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Comput. Methods Appl. Mech. Engrg. 388 (2022) 114223

www.elsevier.com/locate/cma

Adaptation and validation of FFT methods for homogenization of

lattice based materials

S. Lucarinia, L. Cobiana,b, A. Voitusa, J. Seguradob,a,∗

aFundación IMDEA Materiales, C/ Eric Kandel 2, 28906, Getafe, Madrid, Spain

bUniversidad Politécnica de Madrid, Department of Materials Science, E.T.S.I. Caminos, C/ Profesor Aranguren 3, 28040, Madrid, Spain

Received 31 December 2020; received in revised form 2 August 2021; accepted 1 October 2021

Available online xxxx

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.

c

⃝2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: FFT homogenization; Lattice materials; Micromechanics; Image based simulations

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 [1–3]. In addition, unit cells of lattice metamaterials can

∗Corresponding author at: Universidad Polit´

ecnica de Madrid, Department of Materials Science, E.T.S.I. Caminos, C/ Profesor Aranguren

3, 28040, Madrid, Spain.

E-mail address: javier.segurado@upm.es (J. Segurado).

https://doi.org/10.1016/j.cma.2021.114223

0045-7825/ c

⃝2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223

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 it is fundamental to perform accurate and computa-

tionally 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 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 [7,8], 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 [11–13].

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 [14–21]. 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–

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 fulﬁll 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

2

S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223

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 [32–34]. 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]. Two other ﬁnite differentiation schemes which

are not based on the use of modiﬁed frequencies were also proposed in [38,39] 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 [8,43], 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 non linear 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 voids and pores

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.

3

S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223

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 strongly deteriorates 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, such 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 was innovative and smart, the consistent differential operator derived in this work is

very complex to compute and the authors themselves do not use it in the end, 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 different 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.

As a ﬁrst candidate for the lattice material homogenization, the Galerkin FFT approach [22,23] combined with

the use of mixed control [25] and the rotated ﬁnite difference scheme [36] has been chosen. This combination was

selected 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 linear 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 Galerkin FFT method was initially developed by Vondrˇ

ejc 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 Ωthe following equation can be derived

F−1

G(ξ):F{σ(x)}=0, (1)

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

local constitutive equations – and

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 Eq. (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 ,

4

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

of Eq. (1) is a linear system of algebraic equations in which the unknown is the value of the strain ϵat 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)sy m

or macroscopic stress components K L σ=σK L (eK⊗eL)sy m with kl ∩K L = ∅. For these general loading

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

Gi j kl (ξ)=

Is

i j K L if ξ=0for components K L

0i jk l if ξ=0for components kl

0i jk l for Nyquist frequencies

Is

i pkq ξpξq−1

ξjξlfor ξ̸= 0

, (3)

where Isis the fourth order symmetric identity tensor and

Gaccounts for major and minor symmetries.

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

F−1

G(ξ):F{C(x):ε(x)}= −F−1

G(ξ):F{C(x):ε−σ}, (4)

where Cis the local fourth order stiffness tensor. Eq. (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−1

G(ξ):F{C(x): •}(5)

and the right-hand side is the independent term b

b= −F−1

G(ξ):F{C(x):ε−σ}.

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,

rli n =∥A(ε)−b∥L2

∥b∥L2

(6)

reaching the solution when rli n is below a tolerance. Note that Eq. (4) is undetermined independently of the phase

properties because any incompatible ﬁeld εI N C added to the solution still fulﬁlls Eq. (4),

A(ε+εI N 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 overcome this limitation.

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

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

5

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

correspond to

ξ′

p=i2Np

Lp

tan qp/2d

p=1

1

21+eiq pwith 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.

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)

∂εϵ=ϵi−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−1

G(ξ):FCi−1(x):δε(x)= −F−1

G(ξ):Fσεi−1(x)−σt+∆t , (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 Eq. (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 establishes the convergence

criterion on the relative residual, deﬁned as

∥Ai(x)−bi∥L2

∥bi∥L2

,

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 ∥b0∥is 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

rli n =∥Ai(δε)−bi∥L2

∥b0∥L2

.(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 ∥bi∥/∥b0∥is 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

6

Fig. 1. Schematic representation of the domains for the standard FFT and the present approach.

the last deformation gradient correction and the inﬁnity norm of the change in the total strain ﬁeld within each time

increment

rnewt on =∥δε∥∞

εi−εt

∞

.(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.

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 discretization – using

trigonometric polynomials – and derivation in Fourier space, without the need of using alternative discretization

schemes 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 Fig. 1(a)) representing the two phases: Ωm, the matrix,

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

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

Ωm

σ(∇su): ∇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 Ωso any

compatible strain ﬁeld is acceptable for the stress equilibrium in the region Ωvand the system is underdetermined.

7

In the new approach, the standard formulation of the two-phase domain is modiﬁed (see Fig. 1(b)) 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 that 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 which is deﬁned only in

that region and corresponds to

Ωv

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)

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

Ω

(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 Γof the boundary value problem, deﬁned in the second

equation of Eq. (12), should be imposed in Eq. (17). To this aim, the boundary condition on Γis diffused over a

thin volume which is deﬁned by the surface Γwith 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. 1(b). 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

Γ

σ(x)·nΓdΓ=Ω

σ(x)·nΓδ(x)ΓdΩ=Ω

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 Γit is fulﬁlled

σ·n=α∇su(x)·nin Γ

then, the volume counterparts of any of these terms are also identical,

σ(x)· ∇χv=α∇su(x)· ∇χvin Ω,

and any of them can be set to zero in Eq. (17). 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, the displacement ﬁeld is split into two contributions as

u(x)=

u(x)+ε·x, (22)

where

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):∇s

u(x)+εσ+χv(x)∇ · α∇s

u(x)= −∇ · (C(x):ε−σ), (23)

where the ﬁeld

uis the unknown, that has to be solved together with the I J 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 I J components of εσwith the corresponding components of the macroscopic stress,

1

VΩΩC(x):∇s

u(x)+εσI J dΩ= − 1

VΩΩ

[C(x):ε]I J 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−1F{•}⊗sξ=F−11

2(F{•}⊗ξ+ξ⊗F{•})(25)

∇ · (•)=F−1{F{•}·ξ}(26)

To solve (23) in 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(•), proposed for the

DBFFT approach [26], is also used here as preconditioner to improve convergence,

M(•)=∇ · C∇−1

(•) , (27)

where Cis the volume averaged stiffness tensor,

C=1

VΩΩ

C(x)dΩ(28)

and VΩrepresents the volume of the entire domain Ω.

The resulting equilibrium equation written in Fourier space yields

F(1−χv)C:F−1

u⊗sξ+εσ·ξ+FχvF−1α

u⊗sξ·ξ= −F{C:ε−σ}·ξ, (29)

and the Fourier transform of the equation to impose the macroscopic stress components I J (Eq. (24)) corresponds

to

F(1−χv)C:F−1

u⊗sξ+εσ(0)=F{C:ε−σ}(0). (30)

Eqs. (29),(30) are linear and can be solved iteratively using a Krylov solver. For preconditioning the system,

the Fourier space representation of the preconditioner in Eq. (27) is used, which reads

M(∗)=ξ·C·ξ−1

· ∗ , (31)

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

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)),

u

εσ.

9

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

following

Aα

u

εσ=F(1−χv)C:F−1

u⊗sξ+εσ·ξ+FχvF−1α

u⊗sξ·ξ

F(1−χv)C:F−1

u⊗sξ+εσ(0)(32)

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

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

rli n =

Aα

u

εσ−

bα

L2

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 Eq. (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

F(1−χv)Ci−1:F−1

δ

u⊗ξ+δεσ·ξ+FχvF−1α

δ

u⊗ξ·ξ

= −F(1−χv)σF−1

ui−1⊗ξ+εi−1

σ+εt+∆t−σt+∆t·ξ

−FχvF−1α

ui−1⊗ξ·ξ (35)

in which the displacement correction and the average strain correction,

δ

uand δεσ, are the unknowns and the

solution for iteration iis updated as

ui=

ui−1+δ

u;εi

σ=εi−1

σ+δεσ.

In Eq. (35) the macroscopic prescribed strain and stress ﬁelds enter in the deﬁnition of the ﬁrst iteration as

u0=

ut

and ε0

σ=εt

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

Aα

i δ

u

δεσ

and an independent right-hand side vector

bα

i, both deﬁned in the ith 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

rli n =

Aα

i δ

u

δεσ−

bα

i

L2

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).

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

10

Fig. 2. Local densities (phase maps φ) near the cross section of a strut for the different geometrical approaches. In the gray scale φ=1

corresponds to black and 0 to white.

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 ℓ. 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 corresponds 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]=Ω

1

2ℓ2∥∇φ∥2+ϵ

2(φ−ω)2dΩ, (37)

where ℓis the characteristic length, a parameter which deﬁnes the width of the smoothening region and ϵis

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 gradients of φand 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)

ℓ

ϵ

2

∇2φ−φ= −ω(38)

11

under periodic boundary conditions in φ. If the problem is discretized on 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

φ(ξ)=1

1−ℓ2

ϵξ·ξω(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 df ine 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, dcoar se . The ratio

between dcoar se /df i ne is between 2 and 20. The ﬁne discretization is used for 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 ℓ

the half of the length of the voxel in the coarse discretization (ℓ=0.5dcoars e ) 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 the characteristic weight ϵis 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 ℓ/2 with respect to the boundary

of the struts Γ. For those voxels centers whose distance to the boundary is smaller than ℓ/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+ℓ/2

ℓfor 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, ℓ, 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.

12

Fig. 3. Octet lattice with 10% of relative density, FEM model with 15 elements per diameter and FFT discretizations with 543and 2693

voxels.

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. 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 ( fr e f (x)) is used as metric of the error

Local diff. [%]=

f−fre f

L2

fre f

L2

. (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 interesting

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 whom 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 Fig. 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 @ 2.20 GHz with

64 GB 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.

13

Fig. 4. Residual evolution and effective Young’s modulus on an octet truss lattice for the different αs considered.

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

isotropic linear elastic behavior with E=1.7 GPa 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 convergence 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 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

canceled 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. Fig. 4

represents the relative residual values for the different linear equilibrium equations and different αvalues, being

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

14

Fig. 5. Resulting RVE relative density for the different surface smoothening techniques with target relative density of 10%.

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

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 smoothen-

ing (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 Fig. 5. From Fig. 5 it can be observed that the misrepresentation of the density is

limited and the maximum absolute error is below 0.7% in all the cases. If the discretization is reﬁned up to 1623

then 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.

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

15

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

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 ratio have been represented in Fig. 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 sake of clarity, only VA will be considered for the rest

of the discussion.

Fig. 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 of 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 smoothening (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.

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

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

16

Fig. 7. Local stress ﬁelds in loading direction (σzz ) on the deformed conﬁguration (×20) for FEM, Galerkin FFT (VA) and MoDBFFT

(PV).

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

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 ℓconsidered 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.

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 Fig. 9 as function 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

17

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

Fig. 10. Local stress ﬁelds in loading direction (σzz ) on the deformed conﬁguration (×20) for FEM, Galerkin FFT (VA) and MoDBFFT

(PV).

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%.

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 Fig. 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.

18

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

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

time spent on the simulations was obtained for different FFT approaches and FEM and was represented in Fig. 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, a 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 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.

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=70 MPa. 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 (Eq. (11)) 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.

19

Fig. 12. Simulation time for different relative densities, using models with around 15 elements/voxels per diameter.

Fig. 13. Stress–strain curves with non linear material.

The resulting macroscopic stress–strain curves are represented in Fig. 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=t˙

εP:˙

εP1/2dt

with ˙

εPthe plastic strain rate tensor, has been represented in Fig. 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 (×2)

to observe the deformed cell shape. During the simulations, it was observed that the plastiﬁcation started in the

20

Fig. 14. Equivalent plastic strain for FEM, Galerkin FFT with Voigt analytic smoothening and the MoDBFFT with plain voxelized approach.

Fig. 15. Local stress differences in the loading direction (σzz ) for the different relative densities with non linear material.

strut joints and after plastifying these joints behave as ball joints resulting in an almost uniaxial stretching of the

struts, as it can be observed in Fig. 14. The deformed conﬁgurations and stress distributions predicted by both FFT

approaches are very similar to the FEM results.

For a quantitative measure of the local difference between FFT and FEM microscopic results, the L2norm 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 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].

21

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

Table 1

Effective properties extracted from uniaxial tests on cubic-diagonal lattice design and actual geometries.

Galerkin FFT (density map) MoDBFFT (threshold map)

E[MPa] νE[MPa] ν

Design geometry 45.76 0.314 46.25 0.313

3D Tomography 41.06 0.292 43.00 0.291

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

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

is 0.9 mm. 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 Fig. 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 these microscopic ﬁelds, the

stress component in the loading direction σzz obtained in FFT simulations has been represented in Fig. 17. It must

22

Fig. 17. Local stress ﬁelds in loading direction (σzz ) on the deformed conﬁguration (×20) for Galerkin FFT (top) and MoDBFFT (bottom),

design and real geometries.

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.

23

Regarding FFT solvers, after a ﬁrst analysis, two algorithms have been selected as suitable 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 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, 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.

Declaration of competing interest

The authors declare that they have no known competing ﬁnancial interests or personal relationships that could

have appeared to inﬂuence the work reported in this paper.

Acknowledgments

The authors gratefully acknowledge the support provided by 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

Programme.

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