Content uploaded by Christian Gierden
Author content
All content in this area was uploaded by Christian Gierden on Mar 19, 2021
Content may be subject to copyright.
A geometrically adapted reduced set of frequencies for
a FFT-based microstructure simulation
Christian Gierdena,*
, Johanna Waimanna, Bob Svendsenb,c, Stefanie Reesea
aInstitute of Applied Mechanics, RWTH Aachen University, D-52074 Aachen, Germany
bMaterial Mechanics, RWTH Aachen University, D-52062 Aachen, Germany
cMicrostructure Physics and Alloy Design, Max-Planck-Institut für Eisenforschung GmbH
D-40237 Düsseldorf, Germany
Abstract. We present a modified model order reduction (MOR) technique for the FFT-based simula-
tion of composite microstructures. It utilizes the earlier introduced MOR technique (Kochmann et al.
[2019]), which is based on solving the Lippmann-Schwinger equation in Fourier space by a reduced set
of frequencies. Crucial for the accuracy of this MOR technique is on the one hand the amount of used
frequencies and on the other hand the choice of frequencies used within the simulation. Kochmann et al.
[2019] defined the reduced set of frequencies by using a fixed sampling pattern, which is most general
but leads to poor microstructural results when considering only a few frequencies. Consequently, a re-
construction algorithm based on the T V1-algorithm [Candes et al., 2006] was used in a post-processing
step to generate highly resolved micromechanical fields.
The present work deals with a modified sampling pattern generation for this MOR technique. Based
on the idea, that the micromechanical material response strongly depends on the phase-wise material
behavior, we propose the usage of sampling patterns adapted to the spatial arrangement of the individual
phases. This leads to significantly improved microscopic and overall results. Hence, the time-consuming
reconstruction in the post-processing step that was necessary in the earlier work is no longer required. To
show the adaptability and robustness of this new choice of sampling patterns, several two dimensional
examples are investigated. In addition, also the 3D extension of the algorithm is presented.
Keywords: Model order reduction, FFT, Composites, Microstructure simulation, Spectral solver
*Corresponding author: christian.gierden@ifam.rwth-aachen.de
1
arXiv:2103.10203v1 [cs.CE] 18 Mar 2021
1 Introduction
To calculate spatial resolutions for complex microstructural material behaviors within struc-
tural finite element (FE) simulations, a two-scale full field simulation is necessary. To per-
form these highly resolved two-scale simulations, various methodologies have been established
(Geers et al. [2010]). Examples of these are the FE2method (e.g. Smit et al. [1998]; Feyel and
Chaboche [2000]) and the FE-FFT method (e.g. Spahn et al. [2014]; Kochmann et al. [2016]).
In this context, we focus on the FE-FFT-based simulation approach, but restrict ourselves in
this work exclusively to the FFT-based microstructure simulation. Such a microstructure may
be given in terms of a representative volume element (RVE) or a unit cell (e.g. Hill [1963];
Ostoja-Starzewski [2002]).
The FFT-based modelling of periodic microstructures was introduced by Moulinec and Su-
quet [1994, 1998]. Based on fixed-point iterations, it is used for the simulation of different
microstructures, such as composites [Dreyer and Müller, 2000] and polycrystals [Lebensohn,
2001]. In the last two decades improvements of the solution behavior of the FFT-based method
were gained in various ways. For example, by the development of more efficient solvers,
which are numerically more robust and lead to better convergence behavior. Among these are
polarization-based formulations [Eyre and Milton, 1999; Monchiet and Bonnet, 2012; Schnei-
der et al., 2019], formulations based on augmented Lagrangians [Michel et al., 2000, 2001], or
formulations based on conjugate gradients [Zeman et al., 2010; Brisard and Dormieux, 2010;
Gélébart and Mondon-Cancel, 2013; Kabel et al., 2014]. In addition, numerical resolution prob-
lems related to the Gibbs phenomenon [Gibbs, 1898] have been addressed by using e.g. first-
[Willot et al., 2014; Willot, 2015] and higher-order [Vidyasagar et al., 2017] finite difference
approximations of the differential operator. In summary, the FFT-based microstructure simula-
tion is an accurate and efficient solution scheme, which is even more efficient than the common
FE simulation as shown by Michel et al. [1999] or Prakash and Lebensohn [2009]. Neverthe-
less, the computational effort especially in the context of a two-scale FE-FFT-based simulation
is still extremely high. Hence, the development of even more efficient methods is necessary.
One possibility is to use hybrid homogenization methods, which combine numerical simula-
tions and theoretical investigations, such as the uniform [Dvorak, 1992] or non-uniform [Michel
and Suquet, 2003; Fritzen and Böhlke, 2010] transformation analysis or the clustering analysis
[Liu et al., 2016; Wulfinghoff et al., 2017]. Concerning numerically efficient FE-FFT-based ho-
mogenization techniques a straight-forward ansatz is based on using a coarsely discretized mi-
crostructure. Consequently, this leads to coarse microstructural results so that a post-processing
step is necessary to generate the required highly resolved microstructural fields [Kochmann
et al., 2018; Gierden et al., 2021]. Other model order reduction techniques are for example
based on a proper orthogonal decomposition (POD) [Pinnau, 2008] using the strain tensor in
Fourier space ˆ
εto compute the required projection tensor [Garcia-Cardona et al., 2017], low-
rank approximations [Vondrejc et al., 2019] or on compuations using a reduced set of frequen-
2
cies [Kochmann et al., 2019] in Fourier space.
The present paper deals with the model order reduction technique considering a reduced set of
frequencies [Kochmann et al., 2019]. This technique reduces the computational effort of the
spectral solver and is suitable for small [Kochmann et al., 2019] and finite strain kinematics
[Gierden et al., 2019]. The earlier proposed solution procedure uses a fixed sampling pattern
to define the reduced set of frequencies. During the computations, this reduced set of frequen-
cies is used to solve the Lippmann-Schwinger equation in Fourier space. Subsequently, in a
post-processing step, a reconstruction and a compatibility step is performed to generate highly
resolved microstructural fields. Since the selection of frequencies is crucial for the accuracy of
the simulation in our current work, we propose to use a geometrically adapted sampling pattern
instead, which incorporates the microstructural phase distribution. This leads to significantly
better microstructural results, so that the time consuming reconstruction step using the T V1-
algorithm proposed by Candes et al. [2006] is not necessary anymore.
The paper is structured as follows. The microscopic boundary value problem (BVP) is reviewed
in Section 2. The solution strategy to this microscopic BVP by using a spectral solver and fixed-
point iterations is given in Section 3. Section 4 gives an overview of the recently introduced
model order reduction technique based on a reduced set of frequencies with a fixed sampling
pattern and the new technique based on using a geometrically adapted sampling pattern. A
comparison of the results of both methods is presented in Section 5. The paper ends with a
conclusion and an outlook in Section 6.
Notation. A direct tensor notation is preferred throughout the text. Vectors and second-order
tensors are represented by bold letters, e.g. aand A, a tensor of fourth order by, e.g., A. The
linear mapping of a second-order tensor Bby a fourth-order tensor Cis denoted by A=C:B.
The scalar and dyadic products are denoted, e.g. by a·bas well as A:B, and A⊗B,
respectively. Furthermore, || • || represents the Frobenius norm and a bar over any quantity ¯
A
always refers to a macroscopic quantity, while the absence of a bar is related to microscopic
quantities. Additional notation is introduced when needed.
2 Microscopic boundary value problem (BVP)
Let us consider an inhomogeneous periodic microstructure Ω=ΩI∪ΩMwith inclusions ΩI
embedded in a softer matrix material ΩM. One example is a microstructure with one centered
spherical inclusion as shown in Figure 1.
3
A
B
ΩM
ΩI
Figure 1: Microstructure with one centered spherical inclusion.
Considering small strain kinematics, the total strain ε(¯
x,x) = ¯
ε(¯
x) + ˜
ε(x)at the macroscopic
position ¯
xand the microscopic position xis additively split into the macroscopic part ¯
ε(¯
x)
and the microscopic fluctuating part ˜
ε(x), respectively. Subjecting the given microstructure
to a macroscopic strain ¯
ε(¯
x), the total stress σ(¯
x,x)is computed by solving the microscopic
boundary value problem
div σ(¯
x,x) = 0∀x∈Ω
σ(¯
x,x) = σ(¯
x,x,ε(¯
x,x),α(x))
ε(¯
x,x) = ¯
ε(¯
x) + ∇S(˜
u(x))
(1)
for each macroscopic position ¯
x, while body forces are considered only on the macroscopic
level and are thus neglected on the micro level. Within Equation (1), α(x)describes a set of
internal variables, ˜
u(x)is the microscopic fluctuating displacement field and ∇Srepresents
the symmetric gradient operator. In regard of comprehensibility, the dependence of all variable
besides the macroscopic strains on the macroscopic position ¯
xis not shown. The corresponding
macroscopic stress ¯
σ(¯
x)and strain ¯
ε(¯
x)tensors are defined by the volume average of their local
quantities:
¯
σ(¯
x) := 1
VZΩ
σ(x) dΩ and ¯
ε(¯
x) := 1
VZΩ
ε(x) dΩ .(2)
We restrict ourself to microstructures consisting of two phases with linear elastic or linear
elasto-plastic material behavior. Doing that, the total strain can be additively split into an elastic
part εe(x)and a plastic part εp(x):ε(x) = εe(x) + εp(x). The linear-elastic stress-strain rela-
tion reads σ(x) = C(x) : εe(x). The yield condition for the elasto-plastic behavior is defined
as
Φ(σ(x), εacc
p(x),x) = σeq(x)−[σ0
y(x) + H(x)εacc
p(x)] (3)
4
which is the classical von Mises yield condition with an isotropic linear hardening. Here, σ0
y(x)
is the initial yield stress, σeq (x)is the von Mises equivalent stress, H(x)is the hardening
modulus, and εacc
p(x)is the accumulated plastic strain. In terms of an associative flow rule, the
evolution of the plastic strain ˙
εpis given by
˙
εp= ˙γ∂Φ
∂σ(4)
with the plastic multiplier ˙γ. Finally, the Karush-Kuhn-Tucker conditions Φ≤0,˙γ≥0and
Φ ˙γ= 0 need to be fulfilled.
3 FFT-based microstructure simulation using the basic fixed-
point scheme
To solve the inhomogeneous microscopic boundary value problem introduced in Equation 1,
Hashin and Shtrikman [1962] proposed to transfer it into an equivalent homogeneous represen-
tation
div C0:ε(x) + div τ(x) = 0∀x∈Ω
τ(x) := σ(x,ε(x),α(x)) −C0:ε(x)
ε(x) = ¯
ε(¯
x) + ∇S(˜
u(x)) ,
(5)
by defining the polarization stress τ(x). It describes the fluctuation of the microstructural stress
around the stress of a homogeneous reference material with stiffness C0. Introducing Green’s
function G0(x,x0)and Green’s operator
L
(0)(x,x0), respectively, the integral equation
ε(x) = ¯
ε(¯
x)−ZΩ
L
(0)(x,x0) : τ(x0)dx0(6)
enables the solution of Equation 5. Equation 6 is also known as Lippmann-Schwinger equation
[Kröner [1959]; Willis [1981]]. The present convolution intregal is solved by transferring the
Lippmann-Schwinger equation into Fourier space yielding
ˆ
ε(ξ) = (−ˆ
L
(0)(ξ)ˆ
τ(ξ)∀ξ6=0
¯
ε∀ξ=0(7)
with the wave vector ξ. In Fourier space, also Green’s function and Green’s operator
ˆ
G0
ik(ξ) = (C0
ijkl ξjξl)−1(8)
and ˆ
Γ0
ijkl (ξ) = 1
4ˆ
G0
jk,li(ξ) + ˆ
G0
ik,lj (ξ) + ˆ
G0
jl,ki(ξ) + ˆ
G0
il,kj (ξ)(9)
are explicitly known. An iterative solution scheme based on Equation 7 was first introduced by
Moulinec and Suquet [1994, 1998]. The mechanical equilibrium is considered to be achieved
5
when the convergence criterion
||ε(i)(x)−ε(i−1)(x)||
|ε|<tolε(10)
is fullfilled within iteration step i. Considering a fixed-point iteration scheme, the best conver-
gence behavior is achieved by defining the homogeneous reference material behavior based on
the arithmetic average of the spatially varying Lamé constants λ(x)and µ(x)[Moulinec and
Suquet, 1998; Kabel et al., 2014]. This so-called basic fixed-point scheme is used within all
following simulations. In addition we use a first-order finite difference approximation of the
differential operator in Equation 9 to avoid numerical resultion problems related to the Gibbs
phenomenon, as proposed by Willot [2015].
4 Model order reduction technique based on a reduced set of
frequencies
Recently, Kochmann et al. [2019] proposed a model order reduction technique based on a re-
duced set of wave vectors Rξto decrease the computational effort of the spectral solver. The
general idea is that any function, such as the step function in Figure 2, may be approximated
by a Fourier series. Using a reduced set of frequencies with only one or ten frequencies leads
to a very coarse approximation (see Figure 2). Nevertheless, using 100 frequencies already
yields an accurate approximation of the step function. More frequencies would lead to even
better solutions, but the computational effort will also rise with more frequencies. Due to that,
a reduced set of frequencies must be defined in a way, that leads to accurate results but also low
computational costs.
A B C
Rξ={ξ= 0}Rξ={ξ1, ..., ξ10}Rξ={ξ1, ..., ξ100 }
Figure 2: Approximating a step function with a Fourier series consisting of one, ten or 100
frequencies.
Using this idea in terms of the FFT-based method yields a reduced solution of the Lippmann-
Schwinger equation in Fourier space. The resulting reduced fixed-point scheme is given in the
following:
6
while ||ε(i+1)(x)−ε(i)(x)||L2
||¯
ε(¯
x)||L2
≤tolεdo
a) τ(i)(x) = σ(ε(i)(x)) −C0:ε(i)(x)∀x∈Ω
b) ˆ
τ(i)(ξ) = FFT τ(i)(x)
c) Rˆ
ε(i+1)(Rξ) = (−
Rˆ
L
(0)(Rξ) : Rˆ
τ(i)(Rξ)for Rξ6=0
¯
ε(¯
x)for Rξ=0
d) Rε(i+1)(x) = iFFT Rˆ
ε(i+1)(Rξ)
end do
The speed-up in this algorithm is gained from step c) by solving the convolution integral in
Fourier space using the reduced set of wave vectors. It is also possible to use the reduced set
of frequencies and a discrete Fourier transformation (DFT) for the Fourier transformation in
step b) and and the inverse Fourier transformation in step d), but since the FFT with the full
set of frequencies is even faster than the DFT with the reduced set of frequencies, the FFT is
preferably used.
Crucial for the performance of this algorithm is the definition of the reduced set of frequencies.
In the following, a short review of the recently proposed fixed sampling pattern for the choice
of a reduced set of frequencies is given. Subsequently, a new method for the generation of sam-
pling patterns based on the microstructural geometry is presented. For simplicity, both methods
are presented for the 2D case.
4.1 Reduced set of frequencies based on a fixed sampling pattern
The proposed algorithm by Kochmann et al. [2019] is based on the identification of a fixed
sampling pattern, which is used to define the reduced set of wave vectors for the solution of
the Lippmann-Schwinger equation. This sampling is always the same and therefore does not
take the material behavior or the microstructural geometry into account. After performing the
online computations with this reduced set of frequencies, a reconstruction based on the T V1-
algorithm proposed by Candes et al. [2006] is performed. Subsequently, a compatibility step
is necessary to generate highly resolved micromechanical fields. To identify an appropriate set
of wave vectors, a circular, a squared and a radial sampling pattern are investigated. The most
general reduced set of frequencies, which is also most suitable for the reconstruction algorithm,
is the radial sampling pattern, shown in Figure 3.
7
AB
C
D
E
D
E
unreduced set reduced set
sampling pattern
ξ1ξ1
ξ2ξ2
Figure 3: Unreduced set of frequencies (left) and a reduced set of frequencies based on a radial
sampling pattern (right).
Figure 3 shows the full set of frequencies (left) as well as a reduced set of frequencies based
on a radial sampling pattern (right) while the lowest frequencies are in the middle of the shown
grid, beginning with the zeroth frequency, which is the mean of the approximated function.
If only a few percent of frequencies are considered, the usage of such a reduced set of frequen-
cies leads to poor microstructural and overall results. Due to that, a post-processing step is used
after the reduced simulation. This generates highly resolved data by using the T V1-algorithm
proposed by Candes et al. [2006]. In addition, thereafter, a compatibility step is needed to
guarantee a compatible strain field.
4.2 A novel approach for a reduced set of frequencies based on a geomet-
rically adapted sampling pattern
The mechanical behavior strongly depends on the geometrical representation of e.g. inclusions
within the microstructure. As an example, smaller strains occur in stiffer inclusions and cor-
responding higher strains in softer matrix material. The idea of an adapted sampling pattern
is to use the microstructural geometry to define the reduced set of frequencies. Therefore, the
geometry of a two phase material is presented by a step function
g(x) =
0for x∈ΩM
1for x∈ΩI,
(11)
in real space as shown in Figure 4 (left) (with 100 ×100 grid points). Transferring this repre-
sentation of the microstructural geometry into Fourier space results in the plot given in Figure
4 (right), in which the frequencies with the corresponding amplitudes are plotted. Again, the
lowest frequencies are centered.
8
A
BC
D
E
x2x1ξ2ξ1
FFT
Figure 4: Geometry in real space g(x)(left) and geometry transferred into Fourier space ˆg(ξ)
(right).
The highest amplitudes, given in this Fourier representation, correspond to the most needed
frequencies for the approximation of the geometry. So, a reduced set of frequencies should be
defined by taking into account a given percentage of frequencies with the highest amplitudes
to obtain the best approximation. Using for example a reduced set of frequencies with R=
2 % of the highest frequencies leads to the sampling pattern given in Figure 5 (left), while R
describes the percentage of used frequencies. The consideration of these frequencies for the
approximation of the geometry in real space results in the approximated step function in Figure
5 (right).
A
B
C
D
E
ξ2
ξ1
x2x1
iFFT
Figure 5: Geometrically adapted sampling pattern with R= 2 % of frequencies (left) and
geometrical step function approximated by this reduced set of frequencies (right).
The idea of a geometrically adapted sampling pattern is to use the same sampling pattern as
shown in Figure 5 (left) also for the approximation of the microstructural strains and therefore
also for solving the Lippmann-Schwinger equation in Fourier space. Doing that, significantly
better microscopic and overall results are achieved compared to the fixed radial sampling pattern
(see Figure 3 (right)), as shown in Section 5. Due to that, the time consuming reconstruction
algorithm which was necessary for the fixed sampling pattern is no longer needed.
9
5 Numerical examples and comparison of the results with
fixed and adapted sampling patterns
To test the adaptivity and accuracy of a geometrically adapted sampling pattern, first, several
2D two phase microstructures with elastic material behavior and one centered inclusion are
considered and compared to the results generated by the fixed sampling pattern in Section 5.1.
This is followed by considering a microstructure with several inclusions and an elastic or elasto-
plastic material behavior in Section 5.2 and the straight forward extension to the 3D case in
Section 5.3.
All microstructures are assumed to be squared or cubic and discretized by n= 256 ×256 and
n= 256 ×256 ×256 equidistant grid points, respectively. The mechanical equilibrium is
considered to be achieved for the tolerance tolε<10−8. To compare the results to the reference
solution with the full set of frequencies, a macroscopic error ¯
Eand a microscopic error Eis
defined by:
¯
E=||¯
σ−¯
σref||
||¯
σref|| and E=1
nX
n
||σ(n)−σref(n)||
||σref(n)|| ,
where nis the total number of grid points. All computations are performed using MATLAB
with the build-in FFTW-library and pre-compiled material routines on a Dell notebook with a
2.80 GHz Intel i7quad-core processor and 32 GB of RAM.
5.1 Elastic 2D two phase materials with one inclusion
To illustrate the adaptivity of the newly introduced sampling pattern generation and the result-
ing microstructural fields compared to the results with the fixed sampling pattern, a two dimen-
sional microstructure with different centered inclusions as shown in Figure 6 is investigated.
Both, the inclusions and the matrix material are considerd to be elastic with λI= 2.0GPa and
µI= 2.0GPa for the inclusion and λM= 1.0GPa and µM= 1.0GPa for the matrix material,
respectively. The applied macroscopic strain is set to ¯ε11 = 0.01 and ¯ε22 =−0.01.
Figure 6: Microstructures with various kinds of central inclusions. The matrix material is col-
ored black and the inclusion is colored white.
10
First, we investigate the microstructure with one circular inclusion. Note, that the following
observations have been made for all stress and strain fields. The results related to the fixed and
adapted sampling pattern are shown in Figure 7 and Figure 8, respectively. In the top row, the
sampling patterns for two reduced sets of frequencies (R= 0.78 % and R= 1.54 %) are given.
Subsequently, the corresponding stress fields σ11 and the reference solution incorporating the
full set of frequencies are shown. In order to better recognize the difference between the results
for the reduced sets of frequencies compared to the reference solution, the bottom row shows
the point-wise absolute difference in the microstructural stress fields ∆σ11 =|σref
11 −σ11|.
AB
D
E
F
G
H
I
C
R= 0.78 % R= 1.54 %
reference solution
σ11[GPa]
0.03
0.01
∆σ11[GPa]
0.01
0.00
Figure 7: Microstructural fields for the 2D elastic microstructure with one circular inclusion.
Top row: Fixed sampling pattern with two different numbers of wave vectors. Middle row:
Corresponding microstructural stress field σ11 and reference stress field computed with the full
set of frequencies. Bottom row: Absolute difference in the microstructural stress field ∆σ11.
As stated by Kochmann et al. [2019] considering a fixed sampling pattern, the radial sampling
pattern leads to good results in general since it considers a high amount of low frequencies, but
also a certain amount of high frequencies. The low frequencies are necessary to capture the
essential features of the microstructure, while the high frequencies are needed to capture for
11
example microstructures with needle-like inclusions. Nevertheless, the radial sampling pattern
leads to incoherent artifacts in the reduced solution, but which are needed in terms of the sub-
sequent reconstruction algorithm. The difference in the microstructural stress fields compared
to the reference solution is plotted in the last row of Figure 7, while no reconstruction is incor-
porated up to now. It can be seen, that this error goes up to 0.01 GPa in the area close to the
edge of the inclusion, which is about one third compared to the maximum stress in the reference
solution.
Choosing the reduced set of frequencies based on the geometrically adapted sampling pattern
and compared to the fixed sampling pattern, a higher amount of low frequencies is needed to
capture the microstructural geometry (see Figure 8). Using this reduced set of frequencies also
in terms of the microstructure simulation, more accurate results are generated. The highest er-
ror occurs in the transition from matrix to inclusion, which arises from the very high amount of
frequencies which are necessary to capture such a sharp transition.
AB
D
E
F
G
H
I
C
R= 0.78 % R= 1.54 %
reference solution
σ11[GPa]
0.03
0.01
∆σ11[GPa]
0.01
0.00
Figure 8: Microstructural fields for the 2D elastic microstructure with one circular inclusion.
Top row: Adapted sampling pattern with two different numbers of wave vectors. Middle row:
Corresponding microstructural stress field σ11 and reference stress field computed with the full
set of frequencies. Bottom row: Absolute difference in the microstructural stress field ∆σ11.
12
The macroscopic error ¯
Eand the microscopic error Eare shown in Figure 9 depending on the
reduced set of frequencies R. Since a simple microstructure with only one centered circular in-
clusion is investigated in this example, the macroscopic (left) and the microscopic errors (right)
are in both cases quite low. Nevertheless, the error corresponding to the adapted sampling
pattern is always significantly smaller - especially, when the reduced set of frequencies con-
sists only of few frequencies. For a higher number of frequencies, the errors of both solutions
converge towards zero.
Figure 9: Macroscopic error ¯
E(left) and microscopic error E(right) for the 2D elastic mi-
crostructure with one circular inclusion depending on the percentage of used frequencies Rfor
the solution with the fixed and adapted sampling pattern.
Using the reconstruction algorithm and the compatibility step the solution for the fixed sam-
pling pattern is improved as shown in Figure 10 in the left column. Since the microstructural
fields related to the adapted sampling pattern do not have incoherent artifacts, the reconstruction
algorithm does not yield any further improvement of the result and is therefore not needed. In-
stead, just the so-called compatibility step is applied. Thus, the Lippmann-Schwinger equation
is solved once using the full set of frequencies based on the stress and the strain fields from the
reduced simulation. It should be mentioned that in this context the earlier given name might be
misleading, since the solutions are already compatible. The corresponding solutions are shown
in Figure 10 in the centered column. It can be seen, that the micromechanical solution fields
which are related to the adapted sampling pattern are still better compared to the solutions based
on the fixed sampling pattern after the reconstruction.
13
AB
C
G
H
I
D
E
F
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
σ11[GPa]
0.03
0.01
∆σ11[GPa]
0.01
0.00
Figure 10: Microstructural fields for the 2D elastic microstructure with one circular inclusion.
Top row: Fixed and adapted sampling pattern with the same number of wave vectors. Middle
row: Corresponding microstructural stress field σ11 incorporating the reconstruction and com-
patibility step for the solution of the fixed sampling pattern and only the compatibility step for
the solution of the adapted sampling pattern and reference stress field computed with the full
set of frequencies. Bottom row: Absolute difference in the microstructural stress field ∆σ11.
Incorporating this post-processing step, the macroscopic error ¯
Eand microscopic error Eare
shown in Figure 11. For the fixed sampling pattern, the error refers to the solution after the
reconstruction and the compatibility step. For the adapted sampling pattern, the error corre-
sponds to the compatibility step, only. For both cases, Figure 11 shows the error depending
on the percentage of used frequencies R. The difference between the fixed and the adapted
sampling pattern is rather small, while the error of the adapted sampling pattern is smaller for
R<6 % but larger for R ≥ 6 %. The range R<6 % is of much larger interest, since the
highest speed-up is gained by a set with a low number of frequencies.
14
Figure 11: Macroscopic error ¯
E(left) and microscopic error E(right) for the 2D elastic mi-
crostructure with one circular inclusion depending on the reduced set of frequencies Rfor the
solution with the fixed sampling pattern with reconstruction and compatibility step and for the
adapted sampling pattern only with the compatibility step.
Tables 1 and 2 present the total CPU times for the computations with the fixed and adapted
sampling pattern, respectively. These CPU times are subdivided into the mean CPU time per
iteration step for solving the convolution integral and the mean CPU time per iteration step for
solving the constitutive law. The time for solving the constitutive law is almost independent of
the number of considered frequencies. The speed up in the total CPU times is gained by solving
the convolution integral in Fourier space with the reduced set of frequencies. Independent of
the sampling pattern, the speed up factor for R= 1.54 % of frequencies is about 7-8, while
the results with the adapted sampling pattern are more accurate. In addition, Table 1 shows the
CPU times for the time consuming reconstruction step, which is not necessary for the adapted
sampling pattern and the CPU time for the compatibility step, which is almost the same for the
fixed and adapted sampling pattern (see Table 2).
15
elastic CPU time [s] - fixed sampling pattern
R[%] total −ˆ
L
(0) ˆ
τ(ε)(mean) σ(ε)(mean) reconstruction compatibility
1.54 0.448 0.015 0.006 58.295 0.324
3.06 0.558 0.022 0.006 54.679 0.333
6.02 0.759 0.036 0.005 56.764 0.337
11.64 0.958 0.061 0.005 59.208 0.354
21.66 1.253 0.106 0.005 47.910 0.341
36.79 1.869 0.166 0.005 49.136 0.327
.
.
..
.
..
.
..
.
..
.
..
.
.
unreduced 3.196 0.302 0.004 - -
Table 1: Total CPU time with mean CPU time per iteration step for solving the convolution
integral and the constitutive law and CPU times for the reconstruction and the compatibility
step of the simulation with the fixed sampling pattern for the 2D elastic microstructure with one
circular inclusion.
elastic CPU time [s] - adapted sampling pattern
R[%] total −ˆ
L
(0) ˆ
τ(ε)(mean) σ(ε)(mean) reconstruction compatibility
1.54 0.388 0.013 0.005 - 0.341
3.06 0.495 0.022 0.005 - 0.378
6.02 0.641 0.034 0.005 - 0.346
11.64 0.948 0.060 0.005 - 0.322
21.66 1.412 0.099 0.005 - 0.340
36.79 1.912 0.169 0.005 - 0.350
.
.
..
.
..
.
..
.
..
.
..
.
.
unreduced 3.196 0.302 0.004 - -
Table 2: Total CPU time with mean CPU time per iteration step for solving the convolution
integral and the constitutive law and CPU times for the reconstruction and the compatibility
step of the simulation with the adapted sampling pattern for the 2D elastic microstructure with
one circular inclusion.
The same investigations could be made using the microstructures with the annular, the elliptical,
and the quadratic inclusion shown in Figure 6 and would lead to similar results. Due to that, we
just show the sampling patterns and the corresponding microstructural stress fields σ11 for R=
1.54 % of frequencies considering these microstructures in Figures 12 - 14. The results show the
relation between the arrangement of the considered frequencies in the adapted sampling pattern
and the geometry of the inclusion within the matrix. Additionally it can be seen, that the results
16
gained by the new approach are always closer to the reference solution than the solutions based
on the fixed sampling pattern.
In Figure 12, the geometrically adapted sampling pattern of the microstructure with an annular
inclusion is shown. This sampling pattern is similar to the adapted sampling corresponding to
the circular inclusion; see Figure 8. Nevertheless, the amount of high frequencies is slightly
higher. The adapted sampling pattern corresponding to the elliptical inclusion, shown in Figure
13, differs totally from that. It can be seen, that in direction of the major axis of the ellipse
lower frequencies are needed and perpendicular to that higher frequencies are necessary, since
a smaller distance needs to be bridged in this direction. As a last example, Figure 14 shows
the microstructure with a quadratic inclusion. The sharp edges of this last examined type of
inclusion again lead to a totally different set of frequencies.
AB
D
E
F
C
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
σ11[GPa]
0.040
0.015
Figure 12: Microstructural fields for the 2D elastic microstructure with one annular inclusion.
Top row: Fixed and adapted sampling pattern with the same number of wave vectors. Bottom
row: Corresponding microstructural stress field σ11 and reference stress field computed with the
full set of frequencies.
17
AB
C
D
E
F
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
σ11[GPa]
0.040
0.015
Figure 13: Microstructural fields for the 2D elastic microstructure with one elliptical inclusion.
Top row: Fixed and adapted sampling pattern with the same number of wave vectors. Bottom
row: Corresponding microstructural stress field σ11 and reference stress field computed with the
full set of frequencies.
l
AB
C
D
E
F
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
σ11[GPa]
0.035
0.010
Figure 14: Microstructural fields for the 2D elastic microstructure with one quadratic inclusion.
Top row: Fixed and adapted sampling pattern with the same number of wave vectors. Bottom
row: Corresponding microstructural stress field σ11 and reference stress field computed with the
full set of frequencies.
18
5.2 Elastic and elasto-plastic 2D two phase material with several inclu-
sions
In the following, a composite with several circular inclusions is investigated. The elastic con-
stants are the same as in the example before: λI= 2.0GPa and µI= 2.0GPa for the inclusion
and λM= 1.0GPa and µM= 1.0GPa for the matrix material, respectively. Besides the elastic
solution, also a simulation with an elasto-plastic matrix material behavior is performed. Con-
sidering an elasto-plastic material behavior of the matrix, the additional material parameters are
set to HM= 0.01 GPa as hardening modulus and σ0
yM = 0.01 GPa as initial yield stress. The
investigated microstructure and the prescribed macroscopic strain is presented in Figure 15.
¯
ε= 0.01 0.002
0.002 −0.01!
Figure 15: Composite with several circular inclusions and prescribed macroscopic stain.
For the fixed and the adapted sampling pattern the same amount of frequencies is used. Figure
16 shows the microstructural stress fields σ11 as well as the differences ∆σ11 to the reference
solution for the pure elastic case. As already seen for one inclusion, the error in the solution
based on the adapted reduced set of frequencies is significantly lower compared to the solution
of the fixed sampling pattern. Regarding the solution with the adapted set of frequencies, the
highest differences occur again in the transition from matrix to inclusion. Instead, for the fixed
sampling pattern, the highest errors occur within the inclusions.
19
AB
D
E
F
G
H
I
C
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
σ11[GPa]
0.04
0.01
∆σ11[GPa]
0.01
0.00
Figure 16: Microstructural fields for the 2D elastic microstructure with several circular inclu-
sions. Top row: Fixed and adapted sampling pattern with the same number of wave vectors.
Middle row: Corresponding microstructural stress field σ11 and reference stress field computed
with the full set of frequencies. Bottom row: Absolute difference in the microstructural stress
field ∆σ11.
Again, we perform the reconstruction and the compatibility step for the solution based on the
fixed sampling pattern and only the compatibility step for the adapted sampling pattern. The
corresponding microstructural fields are shown in Figure 17. Here, similar effects as described
in Chapter 5.1 for a microstructure with only one inclusion occur: The solution with the adapted
sampling pattern is more accurate.
20
AB
D
E
F
G
H
I
C
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
σ11[GPa]
0.04
0.01
∆σ11[GPa]
0.01
0.00
Figure 17: Microstructural fields for the 2D elastic microstructure with several circular inclu-
sions. Top row: Fixed and adapted sampling pattern with the same number of wave vectors.
Middle row: Corresponding microstructural stress field σ11 incorporating the reconstruction
and compatibility step for the solution of the fixed sampling pattern and only the compatibility
step for the solution of the adapted sampling pattern and reference stress field computed with
the full set of frequencies. Bottom row: Absolute difference in the microstructural stress field
∆σ11.
Since the solution behavior of the elastic microstructures with one or several inclusions is sim-
ilar, we do not present further results on that and focus in the following on the microstructure
with elasto-plastic material behavior.
The generation of a geometrically adapted sampling pattern does not depend on the material
behavior itself, but only on the geometrical representation of the matrix and inclusions. Due
to that, the geometrically adapted sampling pattern for the microstructure with several elastic
inclusions and an elasto-plastic matrix material behavior is the same as for the microstructure
with several inclusions and an overall elastic material behvior. Figure 18 shows the microscopic
stress field σ11 corresponding to the fixed and adapted sampling pattern, the reference solution
21
and the absolute difference in the reduced solution compared to the reference solution ∆σ11.
It can be seen, that the stress difference for the fixed and the adapted sampling pattern is in
general higher considering the nonlinear matrix material behavior instead of the purely linear
material behavior shown in Figure 16. Nevertheless, the error in the solution with the adapted
sampling pattern is again significantly lower than the error in the solution with the fixed sam-
pling pattern. In addition, the error considering the adapted sampling pattern is still related to
the transition from inclusion to matrix material, while the error for the fixed sampling pattern is
again particularly high within the inclusions.
AB
D
E
F
G
H
I
C
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
σ11[GPa]
0.02
0.00
∆σ11[GPa]
0.01
0.00
Figure 18: Microstructural fields for the 2D elasto-plastic microstructure with several circular
inclusions. Top row: Fixed and adapted sampling pattern with the same number of wave vectors.
Middle row: Corresponding microstructural stress field σ11 and reference stress field computed
with the full set of frequencies. Bottom row: Absolute difference in the microstructural stress
field ∆σ11.
Figure 19 shows the macroscopic error ¯
E(left) and the microscopic error E(right) again based
on the reduced set of frequencies Rfor the solution with the fixed and adapted sampling pattern
for the elasto-plastic composite. Incorporating R= 1.54 % and considering the fixed sampling
22
pattern, these errors read ¯
E ≈ 34 % and E ≈ 79 %. Using the same amount of frequencies
and the adapted sampling pattern, the errors reduce to ¯
E ≈ 2 % and E ≈ 14 %, respectively.
In addition, Figure 19 shows that at some point (R ≈ 15 %) the fixed sampling pattern leads
to better results than the adapted sampling pattern. This might be related to the elasto-plastic
material behavior of the matrix which results in a material behavior which is not that uniform
within the matrix as the pure elastic material behavior. The adapted sampling pattern is only
related to the geometrical representation of the matrix material and does not represent the differ-
ent elasto-plastic material states. Due to that, the fixed sampling pattern, which is not bounded
to the geometry, may lead to better results for a high amount of frequencies. Besides that, in
the range of interest with a highly reduced set of frequencies, the adapted sampling pattern is
performing significantly better than the fixed sampling pattern.
Figure 19: Macroscopic error ¯
E(left) and microscopic error E(right) for the 2D elasto-plastic
microstructure with several circular inclusions depending on the percentage of used frequencies
Rfor the solution with the fixed and adapted sampling pattern.
In Figure 20 the accumulated plastic strain field εacc
pfor R= 1.54 % of frequencies is shown.
Also here, significant differences of the solution with the fixed sampling pattern compared to
the reference solution are observed. For example, the accumulated plastic strain in the upper
middle has a value of εacc
p≈0.3in the reference solution, while the accumulated plastic strain
in the reduced solution has a value of εacc
p≈0at the same position. In contrast to that, the
solution based on the adapted sampling pattern and the same amount of frequencies is quite
similar to the reference solution.
23
AB
D
E
F
C
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
εacc
p[-]
0.03
0.00
Figure 20: Microstructural fields for the 2D elasto-plastic microstructure with several circular
inclusions. Top row: Fixed and adapted sampling pattern with the same number of wave vectors.
Bottom row: Corresponding microstructural accumulated plastic strain field εacc
pand reference
accumulated plastic strain field computed with the full set of frequencies.
Incorporating the reconstruction and compatibility step for the solution of the fixed sampling
pattern and only the compatibility step for the solution of the adapted sampling pattern leads to
the results given in Figures 21 and 22. Figure 21 shows the microstructural stress field σ11 and
Figure 22 shows the accumulated plastic strain field εacc
p, respectively. Considering the fixed
sampling pattern, it can be seen, that the calculated stress within the inclusions is improved by
the reconstruction and the compatibility step, while the stress within the elasto-plastic matrix
is not improved significantly. This is related to the accumulated plastic strain field, shown in
Figure 22, which is also not improved by these post-processing steps. As shown in Figure
21, the microstructural stress field related to the solution with the adapted sampling pattern
is slightly improved by solving the Lippmann-Schwinger equation once with the full set of
frequencies.
24
AB
D
E
F
G
H
I
C
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
σ11[GPa]
0.02
0.00
∆σ11[GPa]
0.01
0.00
Figure 21: Microstructural fields for the 2D elasto-plastic microstructure with several circular
inclusions. Top row: Fixed and adapted sampling pattern with the same number of wave vectors.
Middle row: Corresponding microstructural stress field σ11 incorporating the reconstruction and
compatibility step for the solution of the fixed sampling pattern and only the compatibility step
for the solution of the adapted sampling pattern and reference stress field computed with the full
set of frequencies. Bottom row: Absolute difference in the microstructural stress field ∆σ11.
25
l
AB
D
E
F
C
fixed: R= 1.54 % adapted: R= 1.54 %
reference solution
εacc
p[-]
0.03
0.00
Figure 22: Microstructural fields for the 2D elasto-plastic microstructure with several circu-
lar inclusions. Top row: Fixed and adapted sampling pattern with the same number of wave
vectors. Bottom row: Corresponding microstructural accumulated plastic strain field εacc
pincor-
porating the reconstruction and compatibility step for the solution of the fixed sampling pattern
and only the compatibility step for the solution of the adapted sampling pattern and reference
accumulated plastic strain field computed with the full set of frequencies.
The CPU times for both sampling patterns and the different amount of frequencies are given in
Tables 3 and 4. The behavior of the CPU times is similar compared to the elastic case shown in
Tables 1 and 2, while a speed-up factor of 5 - 6 is gained by considering a set of R= 1.54 % of
frequencies in the nonlinear case.
26
elasto-plastic CPU time [s] - fixed sampling pattern
R[%] total −ˆ
L
(0) ˆ
τ(ε)(mean) σ(ε)(mean) reconstruction compatibility
t= 100 t= 100
1.54 316.7 0.009 0.022 91.134 0.334
3.06 446.9 0.015 0.028 105.043 0.442
6.02 555.9 0.027 0.025 76.939 0.327
11.64 797.9 0.050 0.027 73.645 0.340
21.66 1139.4 0.087 0.024 65.870 0.339
36.79 1645.2 0.146 0.024 60.556 0.321
.
.
..
.
..
.
..
.
..
.
..
.
.
unreduced 1866.5 0.241 0.024 - -
Table 3: Total CPU time with mean CPU time per iteration step for solving the convolution
integral and the constitutive law and CPU times for the reconstruction and the compatibility
step of the simulation with the fixed sampling pattern for the 2D elasto-plastic microstructure
with several circular inclusions.
elasto-plastic CPU time [s] - adapted sampling pattern
R[%] total −ˆ
L
(0) ˆ
τ(ε)(mean) σ(ε)(mean) reconstruction compatibility
t= 100 t= 100
1.54 366.7 0.008 0.025 - 0.323
3.06 431.0 0.015 0.020 - 0.342
6.02 533.8 0.026 0.021 - 0.334
11.64 744.4 0.050 0.023 - 0.338
21.66 1081.5 0.087 0.020 - 0.324
36.79 1637.3 0.149 0.023 - 0.342
.
.
..
.
..
.
..
.
..
.
..
.
.
unreduced 1866.5 0.241 0.024 - -
Table 4: Total CPU time with mean CPU time per iteration step for solving the convolution
integral and the constitutive law and CPU times for the reconstruction and the compatibility
step of the simulation with the adapted sampling pattern for the 2D elasto-plastic microstructure
with several circular inclusions.
27
5.3 Elastic 3D two phase material with one inclusion
Finally, a 3D microstructure with one centered spherical inclusion, as shown in Figure 23, is
investigated. The inclusions and the matrix material are considerd to be elastic again with λI=
2.0GPa and µI= 2.0GPa for the inclusion and λM= 1.0GPa and µM= 1.0GPa considering
the matrix material, respectively. The applied macroscopic strain is set to ¯ε11 = 0.01.
Figure 23: Two different intersections of the 3D microstructure with one spherical inclusion.
To generate the geometrically adapted sampling pattern in the 3D case, the same strategy as in
the 2D case is used: First, the geometry is represented by a 3D step function. This function is
transferred into Fourier space. The set of frequencies with the highest amplitudes for the repre-
sentation of the step function in Fourier space is used for the reduced simulation. The resulting
adapted sampling pattern for R= 1.54 % of frequencies and the corresponding microstructural
results are given in Figure 24. Since the 3D case with a spherical inclusion corresponds to the
2D case with a circular inclusion, the resulting sampling patterns are similar, see Figure 8. Also
the behavior of the microstructural fields is similar to the 2D case described in Chapter 5.1.
Using only R= 1.54 % of the frequencies, the micromechanical stress fields of the reduced
simulation already match the reference solution very well. The highest errors are again within
the transition from matrix to inclusion.
28
B
C
C
B
D
E
F
G
H
I
R= 1.54 % R= 1.54 %
reference solution reference solution
σ11[GPa]
0.045
0.020
∆σ11[GPa]
0.002
0.000
Figure 24: Microstructural fields of two different intersections for the 3D elastic microstructure
with one circular inclusion. Top row: Adapted sampling pattern. Middle row: Corresponding
microstructural stress field σ11 and reference stress field computed with the full set of frequen-
cies. Bottom row: Absolute difference in the microstructural stress field ∆σ11.
The macroscopic error ¯
Eand the microscopic error Edepending on the reduced set of frequen-
cies Rfor the 3D case are plotted in Figure 25.
Figure 25: Macroscopic error ¯
E(left) and microscopic error E(right) for the 3D elastic mi-
crostructure with one circular inclusion depending on the percentage of used frequencies Rfor
the solution with the adapted sampling pattern.
29
Using the compatibility step, the microstructural stress field σ11 is again improved as shown in
Figure 26.
B
C
C
B
G
H
I
D
E
F
R= 1.54 % R= 1.54 %
reference solution reference solution
σ11[GPa]
0.045
0.020
∆σ11[GPa]
0.002
0.000
Figure 26: Microstructural fields of two different intersections for the 3D elastic microstructure
with one circular inclusion. Top row: Adapted sampling pattern. Middle row: Corresponding
microstructural stress field σ11 incorporating the compatibility step and reference stress field
computed with the full set of frequencies. Bottom row: Absolute difference in the microstruc-
tural stress field ∆σ11.
Table 5 shows the CPU times for the reduced and reference solution in the 3D case. It can
be seen, that a significant speed up factor of up to approximately 100 is gained by using the
geometrically adapted reduced set of frequencies with R= 1.54 % of frequencies. This speed
up is again only gained by solving the convolution integral in Fourier space with the reduced
set of frequencies.
30
3D composite CPU time [s] - adapted sampling pattern
R[%] total −ˆ
L
(0) ˆ
τ(ε)(mean) σ(ε)(mean) reconstruction compatibility
0.39 301.46 2.21 13.998 - 279.1
0.78 284.79 3.32 13.960 - 284.9
1.54 305.54 5.36 13.981 - 298.7
3.06 351.76 9.59 14.411 - 277.2
6.02 426.57 17.66 13.826 - 278.7
11.64 584.64 32.94 14.368 - 278.1
21.66 853.26 60.02 14.149 - 281.3
36.79 1260.20 100.94 13.920 - 279.8
.
.
..
.
..
.
..
.
..
.
..
.
.
unreduced 28188.0 2795.6 12.241 - -
Table 5: Total CPU time with mean CPU time per iteration step for solving the convolution
integral and the constitutive law and CPU times for the reconstruction and the compatibility
step of the simulation with the adapted sampling pattern for the 3D elastic microstructure with
one circular inclusion.
6 Conclusion and outlook
We presented a novel approach to identify a sampling pattern for a reduced set of frequencies
which is used for the FFT-based microstructure simulation. The approach is based on trans-
ferring the microstructural phase distribution represented by a step function into Fourier space
and identifying the corresponding frequencies with the highest amplitudes. A given percentage
of these frequencies with the highest amplitudes is subsequently used to determine the reduced
set of frequencies. As shown for several two and three dimensional examples, such an adapted
sampling pattern leads to significant better microstructural and overall results compared to the
earlier introduced fixed sampling pattern. Considering only a few frequencies in the reduced
wave vector, the error compared to the reference solution is so small, that a reconstruction is not
necessary anymore. Therefore, the solution algorithm is in addition much easier, especially in
the 3D case. Using the proposed solution strategy, a speed up factor of 5-8 in the 2D case and
up to approximately 100 in the 3D case is obtained.
The proposed solution strategy only reduces the computational effort of solving the convolution
in Fourier space, so that the next step is the combination of the proposed MOR technique with
e.g. a clustering analysis, since the most time consuming part for the simulation of a complex
material behavior is the evaluation of the material law. Using such a microstructural clustering
analysis the stress evaluation in each grid point is reduced to a stress evaluation in a defined
31
number of clusters instead. Based on previous works [Wulfinghoff et al., 2017; Cavaliere et al.,
2020; Waimann et al., 2021] an additional significant speed-up is expected by combining both
methods.
Acknowledgements: The authors gratefully acknowledge the financial support of the research
work by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) within
the transregional Collaborative Research Center SFB/TRR 136, project number 223500200,
subproject M03. In addition Stefanie Reese gratefully acknowledges the financial support of
the research work by the German Research Foundation (DFG, Deutsche Forschungsgemein-
schaft) within the transregional Collaborative Research Center SFB/TRR 280, project number
417002380, subproject A01 and the project “Model order reduction in space and parameter
dimension - towards damage-based modeling of polymorphic uncertainty in the context of ro-
bustness and reliability”, project number 312911604, from the priority program (SPP) 1886.
References
Brisard, S. and Dormieux, L. [2010], ‘FFT-based methods for the mechanics of composites: A
general variational framework’, Computational Materials Science 49 (3), 663–671.
Candes, E. J., Romberg, J. and Tao, T. [2006], ‘Robust uncertainty principles: exact signal
reconstruction from highly incomplete frequency information’, IEEE Transactions on Infor-
mation Theory 52(2), 489–509.
Cavaliere, F., Reese, S. and Wulfinghoff, S. [2020], ‘Efficient two-scale simulations of engi-
neering structures using the Hashin-Shtrikman type finite element method’, Computational
Mechanics 65, 159–175.
Dreyer, W. and Müller, W. [2000], ‘A study of the coarsening of tin/lead solders’, International
Journal of Solids and Structures 37, 3841–3871.
Dvorak, G. J. [1992], ‘Transformation field analysis of inealstic composite materials’, Proceed-
ings of the Royal Society of London. Series A: Mathematical and Physical Sciences 437, 311–
327.
Eyre, D. J. and Milton, G. W. [1999], ‘A fast numerical scheme for computing the response of
composites using grid refinement’, The European Physical Journal 6, 41–47.
Feyel, F. and Chaboche, J.-L. [2000], ‘FE2multiscale approach for modeling the elastovis-
coplastic behaviour of long fibre SiC/Ti composite materials’, Computer Methods in Applied
Mechanics and Engineering 183, 309–330.
32
Fritzen, F. and Böhlke, T. [2010], ‘Three-dimensional finite element implementation of the
nonuniform transformation field analysis’, International Journal of Numerical Methods in
Engineering 84.
Garcia-Cardona, C., Lebensohn, R. and Anghel, M. [2017], ‘Parameter estimation in a thermoe-
lastic composite problem via adjoint formulation and model reduction’, International journal
for numerical methods in engineering 112, 578–600.
Geers, M., Kouznetsova, V. and Brekelmans, W. [2010], ‘Multi-scale computational homog-
enization: Trends and challenges’, Journal of Computational and Applied Mathematics
234, 2175–2182.
Gélébart, L. and Mondon-Cancel, R. [2013], ‘Non-linear extension of FFT-based methods ac-
celerated by conjugated gradients to evaluate the mechanical behavior of composite materi-
als’, Computational Materials Science 77, 430–439.
Gibbs, J. W. [1898], ‘Fourier’s series’, Nature 59, 200–200.
Gierden, C., Kochmann, J., Manjunatha, K., Waimann, J., Wulfinghoff, S., Svendsen, B. and
Reese, S. [2019], ‘A model order reduction method for finite strain FFT solvers using a
compressed sensing technique’, Proceedings in Applied Mathematics and Mechanics 19.
Gierden, C., Kochmann, J., Waimann, J., Kinner-Becker, T., Sölter, J., Svendsen, B. and
Reese, S. [2021], ‘Efficient two-scale FE-FFT-based mechanical process simulation of elasto-
viscoplastic polycrystals at finite strains’, Computer Methods in Applied Mechanics and En-
gineering 374, 113566.
Hashin, Z. and Shtrikman, H. [1962], ‘On some variational principles in anisotropic and non-
homogeneous elasticity’, Journal of the Mechanics and Physics of Solids 10, 335–342.
Hill, R. [1963], ‘Elastic properties of reinforced solids: Some theoretical principles’, Journal of
the Mechanics and Physics of Solids 11, 357–372.
Kabel, M., Böhlke, T. and Schneider, M. [2014], ‘Efficient fixed point and Newton-Krylov
solvers for FFT-based homogenization of elasticity at large deformations’, Computational
Mechanics 54, 1497–1514.
Kochmann, J., Manjunatha, K., Gierden, C., Wulfinghoff, S., Svendsen, B. and Reese, S. [2019],
‘A simple and flexible model order reduction method for FFT-based homogenization prob-
lems using a sparse sampling technique’, Computer Methods in Applied Mechanics and En-
gineering 347, 622–638.
Kochmann, J., Wulfinghoff, S., Ehle, L., Mayer, J., Svendsen, B. and Reese, S. [2018], ‘Efficient
and accurate two-scale FE-FFT-based prediction of the effective material behavior of elasto-
viscoplastic polycrystals’, Computational Mechanics 61, 751–764.
33
Kochmann, J., Wulfinghoff, S., Reese, S., Mianroodi, J. R. and Svendsen, B. [2016], ‘Two-
scale, FE-FFT- and phase-field based computational modeling of bulk microstructure evo-
lution and macroscopic material behavior’, Computer Methods in Applied Mechanics and
Engineering 305, 89–110.
Kröner, E. [1959], ‘Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen’,
Archive for Rational Mechanics and Analysis 4 (1), 273.
Lebensohn, R. A. [2001], ‘N-site modeling of a 3D viscoplastic polycrystal using Fast Fourier
Transform’, Acta Materialia 49, 2723–2737.
Liu, Z., Bessa, M. A. and Liu, W. K. [2016], ‘Self-consistent clustering analysis: An efficient
multi-scale scheme for inelastic heterogeneous materials’, Computer Methods in Applied
Mechanics and Engineering 306, 319–341.
Michel, J. C., Moulinec, H. and Suquet, P. [1999], ‘Effective properties of composite materi-
als with periodic microstructure: a computational approach’, Computer Methods in Applied
Mechanics and Engineering 172, 109–143.
Michel, J. C., Moulinec, H. and Suquet, P. [2000], ‘A computational method based on aug-
mented Lagrangians and Fast Fourier Transforms for composites with high contrast’, Com-
putational Modelling in Engineering Science 1, 79–88.
Michel, J. C., Moulinec, H. and Suquet, P. [2001], ‘A computational scheme for linear and non-
linear composites with arbitrary phase contrast’, International Journal of Numerical Methods
in Engineering 52, 139–160.
Michel, J. C. and Suquet, P. [2003], ‘Nonuniform transformation field analysis’, International
Journal of Solids and Structures 40, 6937–6955.
Monchiet, V. and Bonnet, G. [2012], ‘A polarization-based FFT iterative scheme for computing
the effective properties of elastic composites with arbitrary contrast’, International Journal
for Numerical Methods in Engineering 89, 1419–1436.
Moulinec, H. and Suquet, P. [1994], ‘A fast numerical method for computing the linear and
nonlinear mechanical properties of composites’, Comptes Rendus de l’Académie des Sciences
318, 1417–1423.
Moulinec, H. and Suquet, P. [1998], ‘A numerical method for computing the overall response
of nonlinear composites with complex microstructures’, Computer Methods in Applied Me-
chanics and Engineering 157 (1), 69–94.
Ostoja-Starzewski, M. [2002], ‘Towards stochastic continuum thermodynamics’, Journal of
Non-Equilibrium Thermodynamics 27, 335–348.
34
Pinnau, R. [2008], Model Reduction via Proper Orthogonal Decomposition, Vol. 13, Springer
International Publishing.
Prakash, A. and Lebensohn, R. A. [2009], ‘Simulations of micromechanical behavior of poly-
crystals: finite element versus Fast Fourier Transforms’, Modelling and Simulation in Mate-
rials Science and Engineering 17, 064010.
Schneider, M., Wicht, D. and Böhlke, T. [2019], ‘On polarization-based schemes for the
FFT-based computational homogenization of inelastic materials’, Computational Mechanics
64, 1073–1095.
Smit, R. J. M., Brekelmans, W. A. M. and Meijer, H. E. H. [1998], ‘Prediction of the mechan-
ical behavior of nonlinear heterogeneous systems by multi-level finite element modeling’,
Computer Methods in Applied Mechanics and Engineering 155, 181–192.
Spahn, J., Andrae, H., Kabel, M. and Müller, R. [2014], ‘A multiscale approach for modeling
progressive damage of composite materials using fast Fourier transforms’, Computer Meth-
ods in Applied Mechanics and Engineering 268, 871–883.
Vidyasagar, A., Tan, W. L. and Kochmann, D. M. [2017], ‘Predicting the effective response
of bulk polycrystalline ferroelectric ceramics via improved spectral phase field methods’,
Journal of the Mechanics and Physics of Solids .
Vondrejc, J., Liu, D., Ladecký, M. and Matthies, H. [2019], ‘FFT-based homogenisation ac-
celerated by low-rank tensor approximations’, Computer Methods in Applied Mechanics and
Engineering .
Waimann, J., Gierden, C., Schmidt, A., Svendsen, B. and Reese, S. [2021], ‘Microstructure
simulation using self-consistent clustering analysis’, Proceedings in Applied Mathematics
and Mechanics 20.
Willis, J. R. [1981], Variational and related methods for the overall properties of composites,
Vol. 21.
Willot, F. [2015], ‘Fourier-based schemes for computing the mechanical response of composites
with accurate local fields’, Comptes Rendus Mécanique 343 (3), 232–245.
Willot, F., Abdallah, B. and Pellegrini, Y. P. [2014], ‘Fourier-based schemes with modified
Green operator for computing the electrical response of heterogeneous media with accurate
local fields’, International Journal for Numerical Methods in Engineering 98 (7), 518–533.
Wulfinghoff, S., Cavaliere, F. and Reese, S. [2017], ‘Model order reduction of non-linear ho-
mogenization problems using a Hashin-Shtrikman type finite element method’, Computer
Methods in Applied Mechanics and Engineering 330, 149–179.
35
Zeman, J., Vondˇ
rejc, J., Novák, J. and Marek, I. [2010], ‘Accelerating a FFT-based solver for
numerical homogenization of a periodic media by conjugate gradients’, Journal of Compu-
tational Physics 229 (21), 8065–8071.
36
