional corresponding to the loadingfunction. The trial space Vo(Ω ) corresponds to the Sobolev space of functions from H 1(Ω ), whose trace vanishes onthe Dirichlet boundary such that

Vo = v ∈ H 1(Ω ) : v = 0 on Γu. (4)

Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360 343

For small deformation linear elasticity, the bilinear and linear functions are expressed by the following equations:

a(u, v) =

∫Ω

σ (u(x)) : ε(v(x))dΩ (5)

b(v) =

∫Ω

f(x) · v(x)dΩ +

∫ΓT

t(x) · v(x)dΓ (6)

with

σ (u(x)) = C(x) : ε(u(x)) ∀x ∈ Ω (7)

ε(u(x)) =12

[∇u(x) + (∇u(x))T ] ∀x ∈ Ω (8)

where σ and ε are the stress and strain tensors respectively, C is the stiffness tensor, f describes the body force and tis the boundary traction.

In the Galerkin finite element formulation, the approximate solution on the space of piece-wise polynomialfunctions uh

∈ V h satisfies the weak form of Eq. (3) as:

a(uh, vh) = b(vh) ∀vh∈ V h

o (Ω ). (9)

Here the space of the approximate trial functions V h(Ω ) ⊂ V (Ω ). Assuming that the mesh of discrete elements coversthe entire domain Ω completely, the solution uh(x) is an interpolated approximation of u(x) on the family N

m of minterpolation functions Nα given as:

uh(x) =

m∑α

uhα Nα(x) ∀x ∈ Ω (10)

where m corresponds to the number of nodes in the domain. The interpolation basis constructed with the set N m is

a standard finite element basis that follows partition of unity. The interpolant uhα is given by the nodal values of uh(x),

such that uhα = uh(xα). In this paper, the interpolation function Nα for a given node α is defined over the entire domain

Ω , with the understanding that its value is zero outside of the patch of elements associated with the node. Moreover,the hierarchy of interpolation functions that are implemented in this work are linear in nature.

3.1. Adaptive hierarchical enrichment method

The adaptive method introduces a set of enrichment functions ϕmenr in the hierarchy, which expand the

discretization space V h(Ω ) to an enriched space V henr (Ω ) ⊃ V h(Ω ), while preserving the original set of discretizationfunctions N

m . menr corresponds to the number of additional enrichment nodes that are hierarchically added to theinitial number m. The enriched solution of the approximate weak form on V henr (Ω ) is sought of the form:

uhenr (x) =

m∑α

uhenrα Nα(x) +

menr∑β

wβϕβ(x) ∀x ∈ Ω . (11)

The new set of discretization functions N m

∪ ϕmenr should constitute a complete set of linearly independent basis

functions in Ω . The new basis N m

∪ ϕmenr is not an interpolation basis in the sense that wβ is not the nodal value

of uhenr (xβ). Linear hierarchical enrichment functions are chosen in this adaptation process to enhance the solutionwith optimal increase in the degrees of freedom. Each enrichment function ϕβ should satisfy the delta property witheach other as well as with the set N

m . The enriched space with linear hierarchical enrichment functions constitutethe basis of the hierarchical FEM [42]. Hierarchical enrichments, in which each enrichment degree of freedom maybe associated with a displacement correction, do not necessarily satisfy partition of unity. However, it can lead tosignificant enhancement of convergence rates and matrix conditioning as demonstrated in [42]. Assume that the setφ

n is an arbitrarily large (n → ∞) and sufficient set of multi-scale hierarchical enrichment functions for the coarsediscretization space N

m . The adaptation process is stated as:Find the optimal set ϕ

menr ⊂ φn such that:

∥u − uhenr ∥ ≤ ϵ (12)

344 Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360

where ∥ · ∥ is the L2 norm, u(x) is the exact solution, ehenr (x) = u(x) − uhenr (x) is the approximation error and ϵ is anassigned tolerance. For an arbitrarily small value of ϵ, a finite set of functions ϕ

menr ⊂ φn is expected to satisfy

the inequality (12).In the absence of a known exact solution u, the adaptation process is based on an estimate of the solution u. The

role of this estimate u is only to find the enrichment basis functions and the corresponding additional nodes. Oncethe enrichment basis functions are established, the adapted enriched solution uhenr is generated by solving the weakform of the entire boundary value problem. An iterative approach is adopted to accomplish this estimate-enrichmentprocess as described below.

• Step 1: Initialize:

uhenr(0) = uh , ϕ

menr(0) = ∅. (13)

• Step 2: Iterate for solution estimate. For iteration step (i + 1):

– Compute the estimate u

∗ Define the basis φp

⊂ φn as the next scale of possible hierarchical enrichment functions on which

u is sought.∗ Compute an efficient approximate of u on the basis N

m∪ ϕ

menr(i) ∪ φ

p.

– Select the new enrichment basis ϕmenr(i+1)

∗ Identify a minimal set ϕmenr(i+1) ⊂ ϕ

menr(i) ∪φ

p, such that the projection P(u) of u on N m

∪ϕmenr(i+1)

admits the inequality for a given tolerance ζ :

∥P(u) − u∥ ≤ ζ. (14)

– Solve for the hierarchically enriched solution uhenr(i+1)

a(uhenr(i+1), vh

i+1) = b(vhi+1). (15)

such that uhenr(i+1) ∈ V henr

(i+1) = spanN1..Nm, ϕ(i+1)1 ..ϕ(i+1)

menr

• Step 3: Stop iteration following the criterion:

∃ j ≤ i | ϕmenr(i+1) = ϕ

menr( j) . (16)

Eq. (14) for the enrichment basis differs from Eq. (12), as it does not invoke the exact solution u, nor does itcalculate uhenr during the enrichment basis selection. The rationale is that selecting a basis of enrichment functionson which P(u) approximates u with the tolerance ζ ensures that this enrichment basis improves the current solutionuhenr

(i+1). For instance, let the estimate differ from the exact solution by a tolerance ξ such that ∥u − u∥ = ξ . TheCauchy–Schwarz inequality yields:

∥P(u) − u∥ = ∥P(u) − u + u − u∥ ≤ ∥P(u) − u∥ + ∥u − u∥, or∥P(u) − u∥ ≤ ζ + ξ.

(17)

The Lax–Milgram theorem ensures that the solution obtained in a space V henr is its best approximation with respectto the L2 norm. Hence, the solution uhenr

(i+1) of Eq. (15) is a better approximation of the solution u than P(u):

∥uhenr(i+1) − u∥ ≤ ∥P(u) − u∥ ≤ ζ + ξ. (18)

This has the same form as Eq. (12) with ϵ = ζ + ξ . For the iterative scheme to converge, a necessary condition isthat the estimate u is a better approximation of the solution u than the previous enriched solution uhenr

(i) . This is anapproach to define an upper bound of the approximation error.

The stopping criterion terminates the enrichment process when the set of enrichment functions no longer evolves.Unrefinement is possible with this algorithm, since the minimal set ϕ

menr(i+1) may not contain all the functions selected

in the set ϕmenr(i) . The next section discusses a method to compute the estimate u consistent with the exact solution

profile.

Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360 345

3.2. A computationally efficient estimate of the exact solution

This step involves a computationally efficient method of determining the solution estimate u(x) = G(uhenr(i) , C, x)

∀x ∈ Ω in every iteration of step 2 in the table above. Here G denotes the functional dependency of u on uhenr(i) , C and

x. The algorithm must avoid solving the weak form of a boundary value problem in every iteration to obtain u. Theestimate must moreover satisfy the following condition:

∥u − u∥ ≤ ∥u − uhenr(i) ∥. (19)

An algorithm that uses the modified Jacobi method as discussed in [43] is developed for the evaluation of thesolution estimate. Let uhenr

(i) be the solution to the discrete form of the elastic problem in the previous iterate, writtenas:

[K(i)]

uhenr(i)

=b(i)

(20)

where the symmetric stiffness matrix [K(i)] and the load vectorb(i)

are given as:

[K(i)] =

∫Ω

[B(i)]T [C][B(i)]dΩ and (21)b(i)

=

∫Ω

[N(i)]Tf dΩ +

∫ΓT

[N(i)]Tt dΓ .

Here [K(i)] is the 3(m + menr ) × 3(m + menr ) stiffness matrix andb(i)

is a 3(m + menr ) × 1 load vector. The matrix[B(i)] contains derivatives of interpolation functions, while the matrix [N(i)] contains the interpolation functions withiterative hierarchical augmentation. Both have a compact support over Ω . For simplification of the formulation theseoperators are defined on all the degrees of freedom despite being null outside of their support. The discretization ofu(x) for evaluating the solution estimate in the enhanced interpolation space is given by:

u(x) =

m+menr∑α

uα(i) Nα(i)(x) +

p∑β

qβφβ(x) (22)

withm+menr∑

α

uα(i) Nα(i)(x) =

m∑α

uα Nα(x) +

menr∑γ

wγ ϕγ (i)(x). (23)

Correspondingly, the discrete form of Eq. (9) for the elastic problem is written with this enhancement as:[[K(i)] [C]T

[C] [G]

]u(i)

q

=

bubq (24)

where

[G] =

∫Ω

[Θ]T [C][Θ]dΩ , [C] =

∫Ω

[Θ]T [C]([B(i)])dΩ (25)bq

=

∫Ω

[φ]Tf dΩ +

∫ΓT

[φ]Tt dΓ and

bu

=

∫Ω

[N(i)]Tf dΩ +

∫ΓT

[N(i)]Tt dΓ

where [Θ] is the matrix of derivatives of the enrichment interpolation functions φp, [G] is a symmetric 3p × 3p

matrix and [C] is a 3p × 3(m + menr ) matrix. The elastic problem for u has an additional 3p degrees of freedom q

associated with the hierarchical interpolation functions φp.

The solution uhenr(i) (x) of the previous iteration problem in Eq. (20) may be used to generate an efficient estimate of

the coefficients uα(i), i.e.

uα(i) ≈ uhenrα(i) , where uhenr

(i) (x) =

m+menr∑α

uhenrα(i) Nα(i)(x). (26)

346 Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360

From the coefficients uhenrα(i) , approximate values of qβ are extracted as:

q j ∼

(bq

j −

3(m+menr )∑k=1

C jkuhenrk(i)

)/G j j (27)

where j and k are matrix indexes ranging from 1 to 3p and 1 to 3(m + menr ), respectively. This step is an efficientway of computing the solution estimate that is computationally less expensive than solving the complete problemin Eq. (24). It corresponds to the first iteration of the Jacobi method, discussed in [42,43] for hierarchical finiteelements. The estimate u is found by introducing this approximation of qβ in Eq. (22). Since the Jacobi methodconverges monotonically, the solution u is a better approximation to u than the initial guess uhenr

(i) .An alternate estimate u∗ can also be derived from u for use in the hierarchical enrichment of basis functions. This

uses only the diagonal terms of the stiffness matrix in Eq. (24), expressed as:

u∗(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

m+menr∑α=1

[K3α−2,3α−2(uhenr3α−2(i))

2]Nα(i)(x) +

p∑β=1

[G3β−2,3β−2q23β−2]φβ(x)

m+menr∑α=1

[K3α−1,3α−1(uhenr3α−1(i))

2]Nα(i)(x) +

p∑β=1

[G3β−1,3β−1q23β−1]φβ(x)

m+menr∑α=1

[K3α,3α(uhenr3α(i))

2]Nα(i)(x) +

p∑β=1

[G3β,3βq23β]φβ(x)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭. (28)

This estimate uses the diagonal terms as weights of the energy attributed to each degrees of freedom. Both estimatesu and u∗ are used in the iterative method of Section 3.1.

3.3. Wavelet basis functions for solution enrichment

This section explores the use of wavelet functions for providing an optimal basis of hierarchical enrichmentfunctions that conform to the profile of the estimate u in Eq. (14). Wavelet basis functions possess properties thatare ideal for optimal multi-scale enrichment functions, as given below.

• Compact support: Each wavelet basis has a compact support, spanning a subdomain of Ω . Consequently,wavelet-based solutions do not exhibit spurious instabilities such as the Gibbs phenomena.

• Multi-resolution: Wavelet functions have a multi-resolution characteristic, representing the difference betweentwo consecutive scales. This allows for an optimal multi-scale enrichment basis. For a given resolution of thesolution, the space of basis functions is well defined and finite.

• Compatibility with FE discretization: Use of second-generation wavelets [27,44] allows for the developmentof a scheme that is compatible with any irregular hierarchical FE mesh. Using FE-compatible wavelets forenrichment avoids conformity errors.

The following criteria are followed to generate a robust wavelet-based hierarchical finite element model.

• Reisz basis: Wavelet functions with the Reisz basis property can detect differences between scales on the entiredomain Ω and avoid aliasing.

• Vanishing moments: The integral of the wavelet functions over their respective support should be zero, so that asmall coefficient induces a negligible contribution.

• Hierarchical characteristics: The wavelet family should be based on the hierarchical FE interpolation function.

Second generation wavelets e.g. in [27,44] show conformity to the above criteria needed for the enrichment functions.Wavelet enrichment functions at different scales can be used to construct the estimate u in Eq. (22). Assuming that

φ j corresponds to the enrichment basis with p( j) enrichment functions, the l-scale estimate is written as:

ul(x) = u0(x) +

l∑j=1

p( j)∑β

q jβφ

jβ(x) (29)

Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360 347

where the interpolation at scale 0 corresponds to the coarse-scale solution, given as:

u0(x) =

m∑α

uα N 0α (x) (30)

ul denotes the interpolation of u at the current scale l. The estimate ul at scale l can also be defined in terms of thek-scale estimate uk

: ∀ 0 ≤ k ≤ l − 1:

ul(x) = u0(x) +

k∑j=1

p( j)∑β

q jβφ

jβ(x) +

l∑j=k+1

p( j)∑β

q jβφ

jβ(x)

= uk(x) +

l∑j=k+1

p( j)∑β

q jβφ

jβ(x). (31)

Since hierarchical enrichment functions may be used on the initial mesh, a standard finite element representation ofuk(x) ∀ 0 ≤ k ≤ l may be invoked. This suggests that there exists a k-scale basis N k

αp( j) ( p( j) = m +

∑j≤k p( j)),

for which:

uk(x) =

p( j)∑α

ukα N k

α (x). (32)

The formalism used in these equations is the one used for interpolation wavelet functions. It considers the standardFEM shape functions as scaling functions and hierarchical shape functions as wavelet functions. Using hierarchicalshape functions as wavelets is however not optimal as they do not follow the criteria stated above. This can be remediedthrough the use of second generation wavelets [44]. The so-called lazy wavelets ϕl−1

β are created from the hierarchicalshape functions:

ϕl−1β (x) = αφl

β(x) (33)

where α is a constant. Conventionally the wavelet-scale is associated with the lower scale of scaling functions.However, the set of functions ϕl−1

p(l) do not constitute a Reisz basis of L2(Ω ), and they do not have a zero

mean, as required by the criteria stated above. To acquire these properties, the interpolation wavelets are updatedby a lifting scheme discussed in [27]. The lifting scheme adds a set of R scaling functions, where R ≤ p( j) andp( j) = m +

∑j<l p( j), to the l − 1-scale wavelet function:

ϕl−1β (x) = ϕl−1

β (x) −

R∑λ

aλN l−1λ (x). (34)

The coefficients aλ are chosen to produce vanishing moments and ensure that the wavelets have zero mean, i.e.∫Ω

ϕl−1β (x)dΩ = 0 ∀β ∈ [1, p(l)]. (35)

Adding the functions N l−1λ extends the compact support of the wavelet function ϕl−1

β to the whole domain, or

p(l)⋃β

supp(ϕl−1β ) = Ω . (36)

In practice, the lifting scheme with R = 2 is sufficient to obtain all the properties mentioned above. Eq. (36) makes thebasis ϕl−1

p(l) a Reisz basis of L2(Ω ). The lifting scheme for a 1D domain is presented in Fig. 6 with R = 2, α = 2

and aλ = 0.5. The support of ϕ01 extends to x ∈ [−1.5, 1.5] while ϕ0

1 is non-zero for x ∈ [−0.5, 0.5]. Furthermore,∫ 1.5−1.5 ϕ0

1 (x)dx = 0.The lifting method adds vanishing moments to the wavelets and improves the accuracy of the polynomial wavelet

enrichment functions. The l-scale solution estimates, enriched by the lifted wavelet functions ϕl−1β and scaling

348 Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360

Fig. 6. Generating wavelet bases with the lifting scheme in a 1D hierarchical finite element mesh.

functions N l−1α , are given as:

ul(x) =

p( j)∑α

cl−1α N l−1

α (x) +

p(l)∑β

dl−1β ϕl−1

β (x) (37)

ul(x) =

m∑α

c0α N 0

α (x) +

l∑j

p( j)∑β

d j−1β ϕ

j−1β (x). (38)

The coefficients d and c are evaluated by wavelet transformation, such that clα = ul

α at the finest scale l. The secondgeneration wavelet method allows fast wavelet transform for rapid computation of coefficients without explicitlydefining the wavelet profile. Using a lifting scheme based on polynomial interpolation between odd and even pointsover a 1D regular grid, the wavelet transform induced coefficients are given as:

d jk =

1α

(c j+1

2k+1 −

Q∑l

ωjk,lc

j+12k+2l

), c j

k = c j+12k +

Q∑l

ωjk,ld

j+1k+l . (39)

The coefficients of inverse wavelet transforms are expressed as:

c j+12k = c j

k −

Q∑l

ωjk,ld

j+1k+l , c j+1

2k+1 = αd jk +

Q∑l

ωjk,lc

j+12k+2l (40)

where ωjk,l and ω

jk,l are interpolation coefficients related to aλ and to the interpolation scheme, respectively.

A key property of the wavelet transformation induced decomposition is that wavelets measure the differencebetween two consecutive scales. Since the set of functions ϕl−1

β p(l) is a Reisz basis of L2(Ω ), there exists a constant

C l≥ 0 such that:

p(l)∑β

al−1β ϕl−1

β

2

≤ C l

( p(l)∑1

|al−1β |

2

). (41)

Introducing ϵ as a vector for tolerance in Eq. (38), it yields

ul(x) − P(ul)(x) ≤

l∑j

p( j)∑β | d j−1

β ≤ϵ

ϵϕj−1β (x) (42)

Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360 349

with the projection in Eq. (14) given as:

P(ul)(x) =

m∑α

c0α N 0

α (x) +

l∑j

p( j)∑β | d j−1

β >ϵ

d j−1β ϕ

j−1β (x). (43)

Using Eqs. (41) and (42), an upper bound of the error is found to approximate ul by ignoring wavelets, for whichcoefficients are lower than the magnitude of ϵ, i.e.

∥ul(x) − P(ul)(x)∥2≤ C2|ϵ|

2. (44)

An upper bound for C2 is given by C2 ≤∑l

j Cj p( j), where p( j) is the number of wavelet bases that are omitted at

scale j . Recall that ul= u and the scale index may be dropped. Eq. (44) corresponds to the critical equation (14) in

the iterative algorithm with ζ =√

C2|ϵ|.To interrogate practical basis functions for enrichment of the FEM simulation, Eq. (43) is modified with Eqs. (33)

and (34) with the wavelet functions replaced by hierarchical interpolation functions and standard shape functions.This yields:

P(u)(x) =

m∑α

c0α N 0

α (x) +

l∑j

p( j)∑β | d j−1

β >ϵ

d j−1β

(αφ

jβ(x) −

R∑r

aλN j−1λ (x)

). (45)

Standard FEM shape functions can be exchanged with equivalent hierarchical FEM shape functions as:

∀ j ∈ [1, l], ∀β ∈ [1, p( j)] if xα = xβ then φjβ(x) = N j

α (x). (46)

Eq. (45) is rewritten as:

P(u)(x) =

m∑α

e0α N 0

α (x) +

l∑j

menr ( j)∑β

f jβφ

jβ(x) (47)

with e0α and f j

β expressed as functions of c and d. These coefficients are not important for the method, and only the setof enrichment functions is sought. Eq. (47) provides a set of menr =

∑lj menr ( j) enrichment functions φ

jβ(x). This set

is used as the next scale of enrichment function such that:

ϕmenr =

l⋃j

φ j

menr ( j). (48)

This set is optimal in the sense of the wavelet compression and satisfies a crucial part of the iterative adaptive algorithmdiscussed in Section 3.1.

4. 3D implementation of the wavelet enriched adaptive algorithm

Steps in the 3D implementation of the algorithm described in Section 3 are summarized in this section.

4.1. Hierarchical mesh for wavelet enrichment

: Mesh generation. Mesh generation for the hierarchical linear interpolation function is described in this section. Thehierarchical, multi-scale wavelet enrichment calls for a nested mesh that is constructed from the original FEM coarsemesh by successive element refinement. For a typical four-noded tetrahedral element shown in Fig. 7, nested gridsare constructed by subdividing the element into 8 tetrahedra by inserting nodes at the middle of each edge of thetetrahedron. For each edge, the new inserted node is called child and the two nodes at the vertices of the edge arecalled parents. This construct is important for obtaining nodal connectivity tables necessary to perform the wavelettransform scheme. The refinement strategy does not preserve the aspect ratio, since the inner four tetrahedra sharean inner diagonal. This is however mitigated by always selecting the smallest inner diagonal when subdividing. Theaspect ratio in such a method does not degenerate but disparities in the element sizes exist. This method gives best

350 Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360

Fig. 7. Subdivision of a tetrahedron for the nested mesh in wavelet enrichment. The segments between two capital letters (e.g. AB) denote the edgeof the parent tetrahedron, and the other segments (e.g. ab) denote the edges of the nested children tetrahedra.

results if the initial seed mesh has good aspect ratio and of relatively uniform element size. This nesting method leadsto a straightforward wavelet transform method.

: Quadrature. As discussed before, each enrichment function is constructed with standard linear hierarchical shapefunctions that is composed of standard FE shape functions at different scales. The hierarchical shape functions arelinear on the tetrahedral elements at a given scale. This simplifies the quadrature rule. The integration is handled bya conventional one Gauss point Gauss–Legendre quadrature. However, hierarchical finite element construct involvescomplex discretization function connectivity, where the matrix assembly requires computing integrals of products ofspatial derivatives at dissimilar scales. For instance, in Fig. 7, the spatial derivative of the discretization function atnode A is constant on the entire large tetrahedron ABC D, while the spatial derivative of the discretization function atthe node a is not. It is constant only on a part of the domain. Therefore, integration of the product of these derivativeson element ABC D should be decomposed and computed on the elements for which the derivative of the discretizationfunction at node a is constant and non-null, i.e. on elements Aa f d , ade f , abe f and aebB. On account of this, thehierarchical finite element matrix assembly cannot be easily obtained by a superposition of elemental matrices asfor standard FEM, since the connectivity of shape function in hierarchical FEM is not straightforward. Instead, theassembly is easily obtained using a ‘node-based’ method using the children–parent node relationship. This assemblymethod has the added benefit of being highly parallel, since it can be assembled line by line.

4.2. Generating the first vanishing moment wavelet functions

Fig. 7 presents a subdivided linear tetrahedron with two scales, l = 0 and l = 1. In Fig. 7, the coordinates of nodeson the edge BC of the original large tetrahedron are denoted by x l

j , where l corresponds to the scale of the node and jis the index of the nodal coordinate. Each node is associated with a hierarchical shape function φl

j that have analogousindicial notation.

The simplest wavelet generated by the lifting method is the first vanishing moment wavelet, which guarantees thatthe wavelet functions are of zero mean. With the nested tetrahedral mesh, it is possible to construct a simple schemeby using the parents–child connectivity. In this scheme, the lazy wavelet function in Section 3.3 is defined for a simpletetrahedron in Fig. 7 as:

ϕ01 (x) = αφ1

1 (x) (49)

where ϕ01 (x) is the lazy wavelet and φ1

1 (x) is the hierarchical shape function at a node with coordinates x11 . φl

j (x) is alinear shape function satisfying the relation:

φlj (x

li ) = δ j i (50)

φlj (x

ki ) = 0 ∀k < l

where δi j is the Kronecker delta. Each interpolation function φlj is uniquely associated with a node at x l

j and has azero value for every node at coarser scales k ≤ l. For instance, in Fig. 7, φ1

1 (x11 ) = 1, φ1

1 (x01 ) = 0 and φ1

1 (x00 ) = 0.

In Fig. 7, nodes x00 and x0

1 are the parents of node x11 . The lifting scheme is applied using the two coarse scaling

functions associated with adjacent parent nodes, which have a non-zero value at point x11 . A combination of Eqs. (34)

Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360 351

and (49) yields:

ϕ01 (x) = αφ1

1 (x) − a0φ00 (x) − a1φ

01 (x) (51)

with the coefficients α, a0 and a1 selected such that the wavelet functions have a zero mean. Using Eq. (35):

α

∫Ω

φ11 (x)dΩ − a0

∫Ω

φ00 (x)dΩ − a1

∫Ω

φ01 (x)dΩ = 0 (52)

α is selected a priori, while a0 and a1 are independent lifting scheme coefficients that satisfy Eq. (52). The integral ofa linear shape function, given in terms of the volume coordinate ξ of a four-noded CST element Ωe with volume Ve

is: ∫Ωe

ξdΩ =Ve

4. (53)

The hierarchical shape function φlj has a compact support over a patch of tetrahedral elements denoted as

⋃nl j

k=1Ωel jk ,

where Ωel jk is the kth element that has a non-zero φl

j and nl j is the number of elements in the patch. The volume ofelement Ωe

l jk is Ve

l jk . Since φl

j is a linear interpolation function in each element of its support, the integral of φlj on Ω

is evaluated using Eq. (53) as:∫Ω

φlj (x)dΩ =

nl j∑k=1

∫Ωe

l jk

φlj (x)dΩ =

14

nl j∑k=1

Vel jk . (54)

Eq. (52) can then be written as:

α

n11∑k=1

Ve11k=1 − a0

n00∑k=1

Ve00k − a1

n01∑k=1

Ve01k = 0 (55)

where

a0 =α∑n11

k=1 Ve11k

2∑n00

k=1 Ve00k

, a1 =α∑n11

k=1 Ve11k

2∑n01

k=1 Ve01k

. (56)

Eq. (56) is a solution of Eq. (52), in which a0 and a1 are independent.The parent–child connectivity is used to explicitly write the fast wavelet transform for a tetrahedral mesh with CST

elements. Eq. (39), leads to the following relations:

dlj =

1α

(cl+1

j −

∑i

12

cl+1parent( j,i)

), cl

j = cl+1j +

∑i

achild( j,i)dl+1child( j,i) (57)

where child( j, i) is the index of the i th child of node j and parent( j, i) is the index of the i th parent of node j .achild( j,i) is the lifting scheme coefficient that correspond to weights in the contribution of φl

j to the wavelet ϕlchild( j,i).

Each nested node has two parents, and the number of children on an irregular grid varies with element connectivity.The coefficient 1

2 of cl+1parent( j,i) in Eq. (57) comes from the value of the parent shape function at the child node position.

5. Numerical examples for validation of the wavelet enrichment algorithm

This section evaluates the performance of the adaptive wavelet enhanced FE method and hierarchical FEM codefor two different problems. Results of the adaptive method, using different forms of estimates in Section 3.1, arecompared with those from a highly refined finite element mesh marked as the reference solution u f . Four differentcase studies are conducted and results are analyzed for convergence as delineated below.

1. The mesh is uniformly refined from a coarse configuration with standard FEM solution, without any waveletenrichment.

352 Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360

Fig. 8. (a) The initial coarse mesh and boundary conditions on the unit cube analyzed by the adaptive FE method, and (b) position of 46 nodes forwavelet enrichment functions on the coarse mesh using the estimate u∗.

2. The adaptive wavelet enrichment is conducted with the highly refined mesh solution u f as the solution estimate.A single iteration of the adaptive scheme is sufficient with this estimate. It is the best numerical estimate of thesolution u and is used as a reference, to evaluate how well the other estimates capture the features of the solution.This estimate is however not practical as it is computationally expensive.

3. The adaptive wavelet enrichment is conducted with an iterative solution estimate u in Eq. (22).4. The adaptive wavelet enrichment is conducted with an iterative solution estimate u∗ obtained from the diagonal

terms of the stiffness matrix in Eq. (28).

The first case-study focuses on a simple homogeneous cube with constrained boundaries, while the second exploresa problem with microstructure and mesh conformity effects. For each of these examples, the number of refinementscales is limited to 4. The convergence results are obtained by varying the tolerance vector ϵ = ϵ, ϵ, ϵT .

5.1. A cubic domain with constrained boundary

A homogeneous unit cube of dimensions 1 × 1 × 1 µm is subjected to uniaxial displacement condition in thiscase-study. The material is isotropic with Young’s modulus E = 1 GPa and Poisson’s ratio ν = 0.3. The appliedboundary conditions are uz = −0.1 µm on the top surface, and ux = u y = uz = 0 µm on the bottom surface as shownin Fig. 8(a).

The initial coarse FEM mesh with Ne = 5 four-noded tetrahedral elements and Nn = 8 nodes is shown in Fig. 8(a).The coarse solution uh in Eq. (10) is calculated with the initial coarse mesh. The reference finite element solution u f isobtained from a FE analysis on a uniformly refined mesh with Ne = 20 480 and Nn = 4233. The adaptive enrichmentfor this example is run with a maximum of 4 scales of hierarchical enrichment functions φ

n , defined in Section 3.1.The finest scale requires a total of Ne = 20 480 and Nn = 4233.

Results for the four different case studies with three different estimates for the adaptive method and the uniformlyrefined FEM simulations are compared. For a given tolerance ϵ, the wavelet enriched interpolation function ϕ

menr

follows Eq. (11). The nodal positions of 46 enrichment functions ϕmenr in the hierarchical adaptive grid with the

estimate u∗ with ϵ = 0.01 are shown in Fig. 8(b). Enrichment nodes are located near the base of the cube and near theedges, where the displacement error in the coarse finite element model is large. The displacement error is calculatedas a suitable norm of the difference between the wavelet enriched displacement solution uhenr in Section 3.1 and thefine-scale reference FEM simulation u f . Fig. 9 shows the percentage error in the x-direction displacement norm givenas eux =

|ux −u fx |

|u fx |max

× 100% for (a) the 24 DoF coarse mesh solution ux = uhx , and (b) the wavelet adapted solution

ux = uhenrx with 138 enrichment DoFs. The adaptive wavelet enrichment, which uses the estimate u∗ with ϵ = 0.01,

significantly reduces the error.

Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360 353

Fig. 9. Contour plots of displacement field error eux (%) for simulations with: (a) the coarse mesh with 24 DoFs, and (b) the wavelet adapted modelusing 138 enrichment DoFs.

Fig. 10. (a) Convergence rates on a log–log plot of the error eu f in Eq. (58) as a function of degrees of freedom, (b) evolution of the number ofenrichment functions as a function of the number of iterations (convergence is reached in approximately 7 iterations).

The convergence characteristics of the wavelet enriched adaptive method are compared with that of the uniformlyrefined FEM solutions in Fig. 10(a). The convergence rates are calculated as the percentage error in the L2 norm ofthe displacement as:

eu f =∥uhenr (x) − u f (x)∥2

∥u f (x)∥2 × 100(%), where ∥u(x)∥2=

∫x∈Ω

u(x) · u(x)dΩ . (58)

For all the estimates, the adaptive method consistently converges faster than the uniformly refined FEM simulations.Specifically, the convergence rates are ∼O(N−1.56) with the reference estimate u f , ∼O(N−1.56) for the estimate u andO(N−1.45) for the estimate u∗. The uniformly refined FEM method has a slower convergence rate of ∼O(N−1.22).For the estimate u∗ with ϵ = 0.01, the number of iterations necessary to reach convergence is shown in Fig. 10(b).The adaptive method requires around 7 iterations to reach convergence. Similar number of iterations are necessaryfor alternate values of ϵ, as well as with the estimate u. The number of enrichment functions increases as new scalesare introduced, especially in the first four iterations, until the highest scale of enrichment is reached. Thereafter thenumber evolves more slowly, until no additional positions need to be enriched.

5.2. A rectangular bar with material discontinuity

A rectangular domain of dimensions 1 × 1 × 3 µm with material discontinuity in the form of discretely varyingYoung’s modulus from E = 0.1 GPa to E = 1.8 GPa, is shown in Fig. 11(a). Sharp variation in the Young’s modulus

354 Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360

Fig. 11. (a) Rectangular bar with slanted bands of material discontinuity in the form of discretely varying Young’s modulus, showing loadingconditions, and (b) initial coarse FEM mesh of 15 linear tetrahedral elements showing nodal locations of 142 wavelet enrichment functions on theadaptive grid.

is seen in thin slanted bands, while Poisson’s ratio is fixed at ν = 0.3. The initial coarse mesh shown in Fig. 11(b)does not conform to the material discontinuities. The bar is subjected to a compressive load with boundary conditionsgiven as: uz = −0.3 µm on the top surface and uz = 0 µm on the bottom surface. Rigid body motion is constrainedat the bottom nodes. The initial coarse FEM mesh in Fig. 8(a) has Ne = 15 linear tetrahedral elements with Nn = 16nodes. The fine-scale FE model for the reference solution u f has a uniform mesh of Ne = 61 440 linear tetrahedralelements with Nn = 12 121 nodes. The wavelet adaptation is conducted to a maximum of 4 scales of hierarchicalenrichment functions φ

n . The finest scale requires a total of Ne = 61 440 and Nn = 12 121.As in the previous example, displacement error is calculated in terms of the difference between the wavelet enriched

displacement solution uhenr and the fine-scale reference FEM simulation u f . The non-conformity of the mesh tothe material discontinuities is the prime cause for discretization error that leads to local solution enrichment. Thenodal locations of the enrichment interpolation function by using the estimate u∗ with ϵ = 0.005 are shown inFig. 11(b). The interpolation functions ϕ

menr ⊂ φn are mostly introduced at locations where the modulus is low

and deformation is large in thin bands of compliant material. Results are again compared for the adaptive methodwith three different estimates and the uniformly refined FEM simulations. Fig. 12 shows the percentage error in the

z-direction displacement norm given as euz =|uz−u f

z |

|u fz |max

× 100(%) for (a) the 48 DoF coarse mesh solution uz = uhz ,

and (b) the wavelet adapted solution uz = uhenrz with 426 enrichment DoFs. The adaptive wavelet enrichment uses the

estimate u∗ with ϵ = 0.005. The error is largely corrected by the wavelet enrichment method.The convergence plots in terms of the L2 norm of the displacement error in Eq. (58) are shown in Fig. 13(a).

The convergence rates for the adaptive methods are not constant, but tend towards the optimal convergence rates.The adaptive methods have significantly higher convergence rates than the uniformly refined FEM, with up to 3orders of magnitude of error reduction for similar number of DoFs. The estimate u∗ leads to the best results, thatis comparable with the reference estimate u f . The number of iterations to reach convergence is approximately 8, asshown in Fig. 13(b).

The case-studies presented here show that the wavelet enriched adaptive FE method is sufficiently efficient forheterogeneous materials with non-conforming mesh structures. This is of high interest in simplifying the meshingprocess for complex microstructures.

6. Polycrystalline microstructure with anisotropic elasticity

A polycrystalline microstructure, shown in Fig. 14(a) is investigated for evaluating the effectiveness of theadaptive, wavelet-enriched hierarchical FE model. The simulation box is a cubic polycrystalline domain of dimensions12.32 × 12.32 × 12.32 µm, consisting of 208 grains. The material considered is transversely isotropic and exhibitsstrong anisotropy between the transverse plane and its normal (a principal) direction. The color plots in Fig. 14(a)

Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360 355

Fig. 12. Contour plots of displacement field error eux (%) for simulations with: (a) the initial coarse mesh with 48 DoFs, and (b) the wavelet adaptedmodel using 426 enrichment DoFs.

Fig. 13. (a) Convergence rates on a log–log plot of the error eu f in Eq. (58) as a function of degrees of freedom, (b) evolution of the number ofenrichment functions as a function of the number of iterations (convergence is reached in approximately 8 iterations).

depict the misorientation angle of the principal material direction with respect to the global z axis in radians. Thecomponents of the elastic stiffness tensor Ci j , i = 1. · · · 6, j = 1. · · · 6, expressed in a material coordinate system,are given as C11 = 14.5 GPa, C12 = 7.24 GPa, C13 = 6.5 GPa, C33 = 161 GPa, C55 = 7.1 GPa. The boundaryconditions are given as: uz = −0.2464 µm on the top surface, uz = 0 µm on the bottom surface with all rigid bodyDoFs constrained. The cube is subjected to a 2% uniform compression in the z direction.

Simulations of the polycrystalline microstructure under applied displacement conditions are conducted with thehierarchical FE model. The adaptive enrichments are obtained using the displacement estimate u with ϵ = 0.002.Material heterogeneity and anisotropy with high stiffness ratio, e.g. C33/C11 = 11.1, are leading causes of nonuniformdeformation in the domain. The wavelet-based adaptivity admits 3 scales of enrichment. The coarse-scale solution uh

is computed on a mesh composed of Ne = 9485 tetrahedral elements with Nn = 1778 nodes. The mesh for the fine-scale reference FE solution u f , as well as the adapted FE model with 3 scales of enrichment, both have Ne = 4 856 320tetrahedral elements and Nn = 821 569 nodes. A total of 14 000 enrichment functions are used, with 5568 at scale1, 6730 at scale 2, and 1702 at scale 3. The enrichment nodes are located near sharp and highly misoriented grainboundaries. In Fig. 14(b), the nodal positions of the 14 000 enriched function ϕ

menr are predominantly in regions

356 Y. Azdoud, S. Ghosh / Comput. Methods Appl. Mech. Engrg. 321 (2017) 337–360

Fig. 14. (a) A 208 grain polycrystalline microstructure with color-coded grains to reflect the principal axis orientation with respect to the global zaxis in radians, and (b) node enrichment positions for ϵ = 0.002, where the black spheres, red cubes and green octahedra denote the enrichmentscales 1, 2 and 3 respectively.

Fig. 15. Contour plots of displacement field error el (%) for simulations with: (a) the coarse mesh with 5334 DoFs, and (b) the wavelet adaptedmodel with 42 000 additional enrichment DoFs.

of large error. The local displacement error is calculated as el =

√(uhenr −u f )T (uhenr −u f )

(u f )T (u f )× 100(%). Fig. 15 shows the

magnitude of the displacement error for both the coarse mesh and adapted solutions uh and uhenr , respectively. Theerror that is generally high near the misoriented grain boundaries is significantly reduced for the adapted model.

The convergence rate of the solution with the adaptive hierarchical FE method is compared to that of a uniformlyenriched FE method in Fig. 16(a). The average convergence rate for the adaptive method is ∼O(N−1.22) compared to∼O(N−0.99) for the standard FEM. With additional scales of wavelet enrichment, the convergence rate can be evenfurther improved, closer to the optimal rate of O(N−2). The evolution of the number of enrichment functions with thenumber of iterations is depicted in Fig. 16(b) for ϵ = 0.002. Convergence is generally reached in about 5 iterations.

All simulations are conducted on the Bluecrab cluster at MARCC Maryland Advanced Research ComputingCenter [45] using 48 cores. The CPU time for different simulations are compared as functions of the normalizedL2 norm of displacement error, expressed as:

en =∥uhenr − u f

∥2

∥uh − u f ∥2 . (59)

Fig. 17(a) plots the normalized CPU time tn = t/tc with respect to en , where t is the time to compute uhenr andtc is the time to compute uh , either by the uniformly refined or the adaptive FEM. The figure estimates the rate atwhich the CPU time evolves. The plots are initially nearly flat since the communication and initiation time of the

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Fig. 16. (a) Convergence rates on a log–log plot of the L2 norm of displacement error as a function of degrees of freedom, (b) evolution ofthe number of enrichment functions as a function of the number of iterations for the estimate u∗ with ϵ = 0.002 (convergence is reached inapproximately 5 iterations).

Fig. 17. (a) Log–log graph of the normalized CPU time with respect with the normalized error en , and (b) semi-log graph of the theoreticalasymptotic computational complexity with respect to the normalized error.

cores adds to the CPU cost. At higher resolution, the solver cost becomes preponderant. An important observationis that the asymptotically predicted CPU time of the adaptive method is less than that of the FEM, due to the lowercomputational complexity of the former for a fixed error.

If a direct solver is used, the computational complexity of the linear solver is the leading term for global complexityof the problem, given as:

C ∼ O(SN 3) (60)

where C denotes the complexity of the total algorithm, N is the final number of DoFs and S is the number of adaptiveiterations. Note that S = 1 for classical FEM. The number of adaptive iterations is a function of the number ofscales. Eq. (60) constitutes an upper bound of the algorithmic complexity for the adaptive method. The leading termfor the complexity of the total algorithm only depends on the linear solver complexity for sufficiently large linearelastic simulations. The computational complexity of matrix assembly, solution estimate calculation, and wavelettransformation, are all of ∼O(SN ) and can be ignored for sufficiently large N . Furthermore, the above operations

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are highly granular and their computation time inversely scales with the number of processors, while the linear solverrequires communications between processors. Rewriting Eq. (60) with respect to the normalized error en:

CF E M ∼ O(e−3/pn ) and Cadapt ∼ O(Se−3/q

n ) (61)

where p, q are the convergence rates and CF E M and Cadapt refer to the complexity of the FEM and adaptive methodrespectively. Eqs. (61) imply that if the relative error is divided by two when refining, the algorithmic complexityscales by factors 23/p and 23/q respectively. With the average convergence rate in polycrystalline simulations taken asa reference, i.e. p = 0.992 and q = 1.223, there is a critical relative error for which the complexity of the adaptivemethod is lower than that of the refined FEM. Fig. 17(b) illustrates this point with S = 5.

7. Summary and conclusions

Motivated by the need to overcome shortcomings of the conventional crystal plasticity finite element models(CPFEM) and the fast Fourier transform (FFT)-based methods for analyzing deformation in polycrystalline mi-crostructures, this paper proposes an adaptive hierarchical enrichment induced FE method to augment computationalefficiency and solution accuracy. The paper focuses on heterogeneous, anisotropic elastic materials. The enrichmentstrategy involves projecting the solution field onto a set of scaling and multi-resolution wavelet basis functions. Thewavelet functions augment the solution by estimating residuals at higher scales. The multi-resolution wavelet propertyis advantageous for the selection of an optimal set of functions that can then adaptively enrich the solution space witha prescribed level of accuracy.

Contrary to standard adaptive methods, the proposed adaptive wavelet enrichment method does not need to evaluateerror indicators for mesh refinement. Instead it introduces a discretization space that conforms to the profile of thesolution sought at a given tolerance. The second generation family of wavelets engages a lifting scheme to generatehierarchical interpolation functions. With known scaling functions at coarse and fine scales, the lifting scheme definesthe set of wavelet functions that complement coarse-scale interpolation functions to uniquely project a function ona finer set. An iterative algorithm efficiently calculates estimates of the solution from the previous iterate using amodified Jacobi method. The estimate is decomposed into wavelet interpolation functions necessary at a specifiedprecision, which are then used to form a new solution space. A 3D implementation of this method is detailed.

The adaptive wavelet enriched FE method is subsequently used for two validation tests, with special focus onthe convergence rate and error mitigation. Solutions by the adaptive method are compared with reference solutionsfrom fine-scale FE simulations. The adaptive method, where enrichment functions are added at regions of higherror, converges faster than the uniformly refined FEM. Finally a polycrystalline microstructure with a number ofanisotropic grains is simulated by the adaptive method. The adaptive method shows very good convergence rates.The computational complexity and cost are analyzed with respect to the reduction in error with favorable results. Inconclusion, the proposed adaptive wavelet enrichment method is shown to be robust with clear advantages, such asreduction of computational costs while preserving the accuracy of local fields. This method is highly efficient forproblems where a conforming mesh cannot be obtained, such as in heterogeneous and graded materials. Extensions ofthis method to finite deformation crystal plasticity problems are currently in progress and will be reported in a futurepublication.

Acknowledgments

This work has been supported by the Air Force Office of Scientific through a grant FA9550-13-1-0062, (ProgramManager: Dr. David Stargel and Mr. James Fillerup). Computing support by the Homewood High-Performance Cluster(HHPC) and Maryland Advanced Research Computing Center (MARCC) is gratefully acknowledged. The authorswould like to thank Dr. H. Moulinec for graciously providing access to CraFT:1.0.12 for the FFT method simulations.The authors would like to acknowledge the help of Dr. J. Cheng in solving some of the problems discussed in thispaper.

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