ArticlePDF Available

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.
Available online at www.sciencedirect.com
ScienceDirect
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 [13]. 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 [1113].
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 [1421]. 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 [3234]. 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 the following equation can be derived
F1
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 F1respectively.
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,...,Np1;p=1,2,3 ,
4
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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=2πnpNp/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 (ekel)sy m
or macroscopic stress components K L σ=σK L (eKeL)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 ξpξq1
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
F1
G(ξ):F{C(x):ε(x)}= −F1
G(ξ):F{C(x):εσ}, (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()=F1
G(ξ):F{C(x): •}(5)
and the right-hand side is the independent term b
b= −F1
G(ξ):F{C(x):εσ}.
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)bL2
bL2
(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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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=2πnpNp/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 εi1. Let δε(x)be the strain field correction to be obtained at iteration i, then the
linearized stress corresponds to
σi(x)=σi1(x)+σ(x)
εϵ=ϵi1:δε(x)=σi1(x)+Ci1(x):δε(x), (8)
where Ci1is the material consistent tangent evaluated using the solution of the previous iteration i1. The
equilibrium equation linearized at εi1reads
F1
G(ξ):FCi1(x):δε(x)= −F1
G(ξ):Fσεi1(x)σt+t , (9)
where the solution at the previous time step enters in the first iteration as ε0(x)=εt(x)+εt+tεt.
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 i1 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)biL2
biL2
,
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 b0is 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)biL2
b0L2
.(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/b0is 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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
εiεt
.(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: m, the matrix,
and v, the void. The interphase between the two phases is Γ.
The real boundary value problem is defined only in the domain mand corresponds to finding the displacement
field umsuch that
∇ · σ(su(x)) =0in m
σ(x)·nΓ=0in xΓ
σand εperiodic in
.(12)
Due to the periodicity in , the weak formulation of this problem with free Neumann boundary conditions on Γ
is simply
m
σ(su): ∇sδudm=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 vwhere no material exists and
fields are not defined. On the contrary, the balance equation (13) is not defined in the entire unit cell so any
compatible strain field is acceptable for the stress equilibrium in the region vand the system is underdetermined.
7
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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 , including the free Neumann conditions. To this aim,
first, an artificial elastic energy density is defined in vwhich 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 v.
The first variation of the total energy in vleads to a new term in the weak formulation which is defined only in
that region and corresponds to
v
αsu: ∇sδudv, (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 vto the full domain , the indicator
function of the voided region χv(x) which defines the microstructure, is introduced (Eq. (15))
χv=1xv
0xm
.(15)
Eqs. (13) and (14), defined in mand vrespectively, are then premultiplied by their corresponding indicator
functions to be defined in a unique domain , leading to
(1 χv)σ(u): ∇sδu+χvαsu: ∇sδud=0.(16)
Using the divergence theorem and removing boundary terms because of the periodicity, the corresponding strong
formulation of Eq. (16) follows
∇ · (1χv(x)) σsu(x)+ ∇ · χv(x)αsu(x)=0
and if the chain rule is applied to the previous equation and terms are regrouped, the result is
(1χv)∇ · σ+χv · αsu(x)+ ∇χv·(αsu(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=
σ(x)·nΓδ(x)Γd=
σ(x)· ∇χvd.(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=αsu(x)·nin Γ
then, the volume counterparts of any of these terms are also identical,
σ(x)· ∇χv=αsu(x)· ∇χvin ,
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
∇ · (1χv(x)) σsu(x)+χv(x) · αsu(x)=0(21)
together with periodic boundary conditions in su.
8
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
∇ · (1χv(x)) C(x):s
u(x)+εσ+χv(x) · αs
u(x)= −∇ · (C(x):εσ), (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
VC(x):s
u(x)+εσI J d= − 1
V
[C(x):ε]I J d+σI J .(24)
As usual in FFT methods, the differential operators can be defined by their Fourier space counterparts using
spatial frequencies as
s()=F1F{}sξ=F11
2(F{}ξ+ξF{})(25)
∇ · ()=F1{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()=∇ · C1
() , (27)
where Cis the volume averaged stiffness tensor,
C=1
V
C(x)d(28)
and Vrepresents the volume of the entire domain .
The resulting equilibrium equation written in Fourier space yields
F(1χv)C:F1
usξ+εσ·ξ+FχvF1α
usξ·ξ= −F{C:εσ}·ξ, (29)
and the Fourier transform of the equation to impose the macroscopic stress components I J (Eq. (24)) corresponds
to
F(1χv)C:F1
usξ+εσ(0)=F{C:εσ}(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·ξ1
· ∗ , (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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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(1χv)C:F1
usξ+εσ·ξ+FχvF1α
usξ·ξ
F(1χv)C:F1
usξ+εσ(0)(32)
and the right-hand side can be written as a vector reading as
bα=F{C:εσ}·ξ
F{C:εσ}(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(1χv)Ci1:F1
δ
uξ+δεσ·ξ+FχvF1α
δ
uξ·ξ
= −F(1χv)σF1
ui1ξ+εi1
σ+εt+tσt+t·ξ
FχvF1α
ui1ξ·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=
ui1+δ
u;εi
σ=εi1
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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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 . 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)=1χv(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]=
1
22∥∇φ2+ϵ
2(φω)2d, (37)
where is 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)
ϵ
2
2φφ= −ω(38)
11
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
12
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
the half of the length of the voxel in the coarse discretization (=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)=1D+ℓ/2
for D=d(x,Γ) if xm
d(x,Γ) if xv
, (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.
12
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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. [%]=
ffre 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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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 106.
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=104Eis 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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
iNvox
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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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 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.
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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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=104E)
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 ·103. 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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
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.
References
[1] X. Yu, H. Zhou, H. Liang, Z. Jiang, L. Wu, Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief
review, Prog. Mater. Sci. 94 (2018) 114–173.
[2] V.S. Deshpande, N.A. Fleck, M.F. Ashby, Effective properties of the octet-truss lattice material, J. Mech. Phys. Solids 49 (8) (2001)
1747–1769.
[3] S. Xu, J. Shen, S. Zhou, X. Huang, Y.M. Xie, Design of lattice structures with controlled anisotropy, Mater. Des. 93 (2016) 443–447.
[4] C. Findeisen, J. Hohe, M. Kadic, P. Gumbsch, Characteristics of mechanical metamaterials based on buckling elements, J. Mech. Phys.
Solids 102 (2017) 151–164.
[5] C. Lestringant, D.M. Kochmann, Modeling of flexible beam networks and morphing structures by geometrically exact discrete beams,
J. Appl. Mech. 87 (8) (2020) 081006.
[6] H. Wadley J. Berger, R. McMeeking, Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness, Nature 543 (2017)
533–537.
24
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
[7] Y. Amani, S. Dancette, P. Delroisse, A. Simar, E. Maire, Compression behavior of lattice structures produced by selective laser melting:
X-ray tomography based experimental and finite element approaches, Acta Mater. 159 (2018) 395–407.
[8] P. Lhuissier, C. de Formanoir, G. Martin, R. Dendievel, S. Godet, Geometrical control of lattice structures produced by EBM through
chemical etching: Investigations at the scale of individual struts, Mater. Des. 110 (2016) 485–493.
[9] A. Prakash, R.A. Lebensohn, Simulation of micromechanical behavior of polycrystals: finite elements versus fast Fourier transforms,
Modelling Simulation Mater. Sci. Eng. 17 (6) (2009) 064010.
[10] S. Lucarini, J. Segurado, On the accuracy of spectral solvers for micromechanics based fatigue modeling, Comput. Mech. 63 (2019)
365–382.
[11] R. Ma, W. Sun, FFT-based solver for higher-order and multi-phase-field fracture models applied to strongly anisotropic brittle materials,
Comput. Methods Appl. Mech. Engrg. 362 (2020) 112781.
[12] F. Ernesti, M. Schneider, T. Böhlke, Fast implicit solvers for phase-field fracture problems on heterogeneous microstructures, Comput.
Methods Appl. Mech. Engrg. 363 (2020) 112793.
[13] M. Magri, S. Lucarini, G. Lemoine, L. Adam, J. Segurado, An FFT framework for simulating non-local ductile failure in heterogeneous
materials, Comput. Methods Appl. Mech. Engrg. 380 (2021) 113759.
[14] H. Moulinec, P. Suquet, A FFT-based numerical method for computing the mechanical properties of composites from images of
their microstructures, in: R. Pyrz (Ed.), IUTAM Symposium on Microstructure-Property Interactions in Composite Materials, Springer
Netherlands, Dordrecht, 1995, pp. 235–246.
[15] H. Moulinec, P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure,
Comput. Methods Appl. Mech. Engrg. 157 (1) (1998) 69–94.
[16] D.J. Eyre, G.W. Milton, A fast numerical scheme for computing the response of composites using grid refinement, Eur. Phys. J. Appl.
6 (1) (1999) 41–47.
[17] Sven Kaßbohm, Wolfgang H. Müller, Robert Feßler, Fourier series for computing the response of periodic structures with arbitrary
stiffness distribution, Comput. Mater. Sci. 32 (3) (2005) 387–391.
[18] J. Zeman, J. Vondˇ
rejc, J. Novák, I. Marek, Accelerating a FFT-based solver for numerical homogenization of periodic media by
conjugate gradients, J. Comput. Phys. 229 (21) (2010) 8065–8071.
[19] M. Kabel, T. Böhlke, M. Schneider, Efficient fixed point and Newton–Krylov solvers for FFT-based homogenization of elasticity at
large deformations, Comput. Mech. 54 (2014) 1497–1514.
[20] M. Schneider, On the Barzilai–Borwein basic scheme in FFT-based computational homogenization, Internat. J. Numer. Methods Engrg.
118 (2019) 482–494.
[21] D. Wicht, M. Schneider, T. Böhlke, Anderson-accelerated polarization schemes for fast Fourier transform-based computational
homogenization, Internat. J. Numer. Methods Engrg. 122 (9) (2021) 2287–2311.
[22] J. Vondˇ
rejc, J. Zeman, I. Marek, An FFT-based Galerkin method for homogenization of periodic media, Comput. Math. Appl. 68 (3)
(2014) 156–173.
[23] J. Zeman, T.W.J. de Geus, J. Vondˇ
rejc, R.H.J. Peerlings, M.G.D. Geers, A finite element perspective on non-linear FFT-based
micromechanical simulations, Internat. J. Numer. Methods Engrg. 111 (2017) 903.
[24] T.W.J. de Geus, J. Vondˇ
rejc, J. Zeman, R.H.J. Peerlings, M.G.D. Geers, Finite strain FFT-based non-linear solvers made simple,
Comput. Methods Appl. Mech. Engrg. 318 (2017) 412–430.
[25] S. Lucarini, J. Segurado, An algorithm for stress and mixed control in Galerkin based FFT homogenization, Internat. J. Numer. Methods
Engrg. 119 (2019) 797–805.
[26] S. Lucarini, J. Segurado, DBFFT: A displacement based FFT approach for non-linear homogenization of the mechanical behavior,
Internat. J. Engrg. Sci. 144 (2019) 103131.
[27] J.C. Michel, H. Moulinec, P. Suquet, A computational scheme for linear and non-linear composites with arbitrary phase contrast,
Internat. J. Numer. Methods Engrg. 52 (1–2) (2001) 139–160.
[28] H. Moulinec, F. Silva, Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite
materials, Internat. J. Numer. Methods Engrg. 97 (13) (2014) 960–985.
[29] S. Brisard, L. Dormieux, FFT-based methods for the mechanics of composites: A general variational framework, Comput. Mater. Sci.
49 (3) (2010) 663–671.
[30] Q.D. To, G. Bonnet, FFT based numerical homogenization method for porous conductive materials, Comput. Methods Appl. Mech.
Engrg. 368 (2020) 113160.
[31] M. Schneider, Lippmann-Schwinger solvers for the computational homogenization of materials with pores, Internat. J. Numer. Methods
Engrg. 121 (22) (2020) 5017–5041.
[32] S. Kaßbohm, W.H. Müller, R. Feßler, Improved approximations of Fourier coefficients for computing periodic structures with arbitrary
stiffness distribution, Comput. Mater. Sci. 37 (1) (2006) 90–93, Proceedings of the 14th International Workshop on Computational
Mechanics of Materials.
[33] L. Gélébart, F. Ouaki, Filtering material properties to improve FFT-based methods for numerical homogenization, J. Comput. Phys.
294 (2015) 90–95.
[34] P. Shanthraj, P. Eisenlohr, M. Diehl, F. Roters, Numerically robust spectral methods for crystal plasticity simulations of heterogeneous
materials, Int. J. Plast. 66 (2015) 31–45.
[35] W.H. Müller, Fourier transforms and their application to the formation of textures and changes of morphology in solids, in: Yehia A.
Bahei-El-Din, George J. Dvorak (Eds.), IUTAM Symposium on Transformation Problems in Composite and Active Materials, Kluwer,
1998, pp. 61–72.
[36] F. Willot, Fourier-based schemes for computing the mechanical response of composites with accurate local fields, C. R. Méc. 343 (3)
(2015) 232–245.
25
S. Lucarini, L. Cobian, A. Voitus et al. Computer Methods in Applied Mechanics and Engineering 388 (2022) 114223
[37] K.S. Djaka, S. Berbenni, V. Taupin, R.A. Lebensohn, A FFT-based numerical implementation of mesoscale field dislocation mechanics:
Application to two-phase laminates, Int. J. Solids Struct. 184 (2020) 136–152.
[38] M. Schneider, F. Ospald, M. Kabel, Computational homogenization of elasticity on a staggered grid, Internat. J. Numer. Methods
Engrg. 105 (9) (2016) 693–720.
[39] M. Schneider, D. Merkert, M. Kabel, FFT-based homogenization for microstructures discretized by linear hexahedral elements, Internat.
J. Numer. Methods Engrg. 109 (10) (2017) 1461–1489.
[40] K.S. Eloh, A. Jacques, S. Berbenni, Development of a new consistent discrete Green operator for FFT-based methods to solve
heterogeneous problems with eigenstrains, Int. J. Plast. 116 (2019) 1–23.
[41] M. Kabel, D. Merkert, M. Schneider, Use of composite voxels in FFT-based homogenization, Comput. Methods Appl. Mech. Engrg.
294 (2015) 168–188.
[42] M. Suard, G. Martin, P. Lhuissier, R. Dendievel, F. Vignat, J.-J. Blandin, F. Villeneuve, Mechanical equivalent diameter of single struts
for the stiffness prediction of lattice structures produced by Electron Beam Melting, Addit. Manuf. 8 (2015) 124–131.
[43] Z. Chen, Y.M. Xie, X. Wu, Z. Wang, Q. Li, S. Zhou, On hybrid cellular materials based on triply periodic minimal surfaces with
extreme mechanical properties, Mater. Des. 183 (2019) 108109.
[44] V. Monchiet, G. Bonnet, A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with
arbitrary contrast, Internat. J. Numer. Methods Engrg. 89 (11) (2012) 1419–1436.
[45] D. Wicht, M. Schneider, T. Böhlke, On quasi-Newton methods in fast Fourier transform-based micromechanics, Internat. J. Numer.
Methods Engrg. 121 (8) (2020) 1665–1694.
[46] R.J. Lange, Potential theory, path integrals and the Laplacian of the indicator, J. High Energy Phys. 32 (11) (2012) 1029–8479.
[47] J. Lemaitre, A continuous damage mechanics model for ductile fracture, Trans. ASME, J. Eng. Mater. Technol. 107 (1) (1985) 83–89.
[48] J. Segurado, J. Llorca, A numerical approximation to the elastic properties of sphere-reinforced composites, J. Mech. Phys. Solids 50
(10) (2002) 2107–2121.
26
... The FFT-based homogenization approach and its variants have built an impeccable reputation in property prediction. Nowadays, they are widely used in studying elastic-viscoelastic behaviors of polycrystalline materials [38], crack propagation of composite laminates [39], electrical properties of highly-contrasted composites [40,41], and mechanical performances of 3D printing lattice [42]. To et al. [43] developed the FFT-based numerical homogenization method to obtain effective conductive properties of porous materials. ...
Article
Full-text available
Besides constituting components, the properties of composites are highly relevant to their microstructures. The work proposed a fast Fourier transform (FFT)-based inverse homogenization method implemented by the bi-directional evolutionary structural optimization (BESO) technique to explore the vast potential of cellular materials. The periodic boundary condition of self-repeated representative volume elements can be naturally satisfied in the FFT-based homogenization scheme. The objective function of the optimization problem is the specific moduli or the quadratic difference between the effective value and the target, which are obtained in terms of mutual strain energies. Its sensitivity to the design variable, namely the elemental density, is derived from the adjoint variable method and used as the criterion to remove or add material in local elements. Numerical examples show that the proposed method generates a series of architected cellular materials with maximum modulus, negative Poisson's ratio, and specific elasticity tensor. FFT-based homogenization in the method demands less memory usage but has high efficiency. Thus, it can achieve topology optimization of unit cell with one million hexahedral elements. This approach contributes to the extended application of FFT-based homogenization and can guide the microstructure design of mechanical metamaterials.
... The advantages of Fast Fourier Transform are its excellent numerical performance (the algorithm scales to n log n), its reduced computational cost compared to FE and its ability to increase the speed of simulations by several orders of magnitude [88]. The disadvantage of the FFT method is that its convergence rate and accuracy are highly dependent on the contrast between the phases represented in the domain [89]. It requires that the signal must be smooth. ...
Thesis
Nowadays, heterogeneous materials are in-creasingly used for their superior overall properties, suchas porous media, which are widely used in the electronicsand biomedical industries, so determining the equivalentthermal conductivity (ETC) of heterogeneous materials isessential for the correct design of industrial equipment thatmay be subjected to severe thermal loads during use.The main objective of this thesis is to calculate the ho-mogenization of the thermal conductivity of heteroge-neous materials using the finite difference method. Voxelwas chosen for modeling heterogeneous materials and the Günter scheme will be employed as the primary tech-nique for anisotropic thermal diffusion problems. Thetwo-dimensional Günter system is re-demonstrated in thisthesis, along with an extension to the three-dimensionalmodel, as well as methods for loading periodic and mixeduniform boundary conditions. The three methods (FDM,FEM, and FEM+pixel(voxel)) are compared for 2D RVEssuch as crosses, circles, and ellipses and for 3D RVEs suchas spheres and cylinders. It is discovered that the devel-oped FDM produces results that are consistent with thoseof FEM and FEM+pixel(voxel) and that the FDM outper-forms FEM+pixel(voxel) in terms of convergence speed.This method has also been applied to sintered silver ma-terials for the study of equivalent thermal conductivity.Comparisons between the two methods (FDM and FEM)are carried out for the classical unit cells such as simple cu-bic, body-centered cubic, and face-centered cubic, as wellas the silver-based stochastic model. The developed finitedifference algorithm is valid, and consistent results are ob-tained. In addition to the Günter scheme, a 5-point modeland an integral model have also been developed inspiredby the Günter scheme.For high-performance computing, the Eigen library andthe Pardiso library are also detailed in the thesis. Both li-braries contain both direct and iterative solutions for solv-ing linear equations. However, while Eigen allows for parallel computation of only the iterative solution, Pardisoallows for parallel computation of both approaches, andthe parallelism is significantly superior than that of Eigen.While Eigen is more straightforward to construct and morepowerful, Pardiso is faster at tackling complex problems.
... The Gibbs ringing artifacts should be strongly suppressed for this method, however we observe a checkerboard pattern of different style compared to (b). This checkerboard pattern originates in the well-known [34,35] (odd-even) decoupling into two subgrids over short distances of the central-differences stencil shown in Fig. 1a. The decoupling of the two grids is not complete because of an odd-sized grid (17 × 17) and the oscillations decay with increasing distance from the inhomogeneity. ...
Article
Full-text available
Micromechanical homogenization is often carried out with Fourier-accelerated methods that are prone to ringing artifacts. We here generalize the compatibility projection introduced by Vondřejc, Zeman & Marek [Comput. Math. Appl. 68, 156 (2014)] beyond the Fourier basis. In particular, we formulate the compatibility projection for linear finite elements while maintaining Fourier-acceleration and the fast convergence properties of the original method. We demonstrate that this eliminates ringing artifacts and yields an efficient computational homogenization scheme that is equivalent to canonical finite-element formulations on fully structured grids.
Article
Biomimetic lattice structures are one of the hotspots in current new material/structural technologies, and these lattice structures often have extraordinary mechanical properties. This work proposes a bamboo-inspired porous lattice structure (BPLS) that consists of bamboo-inspired prismatic microstructures (BPMs) and connecting plates. The sample of BPLS is fabricated by 3D printing, and the axial compression characteristic of BPLS is studied through simulation analysis and experiment. The deformation mode of BPLS is very stable that has excellent energy absorption effect. The effects of five geometric parameters on the compression performance of the BPLS are investigated. The deformation modes of its cells are analyzed in-depth to study their compressive behavior, and some meaningful conclusions are obtained. In addition, through reasonable parameter design, the BPLS can realize the novel deformation modes of upper cells contraction, lower cells contraction or the upper and lower cells contraction simultaneously and the middle cells expansion. Their deformation modes are like budding flowers. In summary, the BPLS exhibits extraordinary mechanical properties and deformation modes, and has broad application prospects.
Article
Many traditional thin-walled structures have the disadvantages of excessive force drop after the initial peak and large structural mass (SM). Inspired by bamboo and lotus root, hexagonal, pentagonal, quadrilateral and triangular polycellular structures (PSs) are proposed in this work to overcome these shortcomings. Their energy absorption, deformation modes and equivalent densities under axial compression are studied. For PSs with connecting plates, the crushing mode of cells is analyzed, and many interesting deformation modes of these PSs such as double-headed axe, progressive collapse and single-headed axe shapes are presented. Compared with bamboo-like PSs, lotus root-like PSs are prone to bend during compression. For PSs without connecting plates, a variety of PSs are proposed for each shape. The deformation modes of X, double-headed axe, mixed, two trapezoids, thin waist drum and concave polygon shapes are obtained. Models with greater equivalent densities tend to have better energy absorption efficiency. The compression performance of PSs without connecting plates is significantly better than that of PSs with connecting plates. In addition, a PS made of nylon 11 is manufactured and tested to further verify the performance of PSs. These PSs have uniform structural stiffness, reasonable deformation modes, and efficient energy absorption effects. Their excellent compression properties depend on the gradual collapse of cells.
Article
Full-text available
FFT methods have become a fundamental tool in computational micromechanics since they were first proposed in 1994 by H. Moulinec and P. Suquet for the homogenization of composites. From that moment on many dierent approaches have been proposed for a more accurate and efficient resolution of the non- linear homogenization problem. Furthermore, the method has been pushed beyond its original purpose and has been adapted to many other problems including continuum and discrete dislocation dynamics, multi-scale modeling or homogenization of coupled problems as fracture or multiphysical problems. In this paper, a comprehensive review of FFT approaches for micromechanical simulations will be made, covering the basic mathematical aspects and a complete description of a selection of approaches which includes the original basic scheme, polarization based methods, Krylov approaches, Fourier-Galerkin and displacement-based methods. The paper will present then the most relevant applications of the method in homogenization of composites, polycrystals or porous materials including the simulation of damage and fracture. It will also include an insight into synergies with experiments or its extension towards dislocation dynamics, multi-physics and multi-scale problems. Finally, the paper will analyze the current limitations of the method and try to analyze the future of the application of FFT approaches in micromechanics.
Article
Full-text available
The simulation of fracture using continuum ductile damage models attains a pathological discretization dependence caused by strain localization, after loss of ellipticity of the problem, in regions whose size is connected to the spatial discretization. Implicit gradient techniques suppress this problem introducing some inelastic non-local fields and solving an enriched formulation where the classical balance of linear momentum is fully coupled with a Helmholtz-type equation for each of the non-local variable. Such Helmholtz-type equations determine the distribution of the non-local fields in bands whose width is controlled by a characteristic length, independently on the spatial discretization. The numerical resolution of this coupled problem using the Finite Element method is computationally very expensive and its use to simulate the damage process in 3D multi-phase microstructures becomes prohibitive. In this work, we propose a novel FFT-based iterative algorithm for simulating gradient ductile damage in computational homogenization problems. In particular, the Helmholtz-type equation of the implicit gradient approach is properly generalized to model the regularization of damage in multi-phase media, where multiple damage variables and different characteristic lengths may come into play. In the proposed iterative algorithm, two distinct problems are solved in a staggered fashion: (i) a conventional mechanical problem via a FFT-Galerkin solver with mixed macroscopic loading control and (ii) the generalized Helmholtz-type equation using a Krylov-based algorithm combined with an efficient pre-conditioner. The numerical implementation is firstly validated on simple two-dimensional microstructures, showing identical responses for different spatial discretizations and reproducing a ductility change dependent on the characteristic length. Finally, the robustness and efficiency of the algorithm is demonstrated in the simulation of failure of complex 3D particle reinforced composites characterized by millions of degrees of freedom.
Article
Full-text available
We show that, under suitable hypotheses on the non‐porous material law and a geometric regularity condition on the pore space, Moulinec‐Suquet’s basic solution scheme converges linearly. We also discuss for which derived solvers a (super‐)linear convergence behavior may be obtained, and for which such results do not hold, in general. The key technical argument relies on a specific subspace on which the homogenization problem is nondegenerate, and which is preserved by iterations of the basic scheme. Our line of argument is based in the non‐discretized setting, and we draw conclusions on the convergence behavior for discretized solution schemes in FFT‐based computational homogenization. Also, we see how the geometry of the pores’ interface enters the convergence estimates. We provide computational experiments underlining our claims.
Article
Full-text available
The Fourier series method is used to solve the periodic homogenization problem for conductive materials containing voids. The problems involving voids are special cases of infinite contrast whose full field solution is not unique, causing convergence issues when iteration schemes are used. In this paper, we reformulate the problem based on the temperature field in the skeleton and derive an equation where the temperature field is connected to values on the pore boundary. Iteration schemes based on the new equation show that the convergence is fast, yielding good results both in terms of local fields and effective conductive properties.
Article
Full-text available
This paper presents the application of a fast Fourier transform (FFT) based method to solve two phase field models designed to simulate crack growth of strongly anisotropic materials in the brittle regime. By leveraging the ability of the FFT-based solver to generate solutions with higher-order and global continuities, we design two simple algorithms to capture the complex fracture patterns (e.g. sawtooth, and curved crack growth) common in materials with strongly anisotropic surface energy via the multi-phase-field and high-order phase-field frameworks. A staggered operator-split solver is used where both the balance of linear momentum and the phase field governing equations are formulated in the periodic domain. The unit phase field of the initial failure region is prescribed by the penalty method to alleviate the sharp material contrast between the initial failure region and the base material. The discrete frequency vectors are generalized to estimate the second and fourth-order gradients such that the Gibbs effect near shape interfaces or jump conditions can be suppressed. Furthermore, a preconditioner is adopted to improve the convergence rate of the iterative linear solver. Three numerical experiments are used to systematically compare the performance of the FFT-based method in the multi-phase-field and high-order phase-field frameworks.
Article
Full-text available
Most of the FFT methods available for homogenization of the mechanical response use the strain/deformation gradient as unknown, imposing their compatibility using Green's functions or projection operators. This implies the allocation of redundant information and, when the method is based in solving a linear equation, the rank-deficiency of the resulting system. In this work we propose a fast, robust and memory-efficient FFT homogenization framework in which the displacement field on the Fourier space is the unknown: the displacement based FFT (DBFFT) algorithm. The framework allows any general non-linear constitutive behavior for the phases and direct strain, stress and mixed control of the macroscopic load. In the linear case, the method results in a linear system defined in terms of linear operators in the Fourier space and that does not require a reference medium. The system has an associated full rank Hermitian matrix and can be solved using iterative Krylov solvers and allows the use of preconditioners. A preconditioner is proposed to improve the efficiency of the system resolution. Finally, some numerical examples including elastic, hyperelastic and viscoplastic materials are solved to check the accuracy and efficiency of the method. The computational cost reduction respect the Galerkin-FFT was around 30%.
Article
Full-text available
The physical and mechanical properties of cellular materials not only depend on the constituent materials but also on the microstructures. Here we show that, when the cellular materials are constructed by self-repeated representative volume elements, their effective elastic tensor can be obtained by a fast Fourier transform-based homogenization method. Numerical examples confirm that the bulk modulus of cellular material with the topology of triply periodic minimal surfaces such as Diamond, Gyroid, Neovius, and Schwarz P surfaces can approach to the upper Hashin-Shtrikman bound. However, the high values of their Young's modulus are obtained at the cost of low shear modulus and vice versa. Such conflicting behavior suggests that these two individual moduli may complement each other in a hybrid structure via combining different surfaces in cellular material. It is envisaged that our approach will enable the creation of ideal isotropic materials with large Young's modulus, shear modulus, and bulk modulus. Keywords: Cellular materials, Triply minimal surfaces, Fast Fourier transform-based homogenization, Hybrid
Article
Classical solution methods in FFT‐based computational micromechanics operate on, either, compatible strain fields or equilibrated stress fields. In contrast, polarization schemes are primal‐dual methods whose iterates are neither compatible nor equilibrated. Recently, it was demonstrated that polarization schemes may outperform the classical methods. Unfortunately, their computational power critically depends on a judicious choice of numerical parameters. In this work, we investigate the extension of polarization methods by Anderson acceleration and demonstrate that this combination leads to robust and fast general‐purpose solvers for computational micromechanics. We discuss the (theoretically) optimum parameter choice for polarization methods, describe how Anderson acceleration fits into the picture, and exhibit the characteristics of the newly designed methods for problems of industrial scale and interest.
Article
We demonstrate how a geometrically exact formulation of discrete slender beams can be generalized for the efficient simulation of complex networks of flexible beams by introducing rigid connections through special junction elements. The numerical framework, which is based on discrete differential geometry of framed curves in a time-discrete setting for time- and history-dependent constitutive models, is applicable to elastic and inelastic beams undergoing large rotations with and without natural curvature and actuation. Especially the latter two aspects make our approach a versatile and efficient alternative to higher-dimensional finite element techniques frequently used, e.g., for the simulation of active, shape-morphing, and reconfigurable structures, as demonstrated by a suite of examples.
Article
We study fast and memory-efficient FFT-based implicit solution methods for small-strain phase-field crack problems for microstructured brittle materials. A fully implicit first order formulation of the problem coupling elasticity and damage permits using comparatively few, but large, time steps compared to semi-explicit schemes. We investigate memory-efficient FFT-based solution techniques, and identify the heavy ball scheme as particularly powerful. We discuss the memory-efficient implementation and present demonstrative numerical examples.
Article
This work is devoted to investigating the computational power of Quasi‐Newton methods in the context of fast Fourier transform (FFT)‐based computational micromechanics. We revisit (FFT)‐based Newton‐Krylov solvers as well as modern Quasi‐Newton approaches such as the recently introduced Anderson accelerated basic scheme. In this context, we propose two algorithms based on the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) method, one of the most powerful Quasi‐Newton schemes. To be specific, we use the BFGS update formula to approximate the global Hessian or, alternatively, the local material tangent stiffness. Both for Newton and Quasi‐Newton methods, a globalization technique is necessary to ensure global convergence. Specific to the FFT‐based context, we promote a Dong‐type line search, avoiding function evaluations altogether. Furthermore, we investigate the influence of the forcing term, that is, the accuracy for solving the linear system, on the overall performance of inexact (Quasi‐)Newton methods. This work concludes with numerical experiments, comparing the convergence characteristics and runtime of the proposed techniques for complex microstructures with nonlinear material behavior and finite as well as infinite material contrast.