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Adaptation and validation of FFT methods for homogenization of

lattice based materials

S. Lucarini1, L. Cobian1,2, A. Voitus1and J. Segurado1,2

1Fundaci´on IMDEA Materiales, C/ Eric Kandel 2, 28906, Getafe, Madrid, Spain

2Universidad Polit´ecnica de Madrid, Department of Materials Science,

E.T.S.I. Caminos, C/ Profesor Aranguren 3, 28040, Madrid, Spain

October 5, 2021

Abstract

An FFT framework which preserves a good numerical performance in the case of domains with large

regions of empty space is proposed and analyzed for its application to lattice based materials. Two

spectral solvers specially suited to resolve problems containing phases with zero stiﬀness are considered

(1) a Galerkin approach combined with the MINRES linear solver and a discrete diﬀerentiation rule and

(2) a modiﬁcation of a displacement FFT solver which penalizes the indetermination of strains in the

empty regions, leading to a fully determined equation. The solvers are combined with several approaches

to smooth out the lattice surface, based on modifying the actual stiﬀness of the voxels not fully embedded

in the lattice or empty space. The accuracy of the resulting approaches is assessed for an octet-lattice by

comparison with FEM solutions for diﬀerent relative densities and discretization levels. It is shown that

the adapted Galerkin approach combined with a Voigt surface smoothening was the best FFT framework

considering accuracy, numerical eﬃciency and h-convergence. With respect to numerical eﬃciency it was

observed that FFT becomes competitive compared to FEM for cells with relative densities above ≈7%.

Finally, to show the real potential of the approaches presented, the FFT frameworks are used to simulate

the behavior of a printed lattice by using direct 3D tomographic data as input. The approaches proposed

include explicitly in the simulation the actual surface roughness and internal porosity resulting from the

fabrication process. The simulations allowed to quantify the reduction of the lattice stiﬀness as well as

to resolve the stress localization of ≈50% near large pores.

1 Introduction

The latest improvements in additive manufacturing techniques (AM) have made possible the fabrication

of micro- and nano-architected metamaterials for mechanical applications with tailored stiﬀness, strength,

toughness and energy absorption capacity [1]. In most cases, the topology of these architected materials at

the lower length scale is based on the periodic repetition of a unit cell made up of bars or shells forming a

lattice. These lattice materials can reach high strength-to-weight ratios and other speciﬁc properties as well

as achieving non-standard elastic responses such as negative compressibility or zero Poisson’s ratio [1, 2, 3].

In addition, unit cells of lattice metamaterials can be designed using unstable structural elements to achieve

energy dissipation with fully reversible deformation or programmable/switchable properties [4].

In order to design an optimal lattice for a speciﬁc property is fundamental to perform accurate and

computationally eﬃcient simulations of the eﬀective response and the local ﬁelds developed within the

microstructure. Although discrete mechanical models are a good approach for capturing the overall response

[5], full-ﬁeld homogenization approaches allow obtaining a more complete, and often more accurate, result.

Under this approach, a micromechanical problem is solved on a representative volume element (RVE) which

contains a unit cell or a collection of unit cells in which the lattice geometry is explicitly represented. In

particular, the Finite Element Method (FEM) is the most common approach to carry out these simulations

since the designed ideal microstructure can be accurately reproduced with adaptative meshes [6]. However,

the real microstructures arising from an AM process can diﬀer from the ideal ones designed in-silico, and

usually present geometrical deviations from the target geometry, porosity and surface roughness [7]. Although

these features can be neglected for the lattice design phase, a reliable simulation of the material deformation

1

arXiv:2110.00733v1 [cond-mat.mtrl-sci] 2 Oct 2021

in the non linear regime which considers its irreversible response or fracture should take these diﬀerences

into account. FEM models of realistic geometries including surface roughness or porosity are very diﬃcult to

generate and mesh. Moreover, the real geometrical data is usually obtained from tomographic images [8, 7],

and transforming this data into a FEM model is a very complex process that results in a very large number

of elements.

In this context, the use of spectral methods, based on the Fast Fourier Transform (FFT) algorithm, can

be an interesting alternative. FFT methods do not require meshing because the calculations are carried

out on a regular grid and the phase belonging to each point of the grid can be obtained directly from

digital images or tomographic data. In addition, FFT approaches are very eﬃcient, with a computational

cost which grows as nlog(n), improving the FEM computational eﬃciency in the homogenization of bulk

heterogeneous materials by orders of magnitude [9, 10]. There are also other tangential beneﬁts coming from

the use of spectral approaches for studying lattice materials like the possibility of studying brittle fracture or

ductile damage using phase-ﬁeld fracture or gradient damage approaches, methods that show a very eﬃcient

performance in FFT [11, 12, 13].

The basic ideas of FFT homogenization were proposed by Moulinec and Suquet [14]. Since then, diﬀerent

approaches have been proposed to improve the convergence rate of the method. Some approaches are derived

from the original method, based on the use of Green’s functions for a reference medium and the solution

of the Lippmann-Schwinger equation [14, 15, 16, 17, 18, 19, 20, 21]. Alternative approaches have also been

proposed, based on either using a Galerkin approximation of the equilibrium using trigonometric polynomials

to discretize test and trial functions, [22, 23, 24, 25] or on solving the strong form of equilibrium using

displacements as unknown [26]. Despite the clear potential beneﬁts of FFT solvers for lattice materials,

two main limitations arise that have prevented their extensive use in this ﬁeld. First, FFT approaches

present a convergence rate and accuracy strongly dependent on the contrast between the phases represented

in the domain. In the present case, the contrast is inﬁnite because a large amount of the RVE voxels are

empty space, with zero stiﬀness. Second, although the voxelized representation is very convenient for using

digital microstructure data, its use for representing smooth geometries might result in poor results near the

boundaries. In addition, since FFT and inverse FFT transformations should include all the points of the

RVE including the empty space, the eﬃciency of the method with respect to FEM will depend on the relative

volume fraction, and it will not be competitive for very small relative densities.

Regarding the ﬁrst limitation, several studies can be found which aim at overcoming the problems with

the high phase stiﬀness contrast in FFT solvers and try to extend their use to study materials with voids.

Michel et al. [27] developed an Augmented Lagrangian formulation to solve a non linear problem including

non-compatible ﬁelds. Although this formulation allows introducing zero stiﬀness phases, it might require a

very large number of iterations to fulﬁl both stress equilibrium and strain compatibility, as noticed in [28].

Brisard and Dormieux [29] developed a variational formulation based on the energy principle of Hashin and

Shtrikman applied to a porous media. Their approach allows to accurately predict the overall response of

porous materials but it involves the pre-computation of a consistent Green operator which is computationally

very expensive. More recently, a method for solving the conductivity problem in the presence of voids has

been developed by To and Bonnet [30]. This approach is focused on solving the equilibrium only in the bulk

phases including a ﬂux term at the inter-phase between bulk and void phases. This method is suitable for

scalar ﬁelds, but cannot be directly extended to the vector and tensor ﬁelds that arise in the mechanical

problem since the ﬂux term in the internal boundaries does not restrict tensor components parallel to the

interphase. Another recent method proposed by Schneider [31] consists in searching solutions in a subspace of

solutions on which the homogenization problem is nondegenerate for the resolution of a material with pores.

In parallel to these techniques speciﬁcally suited for porous materials, an eﬃcient and simple alternative to

improve the convergence rate under very large phase contrast is the use of methods to reduce the numerical

oscillations that may occur due to Gibb’s phenomenon or aliasing eﬀects. A ﬁrst possibility is ﬁltering the

high frequencies, as proposed for example in [32, 33, 34]. A second possibility is replacing the continuum

diﬀerential operators in the formulation of the partial diﬀerential equations in the real space by some ﬁnite

diﬀerence diﬀerentiation rule. This idea was ﬁrst introduced in [35] who incorporated the ﬁnite diﬀerence

deﬁnitions of the derivatives in the real space using the FFT algorithm and the deﬁnition of Fourier derivation

using modiﬁed frequencies. The ﬁnite diﬀerence stencil used in the real space deﬁnes the particular form

of the modiﬁed frequencies to be used. Among the diﬀerent discrete diﬀerential approaches, the so-called

rotated scheme [36] shows a signiﬁcant reduction of noise and improves the convergence [10, 37]. Other two

ﬁnite diﬀerentiation schemes which are not based on the use of modiﬁed frequencies were also proposed in

2

[38] and [39] showing a clear improvement of the spurious oscillations. A third approach to reduce the noise

in the solution was proposed by Eloh et. al, [40] who instead of using the DFT as the discrete counterpart

of the continuous Fourier transform, considered the continuous Fourier transform of a piecewise constant

operator in the real space to derive consistent periodized discrete Green operators.

The second disadvantage of using FFT homogenization for lattice materials arises when, instead of using

direct images or tomographic data to construct the model, the objective is representing smoothly the lattice

material boundaries for general geometries deﬁned analytically or through CAD models. In these cases,

the voxelized representation can be inaccurate for small number of voxels, not achieving the actual relative

density of the lattice. This can lead to local inaccurate results due to the combination of a high phase

contrast and a non-smooth interface. This issue has been treated by smoothening the sharp inter-phase

using composite voxels [33, 41]. These approaches are based on deﬁning the material response for the voxels

crossed by an internal interphase as the homogenization of the two phases contained.

Due to these limitations, only a few previous attempts of using FFT to study lattice materials can be

found on the literature. In all the cases the motivation was to describe the actual lattice geometry, including

imperfections, using data obtained from tomographic images. In [42], the augmented Lagrangian approach

[27] was used only as a preprocessing step, using tomographic images of a single strut as data, in order to

determine an equivalent diameter of the struts. Then, the simulation of full lattices was done in FEM using

ideal geometries with equivalent radius. In [8] and [43], FFT was used to determine directly the response of

lattice materials. In both cases, the studies were limited to the linear elastic regime and the FFT approaches

used did not consider an inﬁnite compliant phase for the empty space, but used a material with small but

ﬁnite stiﬀness to achieve convergence instead. None of these works included an assessment of the accuracy

of the FFT approach in order to determine the eﬀect of the artiﬁcial stiﬀness used to represent the empty

areas.

The present work presents a systematic and critical assessment of the accuracy and eﬃciency of FFT

approaches for predicting the mechanical response of lattice based materials, both in the linear and the

nonlinear regime, in order to establish an optimal framework for the homogenization of this type of materials.

After a preliminary study, two linear FFT approaches are selected as potential candidates, including a novel

algorithm for RVEs including pores. In parallel, diﬀerent geometrical approaches to represent smoothly the

lattice geometries are combined with each approach. The accuracy and eﬃciency of the diﬀerent combinations

of FFT solvers and geometrical approaches are compared against FEM to model both the linear and non

linear responses of an ideal octet lattice. Finally, both frameworks are applied to study a real octet lattice

cell, including fabrication defects, obtained with 3D-tomography.

2 FFT homogenization for RVEs with empty areas

As previously discussed, a clear limitation for the use of FFT techniques for homogenization is the deﬁcient

convergence rate in the case of microstructures with a large contrast in the phase properties. This problem

becomes critical in the case of materials with voids where one of the phases is inﬁnitely compliant. In

this case, the problem of the low convergence rate (or no convergence in many cases) is superposed to the

singularity of the problem: the solution is not unique since any compatible strain ﬁeld in the empty phase

is admissible. The problem has been circumvented on many occasions by setting a very compliant elastic

behavior for the void phase, but this artiﬁcial stiﬀness might have an eﬀect on the cell response, especially

for large volume fractions of empty voxels.

The proper adaptation of FFT homogenization algorithms to account for actual zero stiﬀness has been

studied almost from the ﬁrst developments of this numerical technique. The augmented Lagrangian approach

[27] is a modiﬁcation of the original basic scheme to improve convergence for RVEs containing phases with

very low or zero stiﬀness. The algorithm is based on the combination of two strain ﬁelds, one of them forced

to be compatible, and two stress ﬁelds, one of them forced to be in equilibrium. Then, the solution is obtained

by the iterative minimization of a Lagrangian. Although the method is in theory able to resolve cases with

inﬁnite phase contrast, the convergence rate becomes really small when controlling a residual which enforces

both stress equilibrium and strain compatibility in addition to the macroscopic constraints [28]. Other

acceleration methods also based on using additional ﬁelds have been developed to account for inﬁnite phase

contrast as [44]. Nevertheless, these methods present a similar convergence rate when small tolerances are

imposed for both equilibrium and compatibility [28]. Another potential method speciﬁcally developed for

considering voids is the variational approach by Brisard et.al. [29]. Although the idea presented is very

3

smart, the consistent diﬀerential operator derived is very complex to compute and the authors themselves

do not use it ﬁnally, but an approximate version based on discrete derivatives.

As mentioned in the introduction, an alternative (or additional) way of improving the convergence rate

under large phase property contrast is using methods that reduce the numerical oscillations. In particular,

the standard continuous diﬀerentiation scheme can be replaced by ﬁnite diﬀerence schemes through diﬀer-

ent approaches as using staggered grids [38] or modifying the Fourier derivative deﬁnition with modiﬁed

frequencies which corresponds to diﬀerent ﬁnite diﬀerence stencils [35, 36]. These alternative diﬀerentiation

schemes can be combined with diﬀerent FFT solvers, improving the convergence of the original ones.

In this paper the Galerkin FFT approach [22, 23] combined with the use of mixed control [25] and the

rotated ﬁnite diﬀerence scheme [36] has been chosen as a ﬁrst candidate for the lattice material homogeniza-

tion. This combination has been made after a preliminary study and is based on its very fast convergence

rate but also on its relative simplicity and the ability of the method to be eﬃciently extended for non lin-

ear cases. This approach does not break the underdetermination of the solution, but allows to converge in

the presence of regions with zero stiﬀness to an equilibrated stress and compatible strain with relatively low

noise. Note also that any other Krylov based approach with the appropriate reference medium and combined

with the same discretization scheme would eventually lead to similar results.

As second candidate, a new method based on a modiﬁcation of the DBFFT approach [26] is proposed.

In this method, the standard equilibrium is augmented with additional conditions for the void regions and

interfaces to break the underdetermination leading to a non-singular discrete system of equations. In the

next subsections, both methods will be presented including their extension for non linear response.

2.1 Galerkin-FFT with discrete diﬀerences and mixed loading control

The FFT-Galerkin method was initially developed by Vondrˇejc et al. [22] to homogenize the elastic behavior

of fully dense heterogeneous materials. This approach presents a very fast convergence rate for limited phase

contrast, but is not able by itself to converge in the presence of inﬁnitely compliant phases. In order to

extend this scheme for RVEs with empty regions, the original method is adapted by changing the iterative

linear solver, introducing an alternative diﬀerentiation scheme [36] and using mixed macroscopic control [26].

Both the original method and these modiﬁcations will be presented below.

Following [22], starting from the weak form of the equilibrium in small strains for a given heterogeneous

periodic domain Ω the next equation can be derived

F−1nb

G(ξ) : F {σ(x)}o=0, (1)

where xrepresents the spatial position, ξthe spatial frequency vector, σis the Cauchy stress —determined

by the local constitutive equations— and b

Gis the Fourier transform of a linear operator which projects any

arbitrary tensor ﬁeld into its compatible part. The Fourier transform and the inverse Fourier transform are

represented in eq. (1) by Fand F−1respectively.

In equation (1) the domain is discretized in a regular grid in which each voxel center x= (x1, x2, x3) is

given by

xp=Lp

2Np

+Lp

Np

np,with np= 0, . . . , Np−1; p= 1,2,3 ,

where Lpand Npstand for the length of the cell edge and number of voxels in direction p. The discrete form

of the equation (1) is a linear system of algebraic equations in which the unknown is the value of the strain

at the center of each voxel. The frequency vector ξis given by

ξp=iqpNp/Lpwith qp= 2πnp−Np/2

Np∈[−π, π] , (2)

where i=√−1 is the imaginary unit. The Fourier transforms correspond to the direct and inverse Discrete

Fourier Transform that are carried out using the FFT algorithm. The ability of the FFT algorithm to reduce

the computational cost of the DFT transforms from O(n2) to O(nlog n) is the main reason behind the high

performance of spectral solvers.

The macroscopic state is provided as a combination of macroscopic strain components kl,ε=εkl(ek⊗

el)sym or macroscopic stress components KL σ=σKL (eK⊗eL)sym with kl ∩K L =∅. For these general

4

loading conditions [25] the expression of the projector operator in the Fourier space follows

b

Gijkl (ξ) =

Is

ijK L if ξ=0for components KL

0ijkl if ξ=0for components kl

0ijkl for Nyquist frequencies

hIs

ijkl ξjξli−1

ξjξlfor ξ6=0

, (3)

where Isis the fourth order symmetric identity tensor and b

Gaccounts for major and minor symmetries.

In the case of linear elastic phases eq. (1) yields

F−1nb

G(ξ) : F {C(x) : ε(x)}o=−F−1nb

G(ξ) : F {C(x) : ε(x)−σ}o, (4)

where Cis the local fourth order stiﬀness tensor. Equation (4) represents a linear system of equations in

which the left-hand side is a symmetric semideﬁnite positive linear operator acting on a discrete strain ﬁeld

A(•) = F−1nb

G(ξ) : F {C(x) : •}o(5)

and the right-hand side is the independent term b

b=−F−1nb

G(ξ) : F {C(x) : ε(x)−σ}o.

The system A(ε) = bcan be solved eﬃciently using the Conjugate Gradient method (CG) method for

domains with a relatively low stiﬀness contrast. The residual of this system of equations is deﬁned as the

L2norm of the diﬀerence between the linear operator applied on the candidate solution and the right-hand

side over the norm of the right-hand side,

rlin =kA(ε)−bkL2

kbkL2

(6)

reaching the solution when rlin is below a tolerance. Note that eq. (4) is undetermined independently of the

phase properties because any incompatible ﬁeld εINC added to the solution still fulﬁls eq. (4),

A(ε+εIN C ) = A(ε) = b.

Nevertheless, the solution of eq. (4) is unique in the subspace of compatible strain ﬁelds, so the CG is able

to handle the underdetermination and recovers the unique compatible solution.

In the case of lattices, there are empty regions in the RVE, which do not transfer stresses and that should

be properly accounted for with zero stiﬀness. The solution of the problem in the full RVE is singular even in

the subspace of compatible strain ﬁelds, because any compatible strain ﬁeld which is zero outside of the empty

region can be added to the solution providing the same equilibrated stress. This singularity is transferred

to the numerical method increasing the underdetermination of the system also to compatible ﬁelds. As a

result, it is observed that the CG method is not able to reach convergence and the Minimal Residual Method

(MINRES), an alternative Krylov subspace solver able to handle eﬃciently singular systems, is used as linear

solver to ovecome this limitation.

Finally, in order to improve both smoothness of the solution and convergence rate, an alternative dis-

cretization scheme is used, the rotated forward ﬁnite diﬀerence rule [36]. This discretization is introduced

through modiﬁed frequencies in the Fourier derivation leading to an alternative projection operator (eq. 3).

The modiﬁed frequencies correspond to

ξ0

p=i2Np

Lp

tan (qp/2)

d

Y

p=1

1

21 + eiqpwith qp= 2πnp−Np/2

Np∈[−π, π] , (7)

where d= 1,2 or 3 is the space dimension. It is important to remark that standard Galerkin approach does

not converge using standard Fourier discretization and a conjugate gradient solver. On the contrary, the

combination of the alternative diﬀerentiation scheme and the use of MINRES allows the Galerkin approach

to reach a solution in a relatively small number of iterations, as it will be shown in the numerical results.

5

2.1.1 Non linear extension

In order to take into account material non linearities on the Galerkin FFT formulation the macroscopic

strain/stress history is applied as function of the time (or pseudo-time for rate independent materials)

in several time increments. The non linear equilibrium at each increment is solved iteratively using the

Newton-Raphson method, as proposed in [23]. If the solution at time tand the macroscopic stress and strain

components applied at t+ ∆t,εt+∆t,σt+∆tare known, the non linear equation at time t+ ∆tis linearized

at each iteration iaround the strain ﬁeld at previous iteration εi−1. Let δε(x) be the strain ﬁeld correction

to be obtained at iteration i, then the linearized stress corresponds to

σi(x) = σi−1(x) + ∂σ(x)

∂ε=i−1

:δε(x) = σi−1(x) + Ci−1(x) : δε(x) , (8)

where Ci−1is the material consistent tangent evaluated using the solution of the previous iteration i−1.

The equilibrium equation linearized at εi−1reads

F−1nb

G(ξ) : FCi−1(x) : δε(x)o=−F−1nb

G(ξ) : Fσεi−1(x)−σt+∆to, (9)

where the solution at the previous time step enters in the ﬁrst iteration as ε0(x) = εt(x) + εt+∆t−εt.

The left-hand side of equation (9) corresponds to a linear operator Ai, acting on the correction strain

ﬁeld δε, that is equivalent to the one deﬁned in eq. (5) using the tangent stiﬀness at iteration i−1 instead

of the elastic one. Similarly, the right-hand side forms an independent vector bi. The solution of the non

linear problem at each time increment is obtained solving the linear equation (9) for each Newton iteration,

and adding the successive solution corrections until the convergence is reached.

Special care has to be paid to the deﬁnition of the residuals in the non linear case. As in the original

approach [23], two residuals are proposed for the non linear solver, but they are redeﬁned in order to avoid

oversolving. The linear residual controls the accuracy of the solution in the linear system resulting from each

Newton iteration. This linear system Ai(δε) = biis solved iteratively up to a given tolerance and the non

linear algorithm becomes an inexact damped Newton method [45]. The standard non-linear approach for

Galerkin FFT establish the convergence criterion on the relative residual, deﬁned as

kAi(x)−bikL2

kbikL2

,

where the norm used is the L2norm. However in this case, since the right-hand-side bichanges at each

Newton iteration and should converge towards 0, the relative residual becomes too restrictive near the

solution and results in additional meaningless linear iterations. To avoid this problem, the norm in the ﬁrst

iteration kb0kis used here to normalize the residual for the rest of the Newton iterations. The convergence

criterion for the linear solver can be then rewritten as

rlin =kAi(δε)−bikL2

kb0kL2

.(10)

Note that if the increment is linear, this expression corresponds to eq. (6). Moreover, since biis zero for an

equilibrated stress ﬁeld, the ratio kbik/kb0kis a relative measure of the internal equilibrium and as a result

the number of iterations required for the linear solver decreases with the number of Newton iterations. The

second residual is the Newton residual for the non linear equation. It is deﬁned as the ratio between the

inﬁnity norm of the last deformation gradient correction and the inﬁnity norm of the change in the total

strain ﬁeld within each time increment

rnewton =kδεk∞

kεi−εtk∞

.(11)

The solution is accepted only when both of the residuals (eqs. (10) and (11)) are below their respective

tolerances. Note that although the choice of the Newton forcing term has not been optimized (as proposed

in [45]), it is observed that when a consistent tangent is used the number of Newton iterations per strain

increment was quite small, always less than ﬁve.

6

2.2 Modiﬁed displacement based FFT for inﬁnite contrast (MoDBFFT)

A modiﬁcation of the displacement based FFT approach presented in [26], called from now on MoDBFFT

for brevity, is proposed here for simulating lattice materials. The objective is to derive a method that

presents accurate results and a good convergence rate for inﬁnite phase contrast maintaining standard con-

tinuum discretization and derivation in the Fourier space, without the need of using alternative discretization

approaches and modiﬁed frequencies.

The starting point of the method is the strong formulation of the conservation of linear momentum on

a periodic domain which can be divided into two sub-domains (see Figure 1a) representing the two phases:

Ωm, the matrix, and Ωv, the void. The interphase between the two phases is Γ.

(a) Two domain scheme

(b) MoDBFFT scheme

Figure 1: Schematic representation of the domains for the standard FFT and the present approach

The real boundary value problem is deﬁned only in the domain Ωmand corresponds to ﬁnding the

displacement ﬁeld u∈Ωmsuch that

∇ · σ(∇su(x)) = 0in Ωm

σ(x)·nΓ=0in x∈Γ

σand εperiodic in Ω

.(12)

Due to the periodicity in Ω, the weak formulation of this problem with free Neumann boundary conditions

on Γ is simply ZΩm

σ(u) : ∇sδudΩm= 0 , (13)

where uand δuare the trial and virtual displacement ﬁelds respectively.

FFT methods need to resolve the ﬁelds in the full cell, including the points in Ωvwhere no material

exists and ﬁelds are not deﬁned. On the contrary, the balance equation (13) is not deﬁned in the entire unit

cell Ω so any compatible strain ﬁeld is acceptable for the stress equilibrium in the region Ωvand the system

is underdetermined.

In the new approach, the standard formulation of the two-phase domain is modiﬁed (see Figure 1b) in

order to extend the weak formulation to every point of the domain Ω , including the free Neumann conditions.

To this aim, ﬁrst, an artiﬁcial elastic energy density is deﬁned in Ωvwhich aims to prevent the indeterminacy

of the displacement in that region. This energy density depends on the square of the displacement gradient

is weighted by a numerical parameter α, with stress dimensions and which represents an artiﬁcial stiﬀness in

the interior of Ωv. The ﬁrst variation of the total energy in Ωvleads to a new term in the weak formulation

7

which is deﬁned only in that region and corresponds to

ZΩv

α∇su:∇sδudΩv, (14)

where αis assumed to be small. Note that when α= 0, the original underdetermined formulation is

recovered.

To extend the weak formulation of the original problem (eq. 13) in Ωvto the full domain Ω, the indicator

function of the voided region χv(x) which deﬁnes the microstructure, is introduced (eq. 15)

χv=(1∀x∈Ωv

0∀x∈Ωm

.(15)

The eqs. (13) and (14), deﬁned in Ωmand Ωvrespectively, are then premultiplied by their corresponding

indicator functions to be deﬁned in a unique domain Ω, leading to

ZΩ

(1 −χv)σ(u) : ∇sδu+χvα∇su:∇sδudΩ = 0.(16)

Using the divergence theorem and removing boundary terms because of the periodicity, the corresponding

strong formulation of eq. (16) follows

∇ · (1 −χv(x)) σ(∇su(x)) + ∇ · χv(x)α(∇su(x)) = 0

and if the chain rule is applied to the previous equation and terms are regrouped, the result is

(1 −χv)∇ · σ+χv∇ · α(∇su(x)) + ∇χv·(α(∇su(x)) −σ)=0.(17)

Second, the stress free condition at the interphase Γ of the boundary value problem, deﬁned in the second

equation of eq. (12) should be imposed in equation (17). To this aim, the boundary condition on Γ is diﬀused

over a thin volume which is deﬁned by the surface Γ with an inﬁnitesimal thickness [30]. This translation

can be done using the surface delta function δ(x)Γ, deﬁned as [46]

δ(x)Γ=nΓ· ∇χv(x) , (18)

where the direction of nΓis represented in Fig. 1b. The gradient of the indicator function vanishes everywhere

except near the surface Γ, where it points in the normal direction [46]. Therefore, multiplying the previous

equation by the normal vector leads to

nΓδ(x)Γ=∇χv(x).(19)

Using the surface delta (eqs. 18 and 19), the stress free condition can be expressed as a volume integral

as ZΓ

σ(x)·nΓdΓ = ZΩ

σ(x)·nΓδ(x)ΓdΩ = ZΩ

σ(x)· ∇χvdΩ.(20)

In order to apply the stress free condition, eq. (20) has to be incorporated to eq. (17). Since for a

non-vanishing αand continuous displacement ﬁeld in the surface Γ it is fulﬁlled

σ·n=α(∇su(x)) ·nin Γ

then, the volume counterparts of any of these terms are also identical and any of them can be set to zero in

eq. (17).

σ(x)· ∇χv=α(∇su(x)) · ∇χvin Ω.

Choosing for simplicity α(∇su(x)) · ∇χv= 0 leads to the ﬁnal expression of the strong formulation

∇ · (1 −χv(x)) σ(∇su(x)) + χv(x)∇ · α∇su(x) = 0(21)

together with periodic boundary conditions in ∇su.

8

In order to impose periodicity conditions in the strain ﬁeld, the displacement ﬁeld is split into two

contributions as

u(x) = e

u(x) + ε·x, (22)

where e

u(x), the ﬂuctuation of the displacement ﬁeld which is periodic and has zero average, becomes the

new unknown of the problem. If this new formulation is particularized for a linear elastic matrix using as

input a general mixed macroscopic state as a macroscopic strain εor macroscopic stress σ, the ﬁnal equation

to solve the ﬂuctuations in the displacement reads

∇ · [(1 −χv(x)) C(x):(∇se

u(x) + εσ)] + χv(x)∇ · (α∇se

u(x)) = −∇·(C(x) : ε−σ) , (23)

where the ﬁeld e

uis the unknown, that has to be solved together with the IJ components of the overall strain

tensor, εσ, which are conjugate of the applied macroscopic stress. For this last unknown, an extra equation

is used linking the IJ components of εσwith the corresponding components of the macroscopic stress,

1

VΩZΩ

[C(x):(∇se

u(x) + εσ)]IJ dΩ = −1

VΩZΩ

[C(x) : ε]IJ dΩ + σI J .(24)

As usual in FFT methods, the diﬀerential operators can be deﬁned by their Fourier space counterparts

using spatial frequencies as

∇s(•) = F−1{F {•} ⊗sξ}=F−11

2(F {•} ⊗ ξ+ξ⊗ F {•})(25)

∇ · (•) = F−1{F {•} · ξ}(26)

To solve (23) in the Fourier space, the spatial domain is discretized in a standard regular grid (section

2.1). The result is a linear system of equations that now is fully determined. Nevertheless, to solve the

system iteratively, the use of a preconditioner is unavoidable for a competitive convergence rate. The linear

operator M(x) (•), proposed for the DBFFT approach [26], is also used here as preconditioner

M(•) = ∇ · C∇−1(•) , (27)

where Cis the volume averaged stiﬀness tensor,

C=1

VΩZΩ

C(x) dΩ (28)

and VΩrepresents the volume of the entire domain Ω.

The resulting equilibrium equation written in Fourier space yields

Fn(1 −χv)C:F−1nb

e

u⊗sξo+εσo·ξ+FnχvF−1nαb

e

u⊗sξ·ξoo=−F {C:ε−σ} · ξ, (29)

and the Fourier transform of the equation to impose the macroscopic stress components IJ (eq.(24)) corre-

sponds to

Fn(1 −χv)C:F−1nb

e

u⊗sξo+εσo(0) = F {C:ε−σ}(0) . (30)

The equations (29,30) are linear and can be solved iteratively using a Krylov solver. To improve the

convergence, a preconditioner b

Mis used for eq. (29), which is the Fourier transform of the preconditioner

deﬁned in eq. (27), and is given by

b

M(∗) = ξ·C·ξ−1· ∗ , (31)

where ∗represents a complex valued vector deﬁned in the Fourier space for all non-zero frequencies. For

preconditioning eq. (30), the inverse of the volume averaged stiﬀness tensor (eq. 28) is directly used.

Contrary to the Galerkin approach using standard Fourier diﬀerentiation, the linear system of equations

deﬁned in eq. (29) is non-singular, and therefore the Conjugate Gradient method is able to converge eﬃciently

and provide the solution of the system. This is a potential beneﬁt of this approach with respect to the

modiﬁed Galerkin which relies on the use of the more memory demanding MINRES solver.

9

For the implementation, the problem unknowns are joined forming a vector composed of the ﬂuctuating

displacement ﬁeld and the components of the macroscopic strain where the stress is imposed (eq. 30),

b

e

u

εσ. The left-hand side of eqs. (29) and (30) can be expressed as a linear operator that acts over the

composed vector following

b

Aα b

e

u

εσ=

Fn(1 −χv)C:F−1nb

e

u⊗sξo+εσo·ξ+FnχvF−1nαb

e

u⊗sξ·ξoo

Fn(1 −χv)C:F−1nb

e

u⊗sξo+εσo(0)

(32)

and the right-hand side can be written as a vector reading as

b

bα=−F {C:ε−σ} · ξ

F {C:ε−σ}(0)(33)

It should be remarked that the linear operator (eq. 32) has a signiﬁcantly higher computational cost

(around 1.5x) compared to the linear operator of the Galerkin method (eq. 3) since it requires performing

the additional Fourier transforms of a vector ﬁeld in the extra term. The equilibrium is reached when a

linear residual deﬁned as

rlin =

b

Aα b

e

u

εσ−b

bα

L2

b

bα

L2

(34)

is lower than a given tolerance.

2.2.1 Non linear extension

In the case of non linear materials, a linearization of equation (21) is done similarly to the Galerkin method

(Section 2.1.1). The stresses and strains are linearized following eq. (8). The non linear problem is divided

into time increments and an iterative Newton method is used at each time increment to solve the problem.

The linearization at each Newton iteration leads to a system of equations

Fn(1 −χv)Ci−1:F−1nc

δe

u⊗ξo+δεσo·ξ+FnχvF−1nαc

δe

u⊗ξ·ξoo

=−F n(1 −χv)σF−1nb

e

ui−1⊗ξo+εi−1

σ+εt+∆t−σt+∆to·ξ

− F nχvF−1nαb

e

ui−1⊗ξ·ξoo (35)

in which the displacement correction and the average strain correction, c

δe

uand δεσ, are the unknowns and

the solution for iteration iis updated as

e

ui=e

ui−1+δe

u;εi

σ=εi−1

σ+δεσ.

In equation (35) the macroscopic prescribed strain and stress ﬁelds enter in the deﬁnition of the ﬁrst

iteration as e

u0=e

utand ε0

σ=εt

σ. The linear equation can be translated into a linear operator applied

to the unknown b

Aα

i δb

e

u

δεσand an independent right-hand side vector b

bα

i, both deﬁned in the i-th

iteration. At each Newton iteration, the preconditioner given by eq. (27) is recomputed using the average

tangent stiﬀness.

Analogous to the Galerkin scheme, two residuals are used to solve the non linear problem. The linear

solver residual is deﬁned as

rlin =

b

Aα

i δb

e

u

δεσ−b

bα

i

L2

b

bα

0

L2

(36)

which normalizes the absolute error in the linear problem by the norm of the right-hand side vector of the

ﬁrst Newton iteration. The second residual is the Newton residual, which is formulated in strains and is

identical to the one used in the Galerkin FFT in eq. (11).

10

3 Geometrical adaptation

FFT methods rely on a regular discretization of a hexahedral domain in which the cell of the lattice material

is embedded. This voxelized representation allows a simple generation of the microstructure and the direct

use of image/tomographic data but presents two disadvantages when considering an ideal cell, especially

for coarse discretizations. First, the ﬁnal density represented can diﬀer slightly from the designed one and,

second, the voxelized representation of the struts might impact the overall behavior of the lattice.

In order to alleviate these problems, the properties assigned to the voxels near the strut surface can be

adapted to better capture the smoothness of the surfaces. In this section, diﬀerent approaches for determining

the properties of the voxels near the interfaces are presented. The geometry deﬁnition of the cross section

of a circular strut is represented schematically in Fig. 2.

3.1 Plain Voxelized representation

The plain voxelized (PV) geometry approach is the most simple representation: it assigns lattice material

or empty space to a voxel based on whether the center of that voxel is inside or outside of the lattice struts.

This method generates sharp boundaries for the struts and the resulting relative densities can slightly diﬀer

from the designed one in the case of coarse discretizations.

3.2 Phase-ﬁeld smoothening

The phase-ﬁeld smoothening (PFS) method consists in creating a smooth property transition, from the

lattice material properties to zero, across the lattice interfaces. The property proﬁle is dictated by the

minimization of a phase-ﬁeld functional and is controlled by a smoothening length scale `. The result of the

phase-ﬁeld minimization is a phase map φ(x) that assigns to each voxel of the cell a phase value between

0 (empty space) and 1 (lattice material). Similar to damage models [47], the stress resulting from applying

the constitutive equation of the pristine material is multiplied by the value of the phase map at that point.

Let ω(x) be a function which represents the exact geometry of the lattice, and which in this case cor-

responds to the indicator function of the lattice material, ω(x)=1−χv(x) (eq. 15) where the value of 1

is attributed to the points belonging to the material and 0 to the empty space. Then, the value of φ(x) is

obtained as the minimizer of the functional deﬁned by eq. (37).

E[φ] = ZΩ

1

2`2k∇φk2+

2(φ−ω)2dΩ , (37)

where `is the characteristic length, a parameter which deﬁnes the width of the smoothening region and is a

weight which penalizes the diﬀerence between the exact geometry ωand its smoothened counterpart φ. The

functional assumes periodicity of all the ﬁelds. The ﬁrst term of the functional penalizes the smoothened

area and it is modulated by the length of the diﬀusion and the second term penalizes the diﬀerence between

the initial ﬁeld and the smoothened one. The result of this minimization corresponds to the solution of the

partial diﬀerential equation described in eq. (38)

`

2

∇2φ−φ=−ω(38)

under periodic boundary conditions in φ. If the problem is discretized in a regular grid and ω(x) is replaced

by its discrete counterpart, deﬁned by the value of the indicator function at the center of each voxel of the

grid, the equation can be explicitly solved on the Fourier space as

b

φ(ξ) = 1

1−`2

ξ·ξbω(ξ) , (39)

where the frequency vector ξis given in eq. (2) and all the ﬁelds involved are periodic.

The numerical implementation of this approach is done using two diﬀerent discretization levels. In a

pre-processing step a very ﬁne grid is used with voxel size dfine to discretize the real geometry and solve

the phase-ﬁeld problem to obtain φ(x). During the simulation of the mechanical problem a coarser grid

is used, dcoarse . The ratio between dcoarse/df ine is between 2 and 20. The ﬁne discretization is used for

11

having an accurate representation of the indicator function ω(x). This ﬁne grid is used for solving eq. (38)

using as characteristic length of diﬀusion `the half of the length of the voxel in the coarse discretization

(`= 0.5dcoarse) which encompasses several voxels of the ﬁne grid. Finally, the resulting ﬁeld φ(x) in the

ﬁne grid is averaged for each voxel of the coarse discretization to deﬁne the phase map to be used during

the mechanical simulations. Note that the values of the phase ﬁeld below 5% are taken as 0, to prevent

the spurious presence of material detached from the truss. In all the phase-ﬁeld smoothening cases th e

characteristic weight is taken equal to 1.

3.3 Voigt approaches

The third approach to smooth out the lattice surfaces is based on the Voigt rule, following [33]. Under this

approach, the stiﬀness of the voxels partially occupied by the lattice material is obtained using the Voigt

homogenization approach. This rule establishes that the eﬀective elastic stiﬀness of that composite voxel

is the volume average of the stiﬀness of the materials present in the voxel. Therefore, since the stiﬀness of

the empty phase is zero, the eﬀective stiﬀness corresponds to the product of the volume fraction of lattice

material in the voxel, φ, multiplied by the stiﬀness of the lattice material (Cm)

C=φCm.

The volume fraction φ∈[0,1] is then equivalent to a phase map, as the one generated using phase-ﬁeld

smoothening. In [33], to compute the volume fraction of each phase contained in the voxels partially occupied

by diﬀerent phases, it was proposed the use of a subgrid to count the number of points in the subgrid belonging

to each phase. We have followed this approach ﬁrst, using the same ratio for coarse and ﬁne grids used in

PFS, and have named this approach as Voigt ﬁne grid (VFG). In parallel, since the surface of the lattice is

known either by its mathematical expression or by an .stl ﬁle, we propose an alternative way to compute the

phase map φwhich does not require the use of a second grid and is just based on the distance of the voxel

center to the lattice surface. This approach is named Voigt analytic (VA) smoothening. The method assigns

φ= 1 or φ= 0 to the voxels whose centers are respectively inside or outside of the geometry considering an

oﬀset of `/2 with respect to the boundary of the struts Γ. For those voxels centers whose distance to the

boundary is smaller than `/2 , the value of φis obtained as a linear function of the signed distance of that

center to the boundary, D, following eq. (40),

φ(x)=1−D+`/2

`for D=d(x,Γ) if x∈Ωm

−d(x,Γ) if x∈Ωv, (40)

where ddenotes the distance between a point and a surface.The characteristic length considered, `, is the

length of one voxel. Note that this deﬁnition of φcorresponds exactly to the volume fraction of lattice

material in the case of a planar interface. Since the strut curvature is normally small with reference to

the voxel length, the values obtained using eq. (40) are almost identical to the VFG in the case of ﬁne

discretizations. This equivalence has been assessed quantitatively and the phase map generated using the

Voigt rule with a ﬁner grid (the same used for the phase-ﬁeld smoothening) was almost identical to the

one obtained by VA (average diﬀerence below 0.2%). For clarity, both methods are only considered for

coarse grids while for ﬁner discretization, where the results are almost identical, only the results of VA are

represented. The practical beneﬁt of the VA deﬁnition of φwith respect to [33] is that the smoothening is

obtained by an analytical expression using the exact geometry and not requiring the use of a smaller grid.

3.4 Combined smoothening

This approach (CS) consists in applying the phase-ﬁeld smoothening (Section 3.2) to the coeﬃcients map

resulting from the Voigt analytic method (Section 3.3). The combination of these two approaches will result

in a very smoothened phase map of weight coeﬃcients that will be multiplied by the stresses within the

diﬀerent algorithms.

4 Validation for elastic materials

The numerical methods proposed to homogenize the mechanical behavior of lattice based materials and their

combination with the diﬀerent geometrical representations are studied in this section for elastic materials.

12

(a) Plain voxelized

representation (PV).

(b) Voigt approaches

(VA and VFG).

(c) Phase-ﬁeld

smoothening (PFS).

(d) Combined

smoothening (CS).

Figure 2: Local densities (phase maps φ) near the cross section of a strut for the diﬀerent geometrical

approaches. In the grey scale φ= 1 corresponds to black and 0 to white.

In order to evaluate the accuracy and eﬃciency of the two FFT methods, several numerical tests have

been carried out, and both the macroscopic result and microscopic ﬁelds have been compared with FEM

simulations. For the microscopic solution, the relative L2norm of the diﬀerence between the local ﬁelds of

the solution in a given method, f(x) compared to a reference solution (fref (x)) is used as metric of the error

Local diﬀ. [%] = kf−fref kL2

kfref kL2

. (41)

4.1 Lattice geometry and simulation parameters

The octet-truss lattice has been selected for the numerical studies. This structure is one of the most inter-

esting lattice based materials since it presents both bending-dominated and stretching-dominated responses,

depending on the strut thicknesses and the loading conditions. Relative densities ranging from 0.5% to 30%

are considered.

All the FFT simulations are performed using the FFTMAD code [10], to which the new algorithms

have been added. FEM simulations are done using the commercial code ABAQUS to serve as reference

solutions in order to assess the accuracy and eﬃciency of FFT approaches. The FEM model consists of a

geometrically conforming mesh of quadratic tetrahedral elements (C3D10 in ABAQUS). The element size is

controlled by the number of elements occupying the lattice strut diameter and is taken to be equal to the

corresponding FFT voxel size. An iterative solver (CG) has been selected to carry out a fair comparison

with FFT approaches which are based on the same type of iterative solver. Periodic boundary conditions

are used and introduced using multipoint linear constraints [48].

An example of a FEM mesh of the unit cell of an octet lattice with 10% relative density is represented in

Figure 3 together with two FFT voxelized models of the same geometry with diﬀerent discretization levels

All simulations have been done in a single node workstation Dual 10 core Intel(R) Xeon(R) CPU E5-2630

v4 @ 2.20GHz with 64GB RAM memory. Both ABAQUS and FFTMAD use parallelization by threading

(20 threads) both for the evaluation of the constitutive equations and for linear algebra operations.

Regarding the elastic properties of the material, the parameters correspond to a typical polyamide PA12

with isotropic linear elastic behavior with E= 1.7GPa and ν= 0.4. The numerical tolerances for the relative

errors in the linear iterative solvers in the Galerkin FFT, MoDBFFT and FEM have been set to 10−6.

4.2 Analysis of the numerical performance of FFT approaches

In this section, the convergence rate of the diﬀerent adaptations of FFT approaches for inﬁnite phase contrast

will be studied. To this aim, the evolution of the residual of the linear iterative solvers of eq. (4) for the

adapted Galerkin FFT (with modiﬁed frequencies and use of MINRES) and of eq. (29) for the MoDBFFT

(using standard frequencies and CG) will be compared.

Although the MoDBFFT method results in a fully determined system of linear equations with a unique

solution, the well-posedness of the resulting coeﬃcient matrix depends on the numerical parameter α.

The system becomes ill-posed when αdecreases and gets very low values with respect to the stiﬀness of

the material domain. On the other hand, larger values of αinduce an artiﬁcial stiﬀness in the BVP that

13

Figure 3: Octet lattice with 10% of relative density, FEM model with 15 elements per diameter and FFT

discretizations with 543and 2693voxels

aﬀects the computed eﬀective properties. In this case, although the null traction at the void interface is

still considered explicitly by eqs. (19-21), the overall results are slightly aﬀected. This eﬀect is only relevant

for large values of αand is due to the components of the stress tensor which are not contained in the

surface traction. Therefore, those components which are not directly cancelled by eqs. (19-21) are indirectly

aﬀecting the overall response.

In this study, a linear elastic test under uniaxial tension has been simulated for diﬀerent alphas in a ρ= 0.1

relative density octet-truss lattice discretized using 2153voxels (20 voxels/diameter) and PV representation.

Figure 4 represents the relative residual values for the diﬀerent linear equilibrium equations and diﬀerent α

values, being Eis the Young’s modulus of the lattice material. Lower values of αin the MoDBFFT lead to

Figure 4: Residual evolution and eﬀective Young’s modulus on an octet truss lattice for the diﬀerent αs

considered

worse convergence compared to the adapted Galerkin FFT method. As a trade-oﬀ, large αvalues aﬀect the

macroscopic properties calculated inducing an artiﬁcial stiﬀness, and in Figure 4 it can be observed that the

induced diﬀerences can go up to 10% in terms of the eﬀective stiﬀness. In this work, α= 10−4Eis selected

as a compromise of convergence rate and accuracy of computed eﬀective properties. It is interesting to note

that the use of discrete frequencies is the most important ingredient for the success of the adapted Galerkin

14

scheme. With discrete frequencies, the use of MINRES improves the performance of the CG version, but the

reduction in computation time is only around 5%. On the contrary, when standard Fourier discretization is

used the diﬀerence in the performance between both solvers becomes substantial.

4.3 Analysis of the surface smoothening approaches

The regular discretization used in FFT can lead to actual densities slightly diﬀerent from the target one,

especially for coarse discretizations. The use of diﬀerent geometrical representations (section 3) to smooth

out the surface also has an impact on the actual value of the relative density of the model. To quantify this

geometrical misrepresentation, the actual relative density considered on the RVE has been calculated for

each geometrical approach as the volume integral of the phase map, corresponding to

ρ=1

Nvox X

i∈Nvox

φi.

A target relative density of ρ= 0.1 has been analyzed for the four geometrical representations, plain voxelized

approach (PV), Voigt ﬁne grid (VFG), Voigt analytic (VA), phase-ﬁeld smoothening (PFS), and combined

smoothening (CS). A range of discretizations from 543to 2693voxels are considered, which corresponds

approximately to a range from 5 to 25 voxels per diameter in this particular case. The resulting densities

for diﬀerent levels of the RVE discretization are represented in Figure 5. From Figure 5 it can be observed

Figure 5: Resulting RVE relative density for the diﬀerent surface smoothening techniques with target relative

density of 10%.

that the misrepresentation of the density is quite limited and the maximum error is below 0.7% in all the

cases. If the discretization is reﬁned up to 1623then the error is reduced below 0.1% in the worst case. The

maximum deviations occur for the plain voxelized representation. On the contrary, the Voigt approaches

give a fairly good approximation of the density for all the mesh sizes considered, always below 0.1%. In the

case of VFG and PFS the small deviations from the target density are caused by the change on the ratio

between the discretization used for the simulation and the ﬁner one to compute the phase map. Finally,

it can be observed that the relative densities obtained using phase-ﬁeld smoothening (before thresholding)

do not modify the relative density of the geometry function used as input. This mass conservation in the

phase-ﬁeld smoothening is due to the periodic boundary conditions which makes that volume integration of

eq. (38) leads to φ=ω. Therefore PFS gives the same density as VFG and CS the same density as VA.

15

4.4 Accuracy of the methods

The accuracy of the macroscopic and microscopic numerical results obtained with the FFT approaches

combined with the diﬀerent smoothening techniques will be assessed for diﬀerent discretization levels, ranging

from 543to 2693voxels (Fig. 3). To this end, the uniaxial tensile deformation of an octet truss lattice with

a relative density of ρ= 0.1 is simulated for the combinations of FFT solvers and surface smoothening. The

RVE is deformed along one of its edges and stress free conditions are imposed in the perpendicular directions.

To assess the result of the simulations, the FFT results are compared with FEM results with an equivalent

discretization level in terms of number of elements per truss diameter.

The macroscopic strain and stress tensors are extracted from the simulation results to obtain the eﬀective

properties. The eﬀective Young’s modulus and Poisson’s ration have been represented in Figure 6 together

with the corresponding FEM results. First, it is observed that the Voigt ﬁne grid (VFG) approach shows

almost identical behavior than the Voigt analytic (VA) and therefore, for the shake of clarity, only VA will

be considered for the rest of the discussion.

Figure 6 shows that the convergence of FFT results with the discretization is slower than FEM results,

except the combination of the Galerkin approach with the VA that provides a solution almost independent

on the grid for model sizes greater than 10 elements per diameter (1083voxels). The largest oscillations

of the eﬀective response with the discretization are obtained for the plain voxelized models (PV) and are

a direct result of the variations in the actual relative density of the cells. The results of the phase-ﬁeld

smothening (PFS) and combined smoothening (CS) converge better than the plain voxelized representations

but worse than the VFG and VA approaches separately. Therefore, it can be concluded that using phase-ﬁeld

smoothening has a non-negligible eﬀect on the resulting stiﬀness for the same average density. In the case of

plain voxel approaches, the MoDBFFT method provides the same tendency as the Galerkin approach with

a small oﬀset, being the ﬁrst one slightly stiﬀer. In the case of Poisson’s ratio, which is less dependent on

the cell density, the convergence with the discretization is faster and smoother. The maximum diﬀerence

between FEM and FFT results is below 0.6% for every discretization. In all the cases, the eﬀective FFT

response converges to the same value that coincides with the FEM results.

Figure 6: Eﬀective Young’s modulus and Poisson’s ratio for diﬀerent discretizations.

The accuracy of the microscopic ﬁelds has also been analyzed and compared with FEM results. In Figure

7, the microscopic stress in the loading direction has been superposed to the deformed geometry, magniﬁed

by a factor ×20, for the two most representative FFT approaches (Galerkin with Voigt analytic smoothening

and MoDBFFT with a plain voxel approach). Qualitatively it can be observed that the deformed shapes are

almost identical and the concentrations of stress ﬁelds are very similar both in location and intensity.

To quantify this diﬀerence, the L2norm of the diﬀerence in the stress in the loading direction σzz (eq.

16

Figure 7: Local stress ﬁelds in loading direction (σz z) on the deformed conﬁguration (x20) for FEM, Galerkin

FFT (VA) and MoDBFFT (PV).

41) is computed with respect to the local ﬁelds of the FEM method and the result has been represented in

Fig. 8. It can be observed that, as it happened with the eﬀective response, FFT results converge to FEM

solutions. It is also remarkable that for discretizations ﬁner than 15 voxels/diameter, the diﬀerences are

always below 10% except for those methods where the phase-ﬁeld smoothening technique is used.

Figure 8: Local stress diﬀerences in the loading direction (σzz ) as function of the discretization level.

The solutions using the plain voxels geometric approach show a slightly better microscopic response but

need a larger number of voxels to accurately predict the overall behavior due to the density variations.

It is interesting to note that, for a PV geometrical representation, both modiﬁed Galerkin (which uses a

rotated scheme) and MoDBFFT —which has a standard discretization— provide very similar microﬁelds.

This result indicates that the terms included in the MoDBFFT to approach the free surface condition have

a similar eﬀect in smoothening the response to the use of discrete frequencies in the modiﬁed Galerkin,

as it can be observed in Fig. 8. Phase-ﬁeld smoothening alleviates the noise eﬃciently but, for the value

of `considered here, induces non-negligible changes in both macroscopic and microscopic responses. As

a summary, Galerkin method combined with Voigt analytic smoothening shows the best combination of

accuracy in the microﬁelds and eﬀective response.

17

4.5 Eﬀect of the relative density and numerical eﬃciency

The eﬀect of the relative density on the accuracy and eﬃciency of the diﬀerent FFT methods is studied

in this section. Relative densities ρranging from 0.5% to 30% are considered for the octet truss lattice.

A discretization of 15 voxels(elements)/diameter is used for every volume fraction, leading to models with

diﬀerent total number of voxels. The loading case applied is uniaxial stress, which is accounted for using

macroscopic mixed boundary conditions. For comparison purposes, ﬁnite element simulations with the same

conditions are also performed for every cell, using in this case 15 elements per diameter.

The macroscopic speciﬁc stiﬀness, E/¯ρ, and Poisson’ s ratio, ν, obtained using the diﬀerent approaches

are represented in Figure 9 as functions of the cell relative density. It can be observed that, in most of

the cases, the speciﬁc Young’s modulus (E/ ¯ρ) is very close to the FEM results. The maximum relative

diﬀerence with respect to FEM is 10% for the elastic modulus in the case of the phase-ﬁeld smoothening

method, showing again that although near-to-surface local ﬁelds are smoothened with this approach, the

macroscopic response is slightly altered. On the other hand, the diﬀerence of the response obtained using

the Galerkin approach with discrete frequencies and Voigt analytic smoothening (VA) with respect to FEM

is always below 2%. The prediction of the Poisson’s ratio was very accurate for all the methods and densities

considered, with maximum diﬀerences below 1.5%.

Figure 9: Eﬀective Young’s modulus and Poisson’s ratio time for diﬀerent relative densities.

The microscopic stresses obtained are also analyzed and compared with the FEM counterparts. The

localization of stresses and their intensity are strongly dependent on the relative density due to the change

from a stretch to a bending dominated behavior. In all the cases the response of all the FFT approaches

considered was very similar to the FEM results. This diﬀerent behavior is also reﬂected in the deformation

modes, which were also accurately captured for the FFT approaches for all the densities. As an illustration,

the diagonal stress component in the loading direction, σzz , is represented in Figure 10 for the cell with

¯ρ= 30% superposed to the deformation of the cell with a magniﬁcation of ×20. Qualitatively, it can be

observed how both stresses and deformed shape of the cell are very similar.

From a quantitative viewpoint, the L2norm of the diﬀerence (eq. 41) between the FFT microscopic stress

component σzz and the FEM value was computed and represented in Fig. 11. The norm of the diﬀerence

was around 15% in most of the cases. Again, the microscopic response of PV approaches are more near to

the FEM results than the other smoothening approaches.

As stated in the study of the cell with ¯ρ= 0.1 for diﬀerent discretization levels, the Galerkin FFT

combined with the Voigt smoothening shows the best compromise between the accuracy of eﬀective and the

local properties.

Finally, the numerical performance of the FFT approaches is analyzed for the diﬀerent relative densities.

18

Figure 10: Local stress ﬁelds in loading direction (σzz ) on the deformed conﬁguration (x20) for FEM,

Galerkin FFT (VA) and MoDBFFT (PV).

Figure 11: Local stress diﬀerences in the loading direction (σzz ) for diﬀerent relative densities.

The time spent on the simulations was obtained for diﬀerent FFT approaches and FEM and was represented

in Figure 12. It can be observed ﬁrst that all FFT approaches were more eﬃcient than FEM method

for relative densities greater than ≈5%. Note that this comparison is made for a particular choice of 15

elements/voxels per diameter and, if this number were increased, better performance of the FFT solver would

be expected. Second, curves in Fig. 12 show that if the number of elements/voxel per strut diameter is kept

constant, the simulation times decrease with the density for both FEM and FFT due to the reduction in the

total number of elements/voxels in the lattice. Nevertheless, the time reduction grows faster in FFT than in

FEM, and the FFT simulations for a relative density of ¯ρ= 0.3 were 4 to 8 times faster than FEM ones. The

improvement of the performance ratio FFT/FEM with the density can be easily explained by the number

of voxels of the full RVE that belong to the interior of the lattice. For low densities most of the voxels of

the RVE belong to the empty space, not contributing to the cell response but having to be considered for

FFT operations. Therefore, the use of FFT for densities below ¯ρ < 7% is not competitive with respect to

FEM. On the contrary, it is remarkable that even with this strong disadvantage, FFT becomes clearly more

eﬃcient for relative densities exceeding 10% making the approaches here proposed very competitive for foams

and porous materials. As a conclusion, among the diﬀerent FFT approaches and smoothening techniques

proposed, the Galerkin FFT combined with the Voigt analytical smoothening is the most interesting one

since it combines very accurate results with the best numerical performance. The displacement approach

19

developed, the MoDBFFT, can provide smooth results which are as accurate as the modiﬁed Galerkin, but

is not competitive in terms of eﬃciency since for obtaining such accurate results a small parameter αis

required (i.e. α= 10−4E) and, for this value, the number of iterations is larger than the adapted Galerkin

approach.

Figure 12: Simulation time for diﬀerent relative densities, using models with around 15 elements/voxels per

diameter.

5 Validation for non linear material response

To assess the non linear extension of the methods proposed, simulations have been made using a Von Mises J2

plasticity model as lattice material behavior. The elastic constants are the same as in the previous section

and perfectly plastic behavior is considered (no strain hardening), being the yield stress σy= 70MPa.

Uniaxial compression tests have been carried out with a maximum strain of 10%. The strain is applied

using a ramp divided in 20 regular strain increments. The tolerance used for the Newton-Raphson method is

5·10−3. The approaches selected for this study are the Galerkin with Voigt analytic smoothening (VA) and

the MoDBFFT with a plain voxelized (PV) representation, considered as the most representative methods

from previous results. Octet truss lattices with relative densities of 10%, 20% and 30% are studied for a

ﬁxed discretization of 15 voxels(elements)/diameter. FEM simulations with equivalent discretization are

performed for comparison purposes.

The resulting macroscopic stress-strain curves are represented in Figure 13. The simulations predict

a large elastic region followed by a plastic regime with a very small hardening rate. The elastic-plastic

transition is smooth and the strength reached increases with the relative density. The results of all the

simulations are very similar, being the maximum diﬀerence between FEM and FFT results smaller than 5%

for all the densities considered.

The microscopic accumulated equivalent plastic strain ﬁeld εp, deﬁned as

εp=Zt˙

εP:˙

εP1/2dt

with ˙

εPthe plastic strain rate tensor, has been represented in Figure 14 for both FFT and FEM approaches

and time corresponding to a total compressive strain of -10%. The iso-plots are represented in the deformed

cell (x2) to observe the deformed cell shape. During the simulations, it was observed that the plastiﬁcation

started in the strut joints and after plastifying these joints behave as ball joints resulting in an almost

20

Figure 13: Stress-strain curves with non linear material

uniaxial stretching of the struts, as it can be observed in Figure 14. The deformed conﬁgurations and stress

distributions predicted by both FFT approaches are very similar to the FEM results.

Figure 14: Equivalent plastic strain for FEM, Galerkin FFT with Voigt analytic smoothening and the

MoDBFFT with plain voxelized approach.

For a quantitative measure of the local diﬀerence between FFT and FEM microscopic results, the L2

norm of the diﬀerences in the stress component in the loading direction is represented in Fig. 15. In all the

cases the agreement is good also from a microscopic viewpoint and the diﬀerences were always below 15%.

These diﬀerences vary with the volume fraction and also with the geometrical representation, since plastic

strain is very localized and a smoothened surface representation might aﬀect the intensity of the localization

in those regions.

6 Application to a real 3D tomography

The principal application of the FFT framework adaptation for lattices is the ability to analyze directly

the actual topology obtained by 3D tomography or a similar approach. In this section, the potential of the

FFT framework proposed will be shown by simulating and comparing the responses of an ideal lattice and

its real counterpart considering the fabrication defects resulting from the additive manufacturing process.

This example illustrates that full-ﬁeld simulation of the cell microstructure obtained by tomography is an

21

Figure 15: Local stress diﬀerences in the loading direction (σzz ) for the diﬀerent relative densities with non

linear material.

extremely powerful technique to quantify the changes in the cell response due to fabrication defects. Under

this framework, the real porosity can be considered explicitly inside the RVE, without performing any post-

processing to obtain averaged porosities and without the need of using homogenization models to account

for the eﬀect of the average porosity [7].

The unit cell selected is a cubic-diagonal lattice with a designed relative density of 14.2% and manufac-

tured in PA12 by Selective Laser Melting powder deposition. The cell edge length is 6.2mm and the trust

nominal diameter is 0.9mm. The cell was fabricated by CIRP https://www.cirp.de/comp/comp\_EN.php5

following standard fabrication parameters.

The cell microstructure was analyzed by 3D tomography using a GE (Phoenix) Nanotom 160 kV with a

Hamamatsu 7942-25SK detector and nanofocus X-ray source. The resolution of each voxel was ≈4µm and

the tomographic data included 1551 ×1557 ×1581 voxels. An image of the full resolution 3D tomography

is shown in Figure 16. The analysis of the 3D tomography data of the actual cell microstructure shows a

volume fraction of porosity of around 3.6%. The presence of that porosity inside the struts might aﬀect the

properties of the cell and the actual behavior can diﬀer from the one expected for the design geometry.

To quantify the eﬀect of the porosity in the elastic response, the design and actual geometries have been

subjected to uniaxial test of 1% of deformation using the two diﬀerent FFT approaches. For the design

geometry, a discretization of 25 voxels per diameter was used (2163voxel RVEs). In the case of the real

tomography, the original 15003pixel 3D tomography image has been compressed to 2563voxelized model by

averaging the densities obtained from the tomography (Fig. 16, right ﬁgure). In the case of the Galerkin FFT,

the smooth map of averaged densities has been directly introduced as the phase-map φinto the simulation.

In the case of the MoDBFFT, thresholding of the densities has been performed, distinguishing as a material

point all the local densities above a value that enforces an average relative density equal to the measured

one, and as empty region all the rest of the points.

The eﬀective response obtained are given in Table 1. The FFT simulations predict a decay in the eﬀective

Young’s modulus of around 10%, a relatively large reduction considering the low porosity volume fraction

measured (3.6%). The prediction of the overall stiﬀness reduction is a very interesting characteristic of the

FFT framework because it cannot be accurately obtained using a mean ﬁeld approach since the location of

the porosity within the struts inﬂuences the macroscopic response of the cell.

In addition to the changes of the macroscopic response, thanks to the resolution of the local ﬁelds,

the FFT analysis can be used to estimate the microscopic ﬁelds and hot spots of the structure and, using

damage indicators, the reduction of the lattice strength due to the presence of defects. As an example of

22

Figure 16: Left: Full resolution 3D tomography of the cell, Right: FFT model from tomographic data.

Galerkin FFT (density map) MoDBFFT (threshold map)

E[M P a]ν E [M P a]ν

Design geometry 45.76 0.314 46.25 0.313

3D Tomography 41.06 0.292 43.00 0.291

Table 1: Eﬀective properties extracted from uniaxial tests on cubic-diagonal lattice design and actual ge-

ometries.

these microscopic ﬁelds, the stress component in the loading direction σzz obtained in FFT simulations

has been represented in Fig. 17. It must be noted that the diﬀerences in local ﬁelds using Galerkin FFT

(using a density map as phase map) and MoDBFFT (using a pure voxelized approach) are below 10%. In

the cell with a perfect microstucture, stress concentrates in the trust joints and varies smoothly through

the geometry of the bars. On the contrary, on the simulations with the actual microstructure, large stress

concentrations localized near bigger pores are observed superposed to the concentrations near the joints. The

maximum stresses found with the real microstructures are approximately 50% larger than the ones obtained

with the designed cell. Due to these stress concentrations, if the maximum local stress were taken as a rough

estimation of the fracture initiation, the real structure would fail at stress level 50% lower than the design

geometry.

7 Conclusions

In this paper, an optimal FFT framework for the homogenization of lattice materials has been searched

and validated. The challenge was ﬁnding an FFT approach that preserves the accuracy, good numerical

performance and ability to use images/tomographies as direct input, in the case of domains with large

regions of empty space. To this aim, two diﬀerent FFT approaches able to solve problems containing phases

with zero stiﬀness were combined with several approaches to smooth out the lattice surface in order to

improve its geometrical representation and reduce the noise in the microscopic solution.

Regarding FFT solvers, after a ﬁrst analysis, two algorithms have been selected as the best options

for inﬁnite phase contrast. The ﬁrst one is an adaptation of the Galerkin FFT approach using MINRES

as linear solver and modiﬁed Fourier frequencies to consider a discrete diﬀerentiation scheme, the rotated

forward approach. The second one, the MoDBFFT, is a method based on the displacement FFT approach

in [26] which eliminates the indeterminacy of strains in the empty regions leading to a fully determined

system of equilibrium equations which allows the use of standard Fourier discretization and diﬀerentiation.

The accuracy of the two FFT solvers considered has been validated by comparison with FEM simulations of

an octet cell for several volume fractions and discretization levels. The homogenized response of both FFT

23

Figure 17: Local stress ﬁelds in loading direction (σzz ) on the deformed conﬁguration (x20) for Galerkin

FFT (top) and MoDBFFT (bottom), design and real geometries.

approaches was almost identical to FEM macroscopic response for linear elastic and elastoplastic materials,

and diﬀerences in microﬁelds were below 20%.

Regarding the surface smoothening, several approaches have been considered based on modifying the

actual stiﬀness of the voxels not fully embedded in the lattice or empty space. The impact of these geometrical

representations in the eﬀective response and local ﬁelds has been analyzed. The Voigt analytic smoothening

technique, which interpolates the stiﬀness of the interfacial voxels with the distance to the real lattice surface,

was the best option since it allowed to represent exactly the relative density of the cell allowing to use coarser

grids with very accurate macroscopic response.

In terms of numerical eﬃciency, both FFT solvers succeed in converging in a relatively small number

of iterations considering actual zero stiﬀness for the empty regions. Nevertheless, the adaptation of the

Galerkin framework convergence rate was slightly superior in all the cases. When comparing the eﬃciency

with respect to FEM with the same discretization in the interior of the cell, FFT became competitive for

relative densities greater than 7%. For relative densities of 30% (70% of porosity), the simulation of this last

FFT approach was 4 to 8 times faster than FEM. As a conclusion, the modiﬁed Galerkin approach combined

with Voigt analytic smoothening was the best FFT framework considering accuracy, numerical eﬃciency,

24

and best h-convergence.

Finally, to show the real potential of the approaches presented, both FFT frameworks are used to simulate

the behavior of an actual printed lattice by using direct 3D tomographic data as input. The simulation volume

element explicitly included the actual surface roughness and internal porosity (around 3.6%) resulting from

the fabrication process. The macroscopic elastic response was around 10% more compliant than the ideal

designed geometry, and local stress concentrations of 50% were found near large pores. As a summary, it

is shown that this technology can help to optimize the lattice fabrication parameters as well as accurately

determine the actual lattice response taking into account the real fabrication defects.

Acknowledgements

The authors gratefully acknowledge the support provided by the Luxembourg National Research Fund

(FNR), Reference No. 12737941 and the European Union’s Horizon 2020 research and innovation programme

for the project “Multi-scale Optimisation for Additive Manufacturing of fatigue resistant shock-absorbing

MetaMaterials (MOAMMM)”, grant agreement No. 862015, of the H2020- EU.1.2.1. - FET Open Pro-

gramme.

References

[1] X. Yu, H. Zhou, H. Liang, Z. Jiang, and L. Wu. Mechanical metamaterials associated with stiﬀness,

rigidity and compressibility: A brief review. Progress in Materials Science, 94:114 – 173, 2018.

[2] V.S. Deshpande, N.A. Fleck, and M.F. Ashby. Eﬀective properties of the octet-truss lattice material.

Journal of the Mechanics and Physics of Solids, 49(8):1747 – 1769, 2001.

[3] S.Xu, J. Shen, S.Zhou, X. Huang, and Y. M. Xie. Design of lattice structures with controlled anisotropy.

Materials & Design, 93:443 – 447, 2016.

[4] C. Findeisen, J. Hohe, M. Kadic, and P. Gumbsch. Characteristics of mechanical metamaterials based

on buckling elements. Journal of the Mechanics and Physics of Solids, 102:151 – 164, 2017.

[5] C. Lestringant and D. M. Kochmann. Modeling of ﬂexible beam networks and morphing structures by

geometrically exact discrete beams. Journal of Applied Mechanics, 87(8), 05 2020. 081006.

[6] H. Wadley J. Berger and R. McMeeking. Mechanical metamaterials at the theoretical limit of isotropic

elastic stiﬀness. Nature, 543:533–537, 2017.

[7] Y. Amani, S. Dancette, P. Delroisse, A. Simar, and E. Maire. Compression behavior of lattice struc-

tures produced by selective laser melting: X-ray tomography based experimental and ﬁnite element

approaches. Acta Materialia, 159:395 – 407, 2018.

[8] P. Lhuissier, C. de Formanoir, G. Martin, R. Dendievel, and S. Godet. Geometrical control of lattice

structures produced by EBM through chemical etching: Investigations at the scale of individual struts.

Materials & Design, 110:485 – 493, 2016.

[9] A. Prakash and R.A. Lebensohn. Simulation of micromechanical behavior of polycrystals: ﬁnite ele-

ments versus fast Fourier transforms. Modelling and Simulation in Materials Science and Engineering,

17(6):064010, 2009.

[10] S. Lucarini and J. Segurado. On the accuracy of spectral solvers for micromechanics based fatigue

modeling. Computational Mechanics, 63:365 – 382, 2019.

[11] R. Ma and W. Sun. FFT-based solver for higher-order and multi-phase-ﬁeld fracture models applied

to strongly anisotropic brittle materials. Computer Methods in Applied Mechanics and Engineering,

362:112781, 2020.

[12] F. Ernesti, M. Schneider, and T. B¨ohlke. Fast implicit solvers for phase-ﬁeld fracture problems on

heterogeneous microstructures. Computer Methods in Applied Mechanics and Engineering, 363:112793,

2020.

25

[13] M. Magri, S. Lucarini, G. Lemoine, L. Adam, and J. Segurado. An FFT framework for simulating

non-local ductile failure in heterogeneous materials. Computer Methods in Applied Mechanics and

Engineering, 380:113759, 2021.

[14] H. Moulinec and P. Suquet. A FFT-based numerical method for computing the mechanical proper-

ties of composites from images of their microstructures. In R. Pyrz, editor, IUTAM Symposium on

Microstructure-Property Interactions in Composite Materials, pages 235–246, Dordrecht, 1995. Springer

Netherlands.

[15] H. Moulinec and P. Suquet. A numerical method for computing the overall response of nonlinear

composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering,

157(1):69 – 94, 1998.

[16] D. J. Eyre and G. W. Milton. A fast numerical scheme for computing the response of composites using

grid reﬁnement. European Physics Journal Applied, 6(1):41–47, 1999.

[17] Sven Kaßbohm, Wolfgang H. M¨uller, and Robert Feßler. Fourier series for computing the response of

periodic structures with arbitrary stiﬀness distribution. Computational Materials Science, 32(3):387–

391, 2005.

[18] J. Zeman, J. Vondˇrejc, J. Nov´ak, and I. Marek. Accelerating a FFT-based solver for numerical homog-

enization of periodic media by conjugate gradients. Journal of Computational Physics, 229(21):8065 –

8071, 2010.

[19] M. Kabel, T. B¨ohlke, and M. Schneider. Eﬃcient ﬁxed point and Newton-Krylov solvers for FFT-based

homogenization of elasticity at large deformations. Computational Mechanics, 54:1497–1514, 12 2014.

[20] M. Schneider. On the Barzilai-Borwein basic scheme in FFT-based computational homogenization.

International Journal for Numerical Methods in Engineering, 118:482–494, 2019.

[21] D. Wicht, M. Schneider, and T. B¨ohlke. Anderson-accelerated polarization schemes for fast Fourier

transform-based computational homogenization. International Journal for Numerical Methods in En-

gineering, 122(9):2287–2311, 2021.

[22] J. Vondˇrejc, J. Zeman, and I. Marek. An FFT-based Galerkin method for homogenization of periodic

media. Computers & Mathematics with Applications, 68(3):156 – 173, 2014.

[23] J. Zeman, T.W.J. de Geus, J. Vondˇrejc, R. H. J. Peerlings, and M.G.D. Geers. A ﬁnite element

perspective on non-linear FFT-based micromechanical simulations. International Journal for Numerical

Methods in Engineering, 111:903–, 09 2017.

[24] T.W.J. de Geus, J. Vondˇrejc, J. Zeman, R.H.J. Peerlings, and M.G.D. Geers. Finite strain FFT-based

non-linear solvers made simple. Computer Methods in Applied Mechanics and Engineering, 318:412 –

430, 2017.

[25] S. Lucarini and J. Segurado. An algorithm for stress and mixed control in Galerkin based FFT homog-

enization. International Journal for Numerical Methods in Engineering, 119:797–805, 2019.

[26] S. Lucarini and J. Segurado. DBFFT: A displacement based FFT approach for non-linear homogeniza-

tion of the mechanical behavior. International Journal of Engineering Science, 144:103131, 2019.

[27] J. C. Michel, H. Moulinec, and P. Suquet. A computational scheme for linear and non-linear composites

with arbitrary phase contrast. International Journal for Numerical Methods in Engineering, 52(1-

2):139–160, 2001.

[28] H. Moulinec and F. Silva. Comparison of three accelerated FFT-based schemes for computing the me-

chanical response of composite materials. International Journal for Numerical Methods in Engineering,

97(13):960–985, 2014.

[29] S. Brisard and L. Dormieux. FFT-based methods for the mechanics of composites: A general variational

framework. Computational Materials Science, 49(3):663 – 671, 2010.

26

[30] Q.D.To and G. Bonnet. FFT based numerical homogenization method for porous conductive materials.

Computer Methods in Applied Mechanics and Engineering, 368:113160, 2020.

[31] M. Schneider. Lippmann-Schwinger solvers for the computational homogenization of materials with

pores. International Journal for Numerical Methods in Engineering, 121(22):5017–5041, 2020.

[32] S. Kaßbohm, W. H. M¨uller, and R. Feßler. Improved approximations of Fourier coeﬃcients for computing

periodic structures with arbitrary stiﬀness distribution. Computational Materials Science, 37(1):90–93,

2006. Proceedings of the 14th International Workshop on Computational Mechanics of Materials.

[33] L. G´el´ebart and F. Ouaki. Filtering material properties to improve FFT-based methods for numerical

homogenization. Journal of Computational Physics, 294:90–95, 2015.

[34] P. Shanthraj, P. Eisenlohr, M. Diehl, and F. Roters. Numerically robust spectral methods for crystal

plasticity simulations of heterogeneous materials. International Journal of Plasticity, 66:31–45, 2015.

Plasticity of Textured Polycrystals In Honor of Prof. Paul Van Houtte.

[35] W. H. M¨uller. Fourier transforms and their application to the formation of textures and changes of

morphology in solids. In Yehia A. Bahei-El-Din and George J. Dvorak, editors, IUTAM Symposium on

Transformation Problems in Composite and Active Materials, pages 61–72. Kluwer, 1998.

[36] F. Willot. Fourier-based schemes for computing the mechanical response of composites with accurate

local ﬁelds. Comptes Rendus M´ecanique, 343(3):232 – 245, 2015.

[37] K. S. Djaka, S. Berbenni, V. Taupin, and R. A. Lebensohn. A FFT-based numerical implementation

of mesoscale ﬁeld dislocation mechanics: Application to two-phase laminates. International Journal of

Solids and Structures, 184:136–152, 2020.

[38] M. Schneider, F. Ospald, and M. Kabel. Computational homogenization of elasticity on a staggered

grid. International Journal for Numerical Methods in Engineering, 105(9):693–720, 2016.

[39] M. Schneider, D. Merkert, and M. Kabel. FFT-based homogenization for microstructures discretized by

linear hexahedral elements. International Journal for Numerical Methods in Engineering, 109(10):1461–

1489, 2017.

[40] K. S. Eloh, A. Jacques, and S. Berbenni. Development of a new consistent discrete Green operator

for FFT-based methods to solve heterogeneous problems with eigenstrains. International Journal of

Plasticity, 116:1–23, 2019.

[41] M. Kabel, D. Merkert, and M. Schneider. Use of composite voxels in FFT-based homogenization.

Computer Methods in Applied Mechanics and Engineering, 294:168 – 188, 2015.

[42] M. Suard, G. Martin, P. Lhuissier, R. Dendievel, F. Vignat, J.-J. Blandin, and F. Villeneuve. Mechanical

equivalent diameter of single struts for the stiﬀness prediction of lattice structures produced by Electron

Beam Melting. Additive Manufacturing, 8:124 – 131, 2015.

[43] Z. Chen, Y. M. Xie, X. Wu, Z. Wang, Q. Li, and S. Zhou. On hybrid cellular materials based on triply

periodic minimal surfaces with extreme mechanical properties. Materials & Design, 183:108109, 2019.

[44] V. Monchiet and G. Bonnet. A polarization-based FFT iterative scheme for computing the eﬀective

properties of elastic composites with arbitrary contrast. International Journal for Numerical Methods

in Engineering, 89(11):1419–1436, 2012.

[45] D. Wicht, M. Schneider, and T. B¨ohlke. On Quasi-Newton methods in fast Fourier transform-based

micromechanics. International Journal for Numerical Methods in Engineering, 121(8):1665–1694, 2020.

[46] R. J. Lange. Potential theory, path integrals and the Laplacian of the indicator. Journal of High Energy

Physics, 32(11):1029–8479, 2012.

[47] J. Lemaitre. A continuous damage mechanics model for ductile fracture. Journal of Engineering Mate-

rials and Technology-transactions of The Asme, 107(1):83–89, 1985.

[48] J. Segurado and J. Llorca. A numerical approximation to the elastic properties of sphere-reinforced

composites. Journal of the Mechanics and Physics of Solids, 50(10):2107 – 2121, 2002.

27