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Deep material networks for fiber suspensions with infinite
material contrast
Benedikt Sterr1, Sebastian Gajek1, Andrew Hrymak2, Matti Schneider3, and Thomas
Böhlke1
1Karlsruhe Institute of Technology (KIT), Institute of Engineering Mechanics
2University of Western Ontario, Department of Chemical and Biochemical Engineering
3University of Duisburg-Essen, Institute of Engineering Mathematics
correspondence to: thomas.boehlke@kit.edu
June 18, 2024
Abstract
We extend the laminate based framework of direct Deep Material Networks (DMNs) to treat suspen-
sions of rigid fibers in a non-Newtonian solvent. To do so, we derive two-phase homogenization blocks
that are capable of treating incompressible fluid phases and infinite material contrast. In particular,
we leverage existing results for linear elastic laminates to identify closed form expressions for the
linear homogenization functions of two-phase layered emulsions. To treat infinite material contrast,
we rely on the repeated layering of two-phase layered emulsions in the form of coated layered materi-
als. We derive necessary and sufficient conditions which ensure that the effective properties of coated
layered materials with incompressible phases are non-singular, even if one of the phases is rigid.
With the derived homogenization blocks and non-singularity conditions at hand, we present a novel
DMN architecture, which we name the Flexible DMN (FDMN) architecture. We build and train
FDMNs to predict the effective stress response of shear-thinning fiber suspensions with a Cross-type
matrix material. For 31 fiber orientation states, six load cases, and over a wide range of shear
rates relevant to engineering processes, the FDMNs achieve validation errors below 4.31% when com-
pared to direct numerical simulations with Fast-Fourier-Transform based computational techniques.
Compared to a conventional machine learning approach introduced previously by the consortium of
authors, FDMNs offer better accuracy at an increased computational cost for the considered material
and flow scenarios.
Keywords: Deep Material Network; Fiber-reinforced composites; Non-Newtonian suspension; Infi-
nite material contrast; Supervised machine learning; Effective viscosity
1 Introduction
1.1 State of the art
Especially because of their high specific stiffness, fiber reinforced composites are widely used for lightweight
design, particularly in the transport and energy sectors [1,2]. During the design process of fiber reinforced
composite components, combinations of molding simulations and structural simulations are central to fa-
cilitate use case appropriate and material oriented design through digital twins [3] and virtual process
1
arXiv:2406.11662v1 [cs.CE] 17 Jun 2024
chains [4–6]. In molding simulations of fiber reinforced composite parts, accurate prediction of the local
fiber suspension viscosity is vital to correctly estimate process parameters [7, 8] and other engineering
quantities. In particular, the fiber orientation and the fiber volume distributions [9], the flow fields, [10],
as well as the final part properties [10–12] depend on the suspension viscosity. However, in component
scale simulations, it is computationally infeasible to fully resolve the microstructure of the whole domain
of interest to compute the suspension viscosity. For this purpose, analytical and computational homog-
enization methods are valuable tools to provide viscosity estimates for molding simulations. However,
significant challenges arise for the holistic analytical modelling of the suspension viscosity, because the
suspension viscosity depends on the local microstructure [13], the fiber volume fraction [14], and the fiber
orientation state [15]. Additionally, the flow field [16,17] and the melt temperature [18] influence the sus-
pension viscosity as well. Furthermore, typical matrix materials in fiber suspensions show non-Newtonian
behavior, which adds additional complexity to the derivation of appropriate models for the suspension
viscosity.
Depending on the fiber concentration of the suspension and the behavior of the matrix material, different
homogenization approaches have been proposed. According to Pipes et al. [19, §1] the fiber concentration
regimes may be defined using the fiber volume fraction cFand the fiber aspect ratio raas follows.
The dilute regime is defined through cF<(1/ra)2, the semi-dilute regime through (1/ra)2< cF<1/ra,
the concentrated regime through 1/ra< cF, and the hyperconcentrated regime through ra>100 [19].
Goddard [20,21], and Souloumiac and Vincent [22] proposed self-consistent analytical models for different
concentration regimes. The models by Goddard are restricted to the dilute and semi-dilute regimes, while
the models by Souloumiac and Vincent are applicable to the dilute, semi-dilute, and concentrated regimes.
Both models agree well with experimental results [23] qualitatively. However, the quantitative prediction
accuracy could be improved and varies strongly depending on the shear rate and fiber volume fraction.
Similarly, a semi-analytical model proposed by Férec et al. [24] for Ellis and Carreau-type matrix behavior
replicates the steady state flow solution of a simple shear flow, but its accuracy varies with the shear
rate and the fiber volume fraction. By incorporating uniformly distributed fiber misalignments with
an orientation averaging approach [25, 26], Pipes et al. [19, 27–29] derived models for the concentrated
and hyper-concentrated regime. The models are only applicable to collimated fiber arrays, but their
predictions agree well with experimental studies by Binding [30]. Combining orientation averaging with a
deformation mode and microstructure dependent informed isotropic viscosity, Favaloro et al. [31] derived a
model that aims to improve available molding simulation solvers with only small modifications. Depending
on the deformation mode and the approximated anisotropic viscosity, the prediction quality of the model
varies. However, molding simulations using the model successfully predicted the shell-core effect that is
common in a wide variety of fiber molding applications [9,11]. Overall, predicting the suspension viscosity
accurately over the wide variety of processing conditions used for engineering systems proves difficult,
which includes recent developments [32,33].
The development of homogenization methods is accompanied by considerable difficulties in studying
the fiber suspension viscosity experimentally. In rheometer studies, certain transient effects like fiber
breakage and fiber reorientation are difficult to quantify during the experiment. Thus, it is hard to link
a specific fiber orientation state and a specific fiber length distribution to a measured viscosity [30,34].
Furthermore, the measurement results may be affected by the interaction between the fibers and the
measurement devices [35].
In view of these difficulties, computational techniques offer an extraordinary approach to study the
effective viscosity of fiber suspensions with non-Newtonian solvents by resolving the local fields of interest
and providing insights, which are hard to obtain otherwise. Combining a mass tracking algorithm for
the free surface representation, a lattice Boltzmann method for fluid flow, and an immersed boundary
2
procedure for the interaction between fluid and rigid particles, Švec et al. [36] studied slump tests of
rigid fibers and rigid spherical particles suspended in a Bingham-type fluid. They compared slump tests
of the suspension with a slump test of the pure matrix material, and observed a smaller spread and an
increased height in the test of the suspension material, which implies an increased effective yield stress in
the suspension. Domurath et al. [37] employed a Finite Element Method (FEM) based approach to study
the transversely isotropic fluid equation by Ericksen [25]. They investigated the rheological coefficients
and found that the model by Souloumiac and Vincent [22] overpredicts the orientation dependence of a
rheological coefficient. Extending work by Bertóti et al. [38] on suspensions with Newtonian solvents,
Sterr et al. [17] used Fast Fourier Transform (FFT) based computational techniques [39] and the RVE
method [40] to study the effective viscosity of fiber suspensions with non-Newtonian solvents. They
investigated the effects of anisotropic shear-thinning on the effective suspension viscosity for varying fiber
volume fractions, shear rates, and flow scenarios.
Accurate computational predictions of the microscopic behavior of a composite often require significant
computational effort [41–45], which motivates the use of fast data based surrogate models for component
scale simulations [46]. Ashwin et al. [47, 48] trained a Multi-Layer-Perceptron, a Convolutional Neural
Network, and a U-Net [49] on particle resolved simulation data to predict fluid forces in dense ellipsoidal
particle suspensions. They restricted to Newtonian matrix behavior and trained the networks on data
for various Reynolds numbers and fiber volume fractions. Depending on the considered combination of
Reynolds number and particle volume fraction, the prediction accuracy of the surrogate models varies.
Boodaghidizaji et al. [50] use a multi-fidelity approach with neural networks and Gaussian processes to
predict the steady state viscosity of fiber suspensions with a Newtonian solvent. To form the training data
set, they combine low-fidelity estimates from constitutive equations with high-fidelity data obtained from
numerically solving the involved partial differential equation system. The prediction accuracy of the multi-
fidelity neural network and the multi-fidelity Gaussian process for simple shear flow depends strongly on
the investigated parameters, especially the fiber volume fraction. Sterr et al. [51] derived four models for
fiber suspensions with a Cross-type matrix fluid by combining FFT-based computational homogenization
techniques with supervised machine learning. They investigated the anisotropic shear-thinning character-
istics of the suspension viscosity for a variety of fiber orientation states via computational experiments,
and formulated model candidates based on the observed phenomena. Using supervised machine learning
techniques, they identified the model parameters from computational data, so that three of the four
models were able to predict the fiber suspension viscosities to engineering accuracy for a wide range of
engineering load cases. Generally, in addition to accurate estimation, extrapolation beyond the training
data, as well as ensuring thermodynamical consistency prove challenging in the construction of surrogate
models.
In the context of solid materials without kinematic constraints, Liu et al. [52,53] proposed Deep Material
Networks (DMNs) as surrogate models for the full-field computational homogenization of microstructured
materials. Their approach is based on nesting rotated laminates in an N-ary tree structure of N-phase
laminates, and thus constructing a micromechanically motivated deep learning architecture. The vol-
ume fractions and rotations of the DMN are then identified via supervised machine learning on linear
elastic data. Remarkably, even if a DMN is trained on data obtained by solving linear homogenization
problems, the predictions of the DMN for non-linear homogenization problems are impressively accurate.
Gajek et al. [54] further developed the DMN architecture into the rotation free direct DMN architec-
ture. Direct DMNs feature a faster and more robust training process, as well as an efficient evaluation
scheme for non-linear problems. Gajek et al. [54] also showed that the thermodynamic consistency of the
laminates is preserved in (direct) DMNs, such that the resulting DMN is thermodynamically consistent
as well. Furthermore, Gajek et al. [54] proved that non-linear homogenization is determined by linear
3
homogenization to first order for two-phase materials. In a later article, Gajek et al. [46] used direct
DMNs to accelerate two-scale FE simulations of fiber reinforced composites by augmenting direct DMNs
with the fiber orientation interpolation concept introduced by Köbler et al. [55]. Alternatively, DMN
parameters may be interpolated with regard to microstructural parameters by a posteriori interpolation
as proposed by Liu et al. [56] and Huang et al. [57], or by augmenting DMNs with neural networks as pro-
posed by Li [58]. To leverage the capabilities of direct DMNs in concurrent thermomechanical two-scale
simulations of composite components, Gajek et al. [59] further extended the direct DMN architecture
to incorporate thermomechanical coupling. Also in a thermomechanical setting, Shin et al. [60] trained
DMNs on linear thermoelastic data instead of linear elastic data, which improved the quality of fit for the
effective thermal expansion properties, but only slightly affected the non-linear prediction quality of the
DMNs. Additionally, they employed DMNs for uncertainty quantification, and for the inverse problem
of optimizing a thermal boundary condition to achieve a desired thermo-elasto-viscoplastic response. By
developing an inelastically-informed training strategy for DMNs, Dey et al. [61] successfully predicted
the creep behavior of fiber reinforced thermoplastics, which involves multiple scales in both space and
time. This enabled the inverse calibration of parameters for creep and plasticity constitutive equations
by using DMNs as surrogates for otherwise costly FFT-based computations [62]. Furthermore, Dey et
al. [63] leveraged DMNs in combination with fiber orientation interpolation [55] to characterize the be-
havior of fiber reinforced thermoplastics including damage, plasticity, and creep. Overall, DMNs were
extended and applied to treat a wide variety of problems, such as interface damage [64], the modeling of
multiscale strain localization [65], problems involving woven materials [66] and porous materials [67], as
well as the architecture independent treatment of multi-phase composites [68].
For composites with kinematically unconstrained solid phases, the presented literature provides com-
pelling evidence for the accuracy and versatility of (direct) DMNs. However, the (direct) DMN architec-
ture has yet to be applied to composites involving fluids or infinite material contrast, which is required
for suspensions of rigid fibers. Because (direct) DMNs are based on laminates of solids, the treatment
of fluids with a DMN architecture requires a different type of building block. Additionally, treatment
of infinite material contrast with DMNs is challenging, because the rank-one laminates of a DMN have
singular effective properties if one of the phases is singular, i.e., rigid. Therefore, the singular effective
properties of the rank-one laminates may propagate through the whole DMN during the training and
evaluation processes, rendering the effective properties of the DMN singular as well. In this article we
address these issues and propose an architecture to treat suspensions of rigid fibers.
1.2 Contributions
In this work, we extend the direct DMN architecture for kinematically unconstrained solid phases with
finite material contrast to an architecture that is able to treat suspensions of rigid fibers. In particular, this
task requires the treatment of fluid phases and infinite material contrast. We name this architecture the
Flexible DMN (FDMN) architecture. To treat fluid materials with FDMNs, we derive homogenization
blocks for layered emulsions that are governed by Stokes flow, and consist of possibly incompressible
phases with finite material contrast. In section 2.1, we establish that the velocity field in this type
of layered material is phase-wise affine, if the dissipation potentials of the phases are strictly rank-one
convex. We use this result and follow an analytical procedure described by Milton [69, §9] to derive the
closed form linear homogenization function of the considered layered emulsions in section 2.2. To treat
infinite material contrast, we study a particular type of layered material, namely, coated layered materials
(CLMs), in section 2.3. We obtain closed form homogenization functions for CLMs with kinematically
unconstrained or incompressible phases. With the objective of using CLMs as FDMN building blocks
4
in the presence of rigid fibers, we derive conditions for the required number of layering directions and
their orientation, such that the effective properties of CLM are non-singular. We do so for CLMs with
incompressible and rigid phases.
In section 3, we present the FDMN architecture as an extension of the (direct) DMN architecture. An
FDMN arises by replacing the rank-one laminates of a (direct) DMN with rank-one layered materials
capable of treating fluids, and non-singular CLMs capable of treating infinite material contrast. Also, we
discuss the material sampling, the offline training procedure, as well as the online evaluation of FDMNs
in the context of incompressible rigid fiber suspensions. We apply FDMNs to predict the viscous response
of fiber polymer suspensions in section 4, demonstrating both the ability to handle incompressible fluid
phases and infinite material contrast. We use FFT-based computational homogenization techniques to
generate linear training data and non-linear validation data for shear-thinning fiber suspensions with a
Cross-type matrix behavior, and a fiber volume fraction of 25%. We consider 31 different fiber orientation
states and a variety of incompressible elongational and shear flows. Using the computational data, we
train FDMNs for all fiber orientation states, and study the prediction accuracy of the FDMNs in the
non-linear case. For all investigated load cases and microstructures, the FDMNs achieve validation errors
below 4.31% over a wide range of shear rates relevant to engineering processes. Not considering the time
required to generate the training data and to train an FDMN, the FDMNs achieve speedup factors between
11785 and 17225 compared to FFT-based computations. Finally, we compare the prediction accuracy of
the FDMN based approach with a different machine learning approach by Sterr et al. [51], and find that
FDMNs offer better accuracy and flexibility at a higher computational cost for the considered material
and flow scenarios.
2 Homogenization of layered emulsions with infinite material con-
trast
2.1 Phase-wise affine displacement and velocity fields in laminates and emul-
sions
Like for (direct) DMNs, the building blocks of an FDMN should admit linear homogenization functions
in closed form, and there should exist efficient solution schemes to compute the stress response of the
building blocks in case their phases are non-linear. In analogy to the laminate based architecture of the
(direct) DMN, we thus consider layered emulsions as the building blocks of the FDMN architecture. Like
laminates, layered emulsions consist of multiple, possibly non-linear, fluid materials, that are arranged
such that every phase boundary between the phases is orthogonal to the layering direction n∈ S2, on
the 2-sphere S2. For the offline training and the online evaluation of an FDMN with layered emulsions
as building blocks, knowing the homogenization functions of layered emulsions with linear and non-linear
phases is necessary. Primarily, we look for similarities between the homogenization of layered emulsions
and solid laminates that can be leveraged. In laminates, the displacement fields are phase-wise affine, and
we wish to know if the velocity fields in layered emulsions belong to a particular class of fields as well. If
the velocity fields are phase-wise affine, established solution techniques on (direct) DMN architectures [54]
can be employed for the online evaluation of the FDMN, and convenient linear homogenization equations
can be employed for the offline training. We follow a procedure detailed by Kabel et al. [70] on the
homogenization of laminates, and establish the existence and uniqueness of phase-wise affine velocity
fields in a particular type of emulsion. We consider a K-phase layered emulsion with layering direction n,
which, similar to a laminate, is constructed by arranging K-phases so that the direction nis orthogonal
5
to all phase boundaries. The emulsion occupies a rectangular and periodic volume Y⊂R3, and consists
of non-Newtonian phases with rank-one convex dissipation potential densities Ψk: (R3)⊗2→R, such
that
Ψk(L+βa⊗b)≤βΨk(L+a⊗b) + (1 −β)Ψk(L)∀β∈[0,1],a,b∈R3\{0},(2.1)
where L∈(R3)⊗2denotes the velocity gradient, and the operator (·)⊗aconstructs a tensor space of a-th
order. Also, Yk⊂Ydenotes the sub volume of the k-th phase, and the phases do not intersect, i.e.,
Yj∩Yk=∅if j=k, (2.2)
and form the volume Y, such that
Y=
K
[
j=1
Yj.(2.3)
Let the operator ⟨·⟩Ydenote volume averaging of a quantity over a volume Y, i.e.,
⟨·⟩Y≡1
|Y|ZY
(·) dY(x)with |Y|≡ ZY
dY(x).(2.4)
Then, we express the velocity field v:Y→R3as
v=¯
Lx +ˆ
v,with ⟨∇ˆ
v⟩Y=0,(2.5)
where ¯
L∈(R3)⊗2stands for the prescribed effective velocity gradient and ˆ
vis the velocity fluctuation
field. Accordingly, the periodic dissipation potential density function Ψ : Y×(R3)⊗2→Rof the emulsion
is defined through
Ψ(x,∇sv) =
K
X
i
χk(x)Ψk(x,∇sv),(2.6)
where χk:Y→ {0,1}refers to the characteristic function of the sub volume Yk, and ∇sstands for the
symmetrized gradient. Let the flow inside the emulsion be governed by Stokes flow, i.e., the advectorial
forces in the layered emulsion are small in contrast to the viscous forces, then the steady state balance
of mass
div (ρv) = 0,(2.7)
where ρ:Y:→Rdenotes the mass density field, is satisfied. Additionally, the stress field
σ=S(·,¯
D+∇sˆ
v)(2.8)
in terms of the effective strain rate tensor ¯
D= ( ¯
L+¯
LT)/2, and the associated stress operator Son the
space of symmetric second order tensors Sym(3)⊂(R3)⊗2[71]
S:Y×Sym(3)→Sym(3),(2.9)
(x,D)7→ ∂Ψ
∂D(x,D),(2.10)
satisfies the balance of linear momentum without inertial effects and volume forces
div σ=0.(2.11)
Therefore, the effective response of the emulsion is obtained from the minimization problem
Ψ(x,¯
D+∇sˆ
v)Y−→ min
ˆ
v∈A,(2.12)
6
where Astands for the admissible set
A=ˆ
v:Y→R3div (ρv)=0 and ⟨∇ˆ
v⟩Y=0(2.13)
Hence, assuming there is no slip between the phases, the proof by Kabel et al. [70, §2] extends to layered
emulsions with finite material contrast. Therefore, if the dissipation potential densities Ψkare strictly
rank-one convex, there exists a unique minimizer ˆ
vin the class of phase-wise affine velocity fields. As a
special case, this is also true for incompressible Stokes flows, for which the balance of mass simplifies to
div v= 0.(2.14)
In the following, we use the term layered emulsion only for the type of emulsion defined above.
2.2 Two-phase layered emulsions with Newtonian phases
Similar to two-phase laminates of linear elastic solid materials, we wish to find a convenient closed form
expression for the linear homogenization function of two-phase layered emulsions. With the goal of using
layered emulsions as building blocks of an FDMN, we dedicate this section to deriving such a closed form
expression for two-phase layered emulsions with kinematically unconstrained or incompressible Newtonian
phases. For this purpose, we employ the results of the previous section 2.1, and extend an existing
homogenization approach for two-phase laminates of linear elastic phases by Milton [69, §9]. To emphasize
the link between the linear homogenization of laminates and layered emulsions we define the class of linear
materials M, and study layered materials of solids and fluids in parallel. A material mkin the class of
linear materials Mis characterized by the primal material tensor Mkor its dual tensor Kk≡M−1
k, such
that
(Mk,Kk)∈ {(M+)2,(M+
0)2}.(2.15)
Here, the convex cones M+⊂(R3)⊗4and M+
0⊂(R3)⊗4comprise all fourth order tensors X∈(R3)⊗4,
which are positive definite on the vector spaces of symmetric second order tensors Sym(3)and traceless
symmetric second order tensors Sym0(3)⊂Sym(3), respectively. Additionally, a tensor X∈ {M+,M+
0}
has minor and major symmetries
X=XTH=XTL=XTR,(2.16)
and a tensor X∈M+
0maps the second order unit tensor Ito zero, such that
ker X=I,i.e., X[I] = 0.(2.17)
Here, we define the map T[Y]in component form and in the standard basis of R3as
T[Y]=
∧
Tijkl Ykl ,for some Y∈(R3)⊗2,T∈(R3)⊗4.(2.18)
Any material mk∈ M follows the linear constitutive equations
σco =Mk[∇sw]and ∇sw=Kk[σco],(2.19)
if the material mkis not subjected to kinematic constraints, or the constitutive equation
σico =τ−pI=Mk[∇sw]−pI,and ∇sw=Kk[τ],(2.20)
if the material mkis incompressible. Here, wdenotes the field associated with the type of material mk,
i.e., the displacement field ufor linear elastic solids or the velocity field vfor linearly viscous fluids. For
the constitutive equation (2.20) we use an additive split of the stress field σ
σ=τ−pI,(2.21)
7
into the deviatoric stress field τ:Y→Sym0(3)and the pressure field p:Y→R. Here, we use Y⊂R3
to denote the volume occupied by a two-phase layered material, i.e., a laminate or a layered emulsion,
and Y1, Y2⊂Yto denote the volume occupied by the two phases m1, m2∈ M of the layered material.
We define the effective properties (¯
M,¯
K)∈ {(M+)2,(M+
0)2}of the layered material in the sense of the
Hill-Mandel condition [72,73]
⟨σ· ∇sw⟩Y=⟨σ⟩Y· ⟨∇sw⟩Y≡¯
M·(⟨∇sw⟩Y⊗ ⟨∇sw⟩Y).(2.22)
Then, for perfectly bonded laminates of linear solid materials without kinematic constraints, a closed
form expression for the primal or dual effective properties of the laminate ¯
A∈ { ¯
M,¯
K}can be derived
using the procedure detailed by Milton [69, §9], such that
¯
A−A0−1+
L
A0(n)−1=(A(x)−A0)−1+
L
A0(n)−1.(2.23)
Here, the inverse is taken on the space M+in which ¯
A∈M+lives, A:Y→M+denotes the field of the
phase wise primal or dual material properties
A:Y→M+,x7→
A1if x∈Y1,
A2if x∈Y2,(2.24)
and A0∈M+refers to an arbitrary reference material tensor such that the term (A(x)−A0)is invertible.
Also, the the operator
L
A0(n)∈(R3)⊗4is defined as
L
A0(n) =
L
I(n) [
L
I(n)A0
L
I(n)]†
L
I(n),(2.25)
where (·)†stands for the Moore–Penrose pseudoinverse, and the operators
L
A0(n)and
L
I(n)project onto
the same subspace with respect to different inner products. More specifically, the operator
L
A0(n)is an
orthogonal projector with respect to the A0-weighted inner product
⟨X,X⟩A0=X·A0[X],X∈Sym(3),(2.26)
and the operator
L
I(n)is an orthogonal projector with respect to the Frobenius inner product
⟨X,X⟩F=X·I[X] = X·X,X∈Sym(3),(2.27)
where I∈(R3)⊗4denotes the fourth order identity tensor. The operator
L
I(n)encodes restrictions im-
posed by the periodicity of the laminate, the perfect bonding of the phases, the momentum balance (2.11),
and the constitutive equation (2.19). Depending on whether we wish to compute the effective primal or the
effective dual properties of the laminate, a different operator
L
I(n)is required. For the homogenization
of the primal effective properties, i.e., ¯
A=¯
M, the operator
L
I(n)equals the operator
L
co
1(n). For the ho-
mogenization of the dual effective properties, i.e., ¯
A=¯
K, the operator
L
I(n)equals the operator
L
co
2(n).
The operators
L
co
1and
L
co
2, are defined through their action on a second order tensor X∈Sym(3)
L
co
1(n)[X] = 2(Xn)⊗sn−(n·Xn)n⊗nand
L
co
2(n)[X] = X−
L
co
1(n)X,(2.28)
where ⊗sdenotes the symmetrized dyadic product. The operators
L
co
1and
L
co
2project onto the subspaces
Eco =Sym(3)∋X=d⊗n+n⊗dd∈R3and Sco ={Sym(3)∋X|Xn =0},(2.29)
and arise from the fact that the jumps of the field J∇swK∈ Eco
J∇swK=J∇w nK⊗sn,(2.30)
8
and the jump of the stress JσK∈ Sco live on mutually orthogonal subspaces [69, §9], because it holds that
JσnK=0.(2.31)
Here, we define the jump JqKof a quantity qacross a phase boundary with normal vector nas
JqK=q+−q−,(2.32)
where q+and q−denote the outer and inner limits of the quantity qwith regard to the outside facing
normal vector n, respectively.
Interestingly, the constitutive equations for kinematically unconstrained linear elastic solids and linearly
viscous fluids share the same structure, see equation (2.19). Additionally, equations (2.30) and (2.31)
are satisfied for layered emulsions by definition, because there is no slip between the phases, and the
momentum balance (2.11) is satisfied. Thus, the homogenization problem for the considered layered
emulsions has the same structure as the homogenization problem for laminates. As a result, the effective
properties ¯
A∈M+of a two-phase layered emulsion of kinematically unconstrained materials m1, m2∈ M
can be computed with equation (2.23). If the kinematic field wis solenoidal such that the incompressible
balance of mass (2.14) is satisfied, it follows from equation (2.30) that
J∇sw nK·n= 0.(2.33)
Also, if the materials m1, m2∈ M are incompressible, it follows from equations (2.21) and (2.31) that
JσnK=0⇐⇒ Jτ nK=JpKn.(2.34)
Therefore, the jumps J∇swK∈ Eico and JσK∈ Sico live in the mutually orthogonal spaces
Eico =Sym0(3)∋X= 2 d⊗sn−2(d·n)n⊗nd∈R3(2.35)
and
Sico ={Sym0(3)∋X|Xn =an, a ∈R}.(2.36)
The two operators
L
ico
1(n)and
L
ico
2(n), defined through their action on a traceless second order tensor
X∈Sym0(3),
L
ico
1(n)[X] = 2(Xn)⊗sn−2(n·Xn)n⊗nand
L
ico
2(n)[X] = X−
L
ico
2(n)[X],(2.37)
project onto the spaces Eico and Sico , respectively. Consequently, the procedure detailed in Milton [69] can
be used to compute the effective properties ¯
A∈M+
0of a layered material with incompressible constituents.
In this context the homogenization equation (2.23) applies if the property field A:Y:→M+
0, the reference
properties A0∈M+
0, and the operator
L
I(n)∈(
L
ico
1(n),
L
ico
2(n)) are selected based on whether the
tensor ¯
Arepresents primal or dual effective properties. In summary, for a layered material consisting of
linear materials m1, m2∈ M, the effective material properties (¯
M,¯
K)∈ {(M+)2,(M+
0)2}can be computed
with the homogenization equation (2.23). The choice of the operator
L
I(n)depends on whether the phases
are compressible or incompressible, and whether the primal or dual effective tensors are sought.
2.3 Coated layered materials
Because we are interested in building FDMNs for suspensions of rigid particles, the architecture of the
FDMN should be capable of handling infinitely viscous, i.e., rigid, phases. In this section, we leverage
the homogenization equation (2.23) to study the homogenization of repeatedly layered materials in the
9
(a) Orthogonal layering directions (b) Non-orthogonal layering directions
Figure 1: Rank-three, coated layered materials with orthogonal layering directions (a), and non-
orthogonal layering directions (b). The core material is shown in red and the coating material is shown
in gray. We hint at the separation of length scales between the material layers through our choice of the
layer thicknesses.
presence of rigid phases. We define and use the term rank of a layered material in the following to better
distinguish between the possible ways to construct layered materials. The rank R∈N≥0of a hierarchical,
layered material defines the number of layering steps that are needed to construct the material using
layering directions nr∈ S2, such that the index rstarts at one and ends at R, i.e., r∈ {1, ..., R}. By
definition, a rank-0layered material is simply a single phase material without any layering. Also, the
materials that are layered at each layering step need not be the same. For example, let the first layer in
a rank-2 layered material consist of the materials m1and m2. Then, the second layer may consist of the
materials m1and m2in the form of the first layer, and a third material m3.
Rank-one two-phase layered materials are unsuitable as DMN building blocks in the case of infinite
material contrast, because their effective properties are always singular if one phase is singular. In search
of a building block capable of handling rigid phases, we study the effective properties of a particular type
of layered material following the terminology and the analytical approach detailed by Milton [69, §9].
We refer to this particular type of layered materials as coated layered materials (CLMs). To construct
a CLM, one material m2, called the coating, is repeatedly layered onto another material m1, called the
core. The layering process is divided into multiple steps, such that the r-th layering step is applied in
direction nr, and the total number of layering steps equals the rank Rof the CLM. As an example, two
rank-three CLMs are depicted in Figure 1, where the layering directions nr∈ N1of the material shown
in Figure 1(a) are
N1=(1,0,0){ei},(0,0,1){ei},(0,1,0){ei},(2.38)
and the layering directions nr∈ N2of the material shown in Figure 1(b) are
N2=(1,0,0){ei},1
√2(1,1,0){ei},1
√2(1,0,1){ei}.(2.39)
As described, the gray coating material m2is repeatedly applied in layers, such that for r > 1, the
r-th sub layer of the rank-three CLM is constructed by layering a rank-(r−1) layered material with
10
the coating material m2. By definition, the 0-th sub layer of the rank-three CLM is purely made of
the core material m1. We wish to compute the effective properties ¯
Aof a coated layered material by
repeatedly applying the homogenization equation (2.23). For the recursive application of the rank-one
homogenization formula (2.23) to be valid, the phases at each homogenization step must be homogeneous
on the length scale of the respective layering step. This requires a separation of length scales, in the
sense that the thickness δrof the material sections in the rank-rsub layer is drastically smaller than the
thickness δr+1 of the material sections in the next sub layer, i.e., δr≪δr+1 . In Figure 1, we hint at
this separation of length scales between the material layers through our choice of the layer thicknesses.
Nonetheless, we retain a visible degree of inhomogeneity, so that the layer structures can be distinguished
for illustration purposes.
Before we repeatedly apply equation (2.23) to derive an expression for the effective properties ¯
A, we
first simplify the expression (2.23) for the case at hand. We let the reference material properties A0in
equation (2.23) approach the properties A2of the coating material m2, resulting in the equation [69, §9]
(1 −f2)(¯
A−A2)−1= (A1−A2)−1+f2
L
A2(n).(2.40)
Here, the coating material m2must not be rigid or a void, the tensor (A1−A2)must be invertible, and f2
denotes the volume fraction of the coating material m2. Also, the fourth order tensor
L
A2(nr)∈(R3)⊗4
is defined by equation (2.25) as
L
A2(nr) =
L
I(nr) [
L
I(nr)A2
L
I(nr)]†
L
I(nr),(2.41)
where the inverse is taken on the subspace onto which
L
Iprojects. To compute the effective properties ¯
A
of a rank-RCLM, we recursively apply equation (2.40) for the effective material properties of rank-one
layered materials, such that [69, §9]
(1 −f2)(¯
A−A2)−1= (A1−A2)−1+f2
R
X
r=1
cr
L
A2(nr).(2.42)
Here, the scalars crare computed from the volume fractions f(r)
1of the core material m1in the rank-r
sub layer of the rank-three CLM, such that
cr=f(r−1)
1−f(r)
1
f2
,(2.43)
i.e., the properties
R
X
r=1
cr= 1 and cr>0,(2.44)
hold. Because the rank-0sub layer consists solely of the core material m1by definition, the volume
fraction f(0) = 1 is one. Equation (2.42) is essential to compute the effective properties of layered
materials with infinite material contrast, and thus for the treatment of such materials using a DMN
architecture. In Appendix A, we employ equation (2.42) to study the conditions under which the effective
properties ¯
Aof a CLM are singular when the core material m1is rigid. With the objective of applying
FDMNs to suspensions of rigid fibers in mind, we summarize the results pertaining to rank-RCLMs with
arigid core material m1and an incompressible coating material m2with the following statements.
1. The effective properties ¯
Aof a CLM of rank R < 3are always singular.
2. For rank-3 CLMs, the effective properties ¯
Aare singular if at least one layering direction is orthog-
onal to two other layering directions, or if at least two layering directions are collinear.
11
3. Rank-3CLMs with mutually non-orthogonal and mutually non-collinear layering directions n1,n2,
and n3have non-singular effective properties. Thus, in the context of this article, we require
0<|n1·n2|<1,0<|n1·n3|<1,0<|n2·n3|<1.(2.45)
The statements 1-3 are of critical importance for the treatment of incompressible suspensions of rigid
particles using a DMN architecture. It follows that CLMs with an incompressible coating material and
rigid core material should be at least of rank R= 3 and have non-collinear and mutually non-orthogonal
layering directions to be non-singular. Therefore, we leverage such non-singular-singular rank-3 CLMs in
the FDMN architecture to treat suspensions of rigid fibers, as we explain in section 3.2.
3 Identifying deep material networks for suspensions of rigid fibers
3.1 Architecture of direct deep material networks
Because the FDMN architecture extends the (direct) DMN framework, we briefly summarize the direct
DMN architecture in this section, before presenting the FDMN architecture in the next section 3.2. For
in depth discussions and explanations of the direct DMN architecture, we refer the reader to articles
by Gajek et al. [46, 54,59]. A two-phase direct DMN is defined as a perfect binary tree, consisting of
two-phase laminates Bi
kas nodes, see Figure 2(a). The laminates Bi
kare defined by a single lamination
direction ni
kand two volume fractions c1
k,1and c2
k,1. The DMN tree is ordered, rooted and has depth K.
We use the letter k= 1, ..., K to denote the depth of a node and the letter i= 1, ..., 2k−1to indicate
the horizontal position of a node in the respective layer. We parametrize a direct DMN by the two
(a) Weight propagation (b) Stiffness propagation
Figure 2: Weight and stiffness propagation (from the bottom to the top) in a two-phase direct DMN [46]
of depth K = 3, by Gajek et al. [59, Fig. 1], licensed under CC BY 4.0
vectors n∈(R3)2K−1
n= (n1
K, ...., n2K−1
K,n1
K−1, ..., n2K−2
K−1, ..., n1
1),(3.1)
and w∈R2K
w= (w1
K+1, ..., w2K
K+1).(3.2)
Instead of the laminate volume fractions, the direct DMN is parametrized in terms of the weights wi
K+1
to improve numerical stability during the training process [52, 53]. The i-th weight in the k-th layer is
computed from the pairwise summation of the weights in the previous layer k+ 1, see Figure 2(a), such
that
wi
k=w2i−1
k+1 +w2i
k+1,(3.3)
12
and the volume fractions ci
k,1and ci
k,2follow from
ci
k,1=w2i−1
k+1
w2i−1
k+1 +w2i
k+1
,and ci
k,2= 1 −ci
k,1.(3.4)
Additionally, the weights wi
K+1 need to be positive and sum to unity, such that
wi
K+1 ≥0and
2K
X
i
wi
K+1 = 1.(3.5)
The vectors nand wfully parametrize a direct DMN of depth Kand are identified by fitting the linear
homogenization function of the DMN to the linear homogenization function of the microstructure of
interest. The fitting process is carried out by using supervised machine learning, and we refer to this
process as offline training. To evaluate the linear homogenization function of the DMN during the offline
training, input stiffness pairs are propagated through the network of homogenization blocks Bi
k, see
Figure 2(b). Consequently, if the stiffness Ci
kof a homogenization block Bi
kis singular, this singular
stiffness can propagate through the whole DMN, which is undesirable for the offline training. Once the
parameter vectors nand ware known, they remain unchanged and are used in the online evaluation to
predict the non-linear response of the studied material. In sections 3.4, 4.2, and 3.5, we discuss the offline
training and the online evaluation for the proposed FDMN architecture.
3.2 Architecture of flexible deep material networks
With the results of the previous sections 2.1-2.3 at hand, we present an extended DMN architecture that
is able to treat fluid phases and infinite material contrast. Compared to the architecture of (direct) DMNs
shown in Figure 2, the lowest layer of laminates is replaced by a layer of rank-RCLMs CR, see Figure 3.
The rank Rof the CLM CRand the restrictions on its layering directions nrare problem dependent, and
(a) Weight propagation (b) Material tensor propagation
Figure 3: Weight and material property propagation (from the bottom to the top) in a two-phase
FDMN [46] of depth K= 3 with rank-3CLMs
chosen such that the non-singularity condition (5.11) is satisfied. This guarantees that only non-singular
material tensors Ai
kare propagated through the network. If all phases are kinematically unconstrained,
solid, and non-singular, rank-1CLMs can be employed to recover the direct DMN architecture. However,
in the case of incompressible suspensions of rigid particles, the CLMs are of rank R= 3 and have
mutually non-collinear and non-orthogonal layering directions nr, as per statements 1-3. Additionally,
13
the homogenization blocks Hi
kimplement the general homogenization function defined by equation (2.23)
in which the projector function
L
Idepends on the physics of the investigated problem. This allows the
treatment of composites consisting of linear materials m∈ M, including incompressible linearly viscous
fluids. For problems involving two-phase materials, R+ 1 weights and two materials are assigned to each
CLM. The number of layering directions Nnand the number of input weights Nwin a two-phase FDMN
of depth Kwith rank-RCLMs are
Nn= 2K−1(R+ 1) −1, Nw= 2K−1(R+ 1).(3.6)
As an example, Figures 3(a) and 3(b) show the weight and material property propagation of a two-phase
FDMN of depth K= 3 with two-phase CLMs of rank R= 3 as bottom layer. Of the four weights of
a rank-3CLM, one is associated with the core material m1, and three are associated with the coating
material m2. This doubles the number of weights for an FDMN with rank-3CLMs as compared to a
DMN of the same depth K. Additionally, each rank-3CLM block C3has three layering directions nr
instead of one layering direction like rank-one laminates. To reduce the number of free parameters and
ensure the CLMs are non-singular, the relative angles between the layering directions nrare fixed, while
their joint orientation in space is determined through the offline training, as we explain in section 3.4
in more detail. Because CLMs are based on the repeated layering of rank-one layered materials, an
FDMN of depth Kcan be considered to be a special case of a (direct) DMN of depth K+ (R−1) with
generalized homogenization blocks. Then, a CLM represents a subtree of the DMN of depth K+ (R−1),
in which the weights wi
kof certain phases are set to zero, such that a CLM emerges, see Figure 4. In
(a) Weight propagation (b) Material tensor propagation
Figure 4: Weight and material property propagation (from the bottom to the top) in a two-phase DMN
of depth K= 3, representing a two-phase rank-3CLM
particular, except for one phase, the weights associated with the singular core material m1are set to
zero. Additionally, Rweights associated with the non-singular coating material m2must be non-zero,
such that the rank Rof the CLM is maintained. Technically, (direct) DMNs are therefore able to treat
infinite material contrast intrinsically. However, the weights wi
kneed to be identified during the offline
training, and might not always have appropriate values to generate CLMs with the required rank, such
that singular material tensors Ai
kmight propagate through the whole DMN. Also, if the layering normals
are not chosen appropriately, the singular material properties may propagate through the whole DMN as
well. Controlling the weights and normals via the implementation of the CLM blocks instead of learning
them is therefore preferable to prevent singularities during the offline training. Furthermore, FDMNs of
depth Khave fewer free parameters compared to equivalent DMNs of depth K+R−1, which aids in
the offline training as well.
14
Hypothetically, the presented FDMN architecture could be applied to problems with more than two
phases as well. Let NRdenote the number of non-singular phases and NSdenote the number of singular
phases. Then, for each singular phase, the FDMN needs to contain at least one CLM to which the phase
is assigned, and which has non-singular effective properties. This requires a minimum of NSCLMs.
Additionally, the coatings of the CLMs can be constructed by mixing the non-singular phases, possibly
by constructing NR-phase layered materials. However, the application of FDMNs to materials with voids
or more than two phases is beyond the scope of this article, because we focus on suspensions of rigid
fibers.
3.3 Material sampling
For materials with solid phases, DMNs are trained by sampling the linear elastic homogenization function
of the microstructure of interest with randomly generated input materials [46,52, 53, 59], and fitting the
DMN parameters to replicate the desired homogenization function. Similarly, in the case of rigid particles
suspended in viscous media, we train FDMNs to learn the linear viscous homogenization function with the
goal of predicting non-linear viscous behavior. Consequently, we sample the linear viscous homogenization
function by using randomly generated, linearly viscous input materials. We base our sampling strategy
on the approach used by Gajek et al. [46]. In the article by Gajek et al. [46] on glass fiber reinforced
polyamide composites, the authors chose the stiffness samples for the fiber material to be isotropic, and
the stiffness samples for to the polyamide matrix to be transversely isotropic. This strategy leads to
a smaller sample space compared to sampling orthotropic stiffnesses for both the fiber and the matrix
materials, as suggested by Liu et al. [52,53]. To reduce the computational cost of sampling even further
for the considered case of fiber suspension, we only generate sample viscosities for the matrix material
and consider the suspended fibers to be rigid in all computations. To generate samples for the matrix
material, we follow Gajek et al. [46] and choose the sample viscosity Vspertaining to the s-th sample to
be transversely isotropic, such that
Vs= 2η2P2−gA′(a)⊗A′(a),(3.7)
where the scalar g∈[0,1) defines the magnitude of the perturbation by the unitary and deviatoric
tensor A′(a)∈Sym0(3), the vector a∈R4collects four angles, and we consider the scalar viscosities
η2= 10pGPa, p ∈[−3,3].(3.8)
For a fiber reinforced polyamide, Gajek et al. [46] employed an ansatz of the form (3.7) to sample the ma-
terial tangents of a J2-elastoplastic matrix material. Similarly, with the objective of applying FDMNs to
fiber suspensions with a shear-thinning polyamide matrix as detailed in section 4.1, we use equation (3.7)
to sample the material tangents of a Cross-type (4.4) matrix material. The tensor function A′is defined
by an eigenvalue decomposition [46]
A′:R4→Sym0(3),(3.9)
(a1, a2, a3, a4)T7→ Q(a2, a3, a4)A(a1)Q(a2, a3, a4)T,(3.10)
into the orthogonal tensor Q(a2, a3, a4)and a diagonal tensor A(a1)in terms of the vector a= (a1, a2, a3, a4)T.
The tensor A(a1)is constructed from the angle a1via [46]
A=1
qξ1+1
2ξ1+1
diag ξ2cos (a1), ξ2sin (a1),1
√2,(3.11)
ξ1= cos (a1) sin (a1), ξ2=−√2
2 cos (a1) + 2 sin (a1),(3.12)
15
and the tensor Q(a2, a3, a4)represents a rotation around the direction n, such that
Q(a2, a3, a4) : R3→R3,x7→ cos (a2)x+ sin (a2)n×x+ (1 −cos (a2))(n·x)n,(3.13)
n= sin (a3) cos (a4)e1+ sin (a3) sin (a4)e2+ cos (a3)e3.(3.14)
Here, the operator diag constructs a second order tensor in the standard basis eiof R3, and the compo-
nents of the vector alive on the intervals
a1∈[0,2π], a2−sin(a2)∈[0, π], a3∈[0, π], a4∈[0,2π].(3.15)
We sample the space of the variables pand ausing the scrambled Sobol sequence [74,75], and employ
an FFT-based computational procedure to compute the resulting effective viscosities ¯
Vs[17,38], see sec-
tion (4).
3.4 Offline training
Using the material sampling procedure described in the previous section 3.3, we wish to optimize the
parameters of the FDMN with regard to the training data set. For the case of rigid fibers suspended in
viscous media, we consider FDMNs with rank-3CLMs as bottom layer, which satisfy the requirements
derived in section A.2. We collect all layering directions of the FDMN in a vector
n= (n1
K, ..., n3(2K−1)
K,n1
K−1, ..., n2K−2
K−1, ..., n1
2,n2
2,n1
1),(3.16)
where we used the fact that the k-th layer contains 2k−1normals if k < K, and the K-th layer con-
tains 3(2K−1)normals. Because the homogenization blocks in the K-th layer are rank-three CLMs, the
number of normals is different from the other layers. Each normal ni
kin the k-th layer for k < K is
constructed using two angles ai
kand bi
kvia
ni
k= sinbi
kcosai
ke1+ sinbi
ksinai
ke2+ cosbi
ke3.(3.17)
However, for the normals ni
Kin the K-th layer, the procedure is different. To avoid singular effec-
tive properties during the evaluation of the FDMN in accordance with the discussion in sections 2.3
and A.2, we ensure that the three layering normals of a rank-three CLM block are not mutually non-
orthogonal and mutually non-collinear as follows. First, for each rank-three CLM block Ci
3, we com-
pute the components Qi
op of the orthogonal tensor Qiin the standard basis {ei}using three Euler
angles ci
K, di
K, ei
K∈[0,2π]via
Qi
op =
1 0 0
0 cosci
K−sinci
K
0 sinci
Kcosci
K
cosdi
K0 sindi
K
0 1 0
−sindi
K0 cosdi
K
cosei
K−sinei
K0
sinei
Kcosei
K0
0 0 1
.(3.18)
Then, we compute the layering directions n3(i−1)+1
K,n3(i−1)+2
K,n3(i−1)+3
Kof the rank-three CLM Ci
3via
n3(i−1)+1
K=qi
1,n3(i−1)+2
K=1
√2qi
1+qi
2,n3(i−1)+3
K=1
√2qi
1+qi
3,(3.19)
where qi
1,qi
2, and qi
3are the first, second, and third column vectors of the tensor Qiin the standard
basis ei, respectively. This leads to a total of Nαangles with
Nα= 5 2K−1−2.(3.20)
16
The angles between the layering directions are fixed for each CLM C3, but the three layering direc-
tions are rotated together during training. Other procedures than the one presented here are vi-
able as well, as long as the constructed vectors n3(i−1)+1
K,n3(i−1)+2
K, and n3(i−1)+3
Kare mutually non-
collinear and non-orthogonal. Alternatively, one could parametrize each of the three layering directions
n3(i−1)+1
K,n3(i−1)+2
K, and n3(i−1)+3
Kwith two angles each, and identify the angles independently. Then,
the effective properties of the CLM Ci
3are non-singular almost surely, because the probability of any two
of the three layering directions being collinear or one vector being orthogonal to the other two is zero.
However, parametrizing each of the three layering directions with two angles requires twice the amount
of angles in the CLM layer of the FDMN as compared to the presented procedure.
To define the linear homogenization function of the FDMN, we collect all Nαangles of the layering
directions in the vector α
α= (c1
K, d1
K, e1
K, ..., c3(2K−1)
K, d3(2K−1)
K, e3(2K−1)
K, a1
K−1, b1
K−1, ..., a2K−2
K−1, b2K−2
K−1, ..., a1
1, b1
1),(3.21)
and collect all Nwinput weights of the FDMN in the vector
w= (w1
K+1, ..., wNw
K+1).(3.22)
We follow Gajek et al. [54] and define the loss function L:R⊗Nα×R⊗Nw→Rin terms of the linear
homogenization function DMNL:A × A × R⊗Nα×R⊗Nw→ A of the FDMN as
L(α, w) = 1
Nb
q
v
u
u
t
Nb
X
s=1
¯
As− DMN L(As
1,As
2, α, w)
p
¯
As
p!q
+λ Nw
X
i=1
wi
K+1 −1!2
,(3.23)
where Nbis the number of samples, As
1,As
2∈ A denote the material tensors of the phases of a sam-
ple, and ¯
As∈ A denotes the effective material tensor of a sample. Additionally, the class of material
tensors A ∈ {M,M0}depends on the considered problem, we follow Gajek et al. [54] and choose the con-
stants p, q ∈N≥0as p= 1 and q= 10, and the operator ∥·∥1refers to the ℓ1-norm of the components in
Mandel notation. The second summand in equation (3.23) encodes the simplex constraint on the weights
of the DMN Nw
X
i=1
wi
K+1 = 1,where wi
K+1 ≥0,(3.24)
in the form of a quadratic penalty term with the penalty parameter λ > 0. We implement the offline
training in the Python programming language and use the machine learning framework PyTorch [76] to
identify the vectors αand w. To do so, we leverage the automatic differentiation capabilities of PyTorch
to solve the minimization problem
L(α, w)→min
α,w ,(3.25)
using the RAdam [77] optimizer. We use mini batches of Nbsamples to compute the parameter updates
αq+1 =αq−κq,α
∂L
∂α (α, w), wq+1 =wq−κq,w
∂L
∂w (α, w)(3.26)
for every epoch q. Also, we follow previous work [52–54] and use perfect binary trees without tree
compression [46,52, 53]
3.5 Online evaluation
In section 2.1, we showed that the velocity fields in layered emulsions are phase-wise affine, and thus
the online evaluation procedure of direct DMNs can be employed for FDMNs as well if the phases
17
are kinematically unconstrained. However, for FDMNs with incompressible phases, the online eval-
uation procedure needs to be adapted, as we discuss in the following. We define the linear opera-
tor A: (R3)⊗Nn→(Sym0(3))Nwwhich maps the emulsion-wise jumps b∈(R3)⊗Nnof the velocity
gradient onto the phase-wise strain rate fluctuations. Also, we express the effective dissipation po-
tential density ¯
Ψ:(Sym0(3))Nw→Rof the FDMN in terms of the phase-wise dissipation poten-
tials Ψi:Sym0(3)→R, the respective weights wi
K+1, and the phase-wise strain rate tensor Di∈Sym0(3)
as
¯
Ψ¯
D+A b=
Nw
X
i=1
wi
K+1Ψi(Di),(3.27)
where all entries of the vector ¯
D∈(Sym0(3))Nwequal the effective strain rate ¯
D, i.e.,
¯
D= ( ¯
D,¯
D, ..., ¯
D).(3.28)
Thus, the minimization problem for the online, non-linear evaluation of the FDMN is
¯
Ψ¯
D+A b→min
b∈(R3)⊗Nn
,(3.29)
and the Euler–Lagrange equation of the problem (3.29) reads
∂¯
Ψ
∂b ¯
D+A b=AT∂¯
Ψ
∂D ¯
D+A b= 0.(3.30)
In terms of a weight matrix W: (Sym0(3))Nw→(Sym0(3))Nwdefined by its action on a vector d∈(Sym0(3))Nw
W d =w1
K+1d1, w2
K+1d2, ..., wNw
K+1dK,(3.31)
we may write the Euler–Lagrange equation (3.30) as
ATW τ ¯
D+A b= 0,(3.32)
with the vector of phase wise stresses τ¯
D+A b∈(Sym0(3))Nw. Therefore, the jump vector bsatisfying
equation (3.32) is determined using a Newton scheme with the update rule
bn+1 =bn+sn∆bn,(3.33)
containing the iteration count n, the backtracking factor sn, and the update direction ∆bn. The update
direction ∆bnsolves the linear system
H∆bn=ATW τ ¯
D+A bA∆bn=ATW τ ¯
D+A b.(3.34)
So far, the procedure is completely analogous to the approaches for (direct) DMNs presented in previous
work [46,54,59]. However, in the case of incompressible phases, the matrix Ais symmetric but singular,
and the solution of the equation system (3.34) is not unique. However, because of incompressibility, we
know that the i-th entry biof the jump vector bis orthogonal to the i-th entry niof the vector of layering
normals n, i.e.,
bi·ni= 0.(3.35)
Hence, we construct the block diagonal matrix N: (R3)⊗Nn→(R3)⊗Nnconsisting of Nwblocks Nion
the main diagonal, which are defined as
Ni=βini⊗ni, i ∈ {1, ..., Nw}(3.36)
18
with the constant βi∈R>0. By definition it holds that
Nibi=0,(3.37)
and the jump vector bis in the kernel of N, such that
N b = 0.(3.38)
Thus, we add equations (3.34) and (3.38), such that the solution ∆bnof the resulting update rule
H+N∆bn=ATW τ ¯
D+A b,(3.39)
is unique. However, floating point precision may cause issues during the addition of the operators H
and N, if numbers of largely different magnitudes are summed. To remedy this issue, the scalars βi
need to be chosen according to the magnitude of the entries in the matrix H. Furthermore, some input
weights wi
K+1 might become equal to, or close to zero during training, which might render the system
matrix (H+N)ill-conditioned. Thus, it might be difficult to obtain accurate update directions ∆bnby
solving equation (3.39). While it is generally possible to prune the FDMN tree [53] by removing phases
with vanishing or almost vanishing volume fractions, this might collapse the rank-3CLMs of the bottom
FDMN layer into layered materials of a lesser rank. In case of infinite material contrast, this could
result in the propagation of singular effective properties through the FDMN. It is therefore preferable
to work with possibly ill-conditioned systems and to employ appropriate methods for the solution of
ill-conditioned systems [78]. Alternatively, multiple FDMNs can be trained and the ones with the best
online evaluation results can be selected. In this article, we follow the latter approach.
4 Application to rigid fibers suspended in polyamide 6
4.1 Material description and computational aspects
Let a fiber orientation state be defined by the fiber orientation distribution function ρ:S2→R, which
encodes the probability that fibers are oriented in the direction p∈ S2. In component scale molding
simulations, the orientation distribution function ρis often spatially inhomogeneous, and it is therefore
prohibitively expensive to compute the spatial and temporal evolution of the function ρ[79,80]. Therefore,
the evolution of surrogate quantities that contain some of the information encoded in the function ρis
often considered instead. One such quantity is the second order fiber orientation tensor [81,82]
N=ZS2
p⊗pρ(n) dS(n),(4.1)
which is commonly used in component scale molding simulations to track the evolution of the fiber
orientation state [79,80]. Because a major prospect of FDMNs is the combination of component scale
molding simulations with high fidelity information from full field viscous computations, we employ the
orientation tensor Nto describe the fiber orientation state of the considered microstructures. The
tensor Nmay be parametrized by its two largest eigenvalues λ1and λ2, as well as a proper orthogonal
tensor Rvia [83,84]
N=Rdiag(λ1, λ2,1−λ1−λ2)RT, λ1≥λ2.(4.2)
By objectivity [51], and because the tensor Nhas unit trace and is positive semi-definite, every fiber
orientation tensor Nrepresents a point in the fiber orientation triangle ST[55,84]
ST=λ= (λ1, λ2)T
1≥λ1≥1
3and min (λ1,1−λ1)≥λ2≥1−λ1
2.(4.3)
19
31 study points
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
λ1
0.0
0.1
0.2
0.3
0.4
0.5
λ2
(a) Fiber orientation triangle ST
100101102103104105106
0
100
200
300
˙γin s−1
ηin Pa s
Data Fit Last data point
100101102103104105106
0
100
200
300
˙γin s−1
ηin Pa s
Data Fit Last data point
(b) Cross-type fit
Figure 5: Fiber orientation triangle STin CMYK coloring with 31 evaluation points (a), and material
data with Cross-type fit for Ultramid®B3K (b).
For detailed discussions regarding the description of fiber orientation states, we refer the reader to
Kanatani [81] and Bauer et al. [84, 85]. To generate the training data and the online validation data
for the FDMN, we use FFT-based computational procedures to compute the effective viscous response of
fiber suspensions [17, 38]. We generate the required fiber suspension microstructures using the sequential
addition and migration method [86] for 31 orientation states λin the fiber orientation triangle ST, see
Figure 7. For the visualization of the orientation triangle ST, we follow Köbler et al. [55], and use a
CMYK coloring scheme. To keep the computational effort for computing the effective viscous responses
feasible, we restrict to microstructures with a fiber volume fraction of 25%, and assign all fibers the
length ℓand the diameter d. Also, we set the aspect ratio ra=ℓ/d of all fibers to 10, and use the results
of Sterr et al. [17] regarding the required resolution and size of the microstructure volume elements. Con-
sequently, we choose a resolution of 15 voxels per fiber diameter, as well as cubic volume elements with
edge size L= 2.2ℓ. Regarding the matrix material, we build on the investigations in Sterr et al. [17,51],
and prescribe a commercially available polyamide 6 [87] as the matrix material. We fit a Cross-type
constitutive equation
η( ˙γ) = η∞+η0−η∞
1+(k˙γ)m,(4.4)
to the available material data in the interval [1.7,16300] s−1at a temperature of 250◦C, and collect the
resulting model parameters in Table 1. For all non-Newtonian considerations, we investigate the six load
η0η∞k m
288.9 Pa s 15.0 Pa s 10.9·10−4s1.1
Table 1: Parameters of the Cross-type constitutive equation (4.4) for a commercially available polyamide
6 [87].
20
cases collected in the matrix ¯
Din Mandel notation
¯
D= ˙γr2
3
1−1
2−1
20 0 0
−1
21−1
20 0 0
−1
2−1
21 0 0 0
0 0 0 q3
20 0
0 0 0 0 q3
20
0 0 0 0 0 q3
2
,(4.5)
or each macroscopic scalar shear rate ˙γin the set of studied shear rates S˙γ, such that
˙γ∈S˙γ=a·10bs−1|a= 1,2,5; b= 1,2,3,4∪105s−1.(4.6)
Consequently, the set ¯
D˙γof all investigated load cases is defined by the equations (4.5) and (4.6). Be-
cause the viscous stress inside the rigid fibers is, constitutively, not well-defined [17], we employ a dual
formulation of the associated homogenization problem for the FFT-based computations. We discretize
the microstructures on a staggered grid [88], and solve the resulting equation systems with the conjugate
gradient method (CG) for linear computations, and with a Newton-CG approach for non-linear computa-
tions. With the goal of training an FDMN for each of the 31 fiber orientation states shown in Figure 5(a),
we generate 32 sample viscosities with the procedure described in section 3.3, and compute the corre-
sponding effective viscosities. To compute a single effective viscosity, five FFT-based computations are
necessary [38], leading to a total of 4960 computations, and 992 computed effective viscosities. In the
previous sections 2.3 and 3.2, we studied the linear homogenization functions of CLMs, and presented
the FDMN architecture to treat infinite material contrast and incompressible materials. In the following,
we discuss the material sampling for the offline training, the offline training procedure, and the online
evaluation of FDMNs for suspensions of rigid particles.
4.2 Offline training
With the linear training data at hand, we wish to train FDMNs for the prediction of the non-Newtonian
viscous behavior of shear-thinning fiber polymer suspensions. We follow previous work [46,54] and choose
the depth of the FDMN as K= 8 to achieve sufficient prediction quality for non-linear computations.
An FDMN of depth K= 8 with CLMs of rank R= 3 has 512 weights and 638 angles as free parameters,
as per equations (3.6) and (3.20). For each microstructure, we define the training data SDas
SD={(Vs
1,VF,¯
Vs)|s∈(1, ..., 32)},(4.7)
where Vs
1denotes the sample viscosities generated with equation (3.7), VFdenotes the infinite viscosity
of the rigid fibers, and ¯
Vsstands for the effective viscosity of the sample s. For each microstructure, we
split the training data SDinto the training set St⊂ SDand the validation set Sv⊂ SD, which consist of
90% and 10% of the total training data, respectively. The training set Stand the validation set Svshare
no samples, i.e., St∩ Sv=∅.
As an initial guess for the FDMN parameters, we uniformly sample all angles αand weights wfrom their
respective intervals, and rescale the weights wsuch that they sum to unity. Because we use a relatively
small set of 32 samples as training data, we dedicate a large portion of it to the training set St, and choose
to train 20 FDMNs per fiber orientation state λ. Furthermore, we train the FDMNs on mini batches with
size Nb= 8, for which we draw randomly from the training set St. In case the last batch is smaller than
eight samples, we drop the batch. Because the initialization of the parameter vectors αand w, as well
21
as the offline training process are random, the FDMNs differ in their parameters and quality of fit. By
training 20 FDMNs per fiber orientation state, we leverage this randomness with the goal of obtaining
FDMNs with a high quality of fit. This strategy aims to reduce the total computational effort required to
obtain sufficiently accurate FDMNs, because repeatedly training FDMNs on 32 samples requires less com-
putational resources than conducting full field simulations for larger sample sizes. For the minimization of
the loss function L, we choose the RAdam algorithm [77] and use a learning rate sweep [89] to determine
the learning rates κ0,α and κ0,w. With the learning rate sweep we obtain highly similar learning rates for
both parameter groups αand w, such that κ0,α =κ0,w = 1 ·10−2. As the learning process advances, we
use PyTorch’s StepLR learning rate scheduler to improve convergence towards minima [90,91]. To do so,
we multiply the learning rates κ0,α and κ0,w with a constant factor fLR = 0.75 every 150 epochs. Other
than the learning rates, we used standard hyperparameters for the two momentum coefficients β1= 0.9
and β= 0.999, as well as the stabilization constant ε= 10−8. Overall, we train every FDMNs for a total
of 2000 epochs each.
0 500 1,000 1,500 2,000
10−1
100
101
102
epochs
Error in %
min(emean
t) or max(emean
t) avg(emean
t)
(a) Training set St
0 500 1,000 1,500 2,000
10−1
100
101
102
epochs
Error in %
min(emean
v) or max(emean
v) avg(emean
v)
(b) Validation set Sv
Figure 6: Minimum, maximum, and average mean error emean of all trained FDMNs for the training
set St(a) and the validation set Sv(b).
To measure the quality of fit, we define the mean error emean over a subset Sof the training data SDas
emean =1
|S| X
(Vs
1,VF,¯
Vs)∈ S
∥DMNL(Vs
1,VF, α, w)−¯
Vs∥1
∥¯
Vs∥1
,(4.8)
where |S| is the cardinality of the set S. Depending on the fiber orientation state λand the initial
guesses of the angles αand the weights w, the quality of fit varies per DMN. Therefore, for the whole
training process of all 620 FDMNs, we visualize the smallest mean error min(emean), the largest mean
error max(emean), and the average mean error avg(emean)for the training and validation sets Stand Sv
per epoch in Figure 6. For the training and the validation set, we observe that the three considered error
measures drop rapidly at the beginning of the training, and continue to improve as the training continues.
As the learning rates are small for epochs larger than 1500, the errors do not change significantly at the
end of the training, and convergence is ensured. Evidently, the largest mean error max(emean
t)on the
training set Stdrops below 1% at the end of the training process, while the smallest mean error min(emean
t)
falls below 0.3%. This indicates a high quality of fit on the training set Stfor all FDMNs. Compared
to the training set St, the spread between the largest mean error max(emean
v)and the smallest mean
error min(emean
v)for the validation set Svis larger. However, the largest mean error max(emean
v)stays
22
below 2% for all considered FDMNs and does not fluctuate by a large margin. Additionally, the average
mean error avg(emean
v)decreases continuously during training. Consequently, the prediction quality on
both the training and validation set improve on average as the training progresses, although the largest
mean validation error max(emean
v)does not improve significantly after 1250 training epochs. Overall, this
indicates that no pronounced overfitting to the training data set Stoccurs.
4.3 Online evaluation
To measure the online performance of an FDMN, we define the online error function eon :Sym0(3)→R
as
eon(¯
D) = ∥σDM N (¯
D)−σFFT(¯
D)∥2
∥σFFT(¯
D)∥2
,(4.9)
where ∥·∥2denotes the ℓ2-norm of the components in Mandel notation, ¯
Dstands for the prescribed
effective strain rate tensor, and σDM N :Sym0(3)→Sym0(3)and σFFT :Sym0(3)→Sym0(3)are the
stress functions of the FDMN and the FFT-based homogenization, respectively. We then choose the
largest error over all load cases emax
on and the mean error over all load cases emean
on
emax
on = max
¯
D∈¯
D˙γ
eon, emean
on =1
|¯
D˙γ|X
¯
D∈¯
D˙γ
eon(¯
D),(4.10)
to study the performance of the different FDMNs. For each investigated orientation state λ, we identify
the FDMN with the smallest maximum error emax
on and visualize the associated errors emean
on and emax
on over
the fiber orientation triangle, see Figures 7(a) and 7(b)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
λ1
0.0
0.1
0.2
0.3
0.4
0.5
λ2
1.16%
1.32%
1.49%
1.65%
1.82%
(a) Mean error emean
on
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
λ1
0.0
0.1
0.2
0.3
0.4
0.5
λ2
2.88%
3.23%
3.59%
3.95%
4.31%
(b) Largest error emax
on
Figure 7: Mean online error emean
on (a) and largest online error emax
on (b) for the best FDMNs over the fiber
orientation triangle.
The maximum error emax
on of the best FDMNs range between 2.88% for the orientation
state λ= (0.4444,0.3611)T, and 4.31% for the fiber orientation state λ= (0.8055,0.1389)T. Thus, for the
investigated orientation states, the FDMNs yield sufficient prediction accuracy in terms of engineering
requirements over a wide range of shear rates ˙γ∈[10,105] s−1. Additionally, the relatively low maximum
errors emax
on show that with fewer samples than in previous studies for solid materials, for example by
Liu et al. [52, 53] or Gajek et al. [46, 54,59], appropriately accurate FDMNs can be produced for sus-
pensions of rigid fibers. Generally, the maximum error emax
on is larger for more strongly aligned oriented
orientation states towards the lower right-hand side of the fiber orientation triangle than for less strongly
23
aligned oriented orientation states towards the upper and the left-hand side. The mean error emean
on
ranges between 1.16% for the orientation state λ= (0.8334,0.0833)T, and 1.82% for the orientation
state λ= (0.6528,0.3194)T, underlining the prediction accuracy of the FDMNs.
To discuss one specific example, we take a more detailed look at the performance of the best FDMN at
the fiber orientation state λ= (0.8055,0.1389)Twhere the largest maximum error emax
on = 4.31% occurs.
Let the components of the strain rate tensor ˜
Di∈Sym0(3), in the standard basis {ei}and in Mandel
notation, be given by the i-th column of the matrix ¯
D/˙γ, see equation (4.5). Then, the tensor ˜
Di
represents a load with unit norm, and we refer to the tensor ˜
Dias load case. In the following, we discuss
the observed stress responses and the prediction quality of the FDMN for all considered shear rates (4.6)
and the six load cases defined in equation (4.5), see Figure 8. Because of the strong alignment of the
fibers in the coordinate direction e1, the stress norms are the largest for the load case ˜
D1, which encodes
incompressible elongational flow in the coordinate direction e1, see Figure 8(a).
FFT DMN
˜
D1˜
D2˜
D3˜
D4˜
D5˜
D6
101102103104105
10−2
10−1
100
101
102
˙γin s−1
¯
σ(¯
D)
2in MPa
(a) ℓ2-norm of the effective stress response ¯
σ(¯
D)
101102103104105
0
1
2
3
4
5
˙γin s−1
eon(¯
D) in %
(b) Online error eon(¯
D)
Figure 8: ℓ2-norm of the effective stress response ¯
σ(¯
D)over the shear rate ˙γ∈[10,105] s−1at the ori-
entation state λ= (0.8055,0.1389) as computed with FFT-based homogenization and the best identified
FDMN (a). Online error eon(¯
D)of the FDMN over the shear rate ˙γ∈[10,105] s−1(b).
The norms of the stress responses to elongational flow in the coordinate directions e2and e3, i.e., the load
cases ˜
D2and ˜
D3, are comparatively small. For the largest considered shear rate of ˙γ= 105s−1, we observe
a stress norm of 21 MPa for the load case ˜
D1, as well as 11.8 MPa and 12.1 MPa for the load cases ˜
D2
and ˜
D3, respectively. Because of the similar degrees of fiber alignment in the coordinate directions e2
and e3, the stress norms associated with the load cases ˜
D2and ˜
D3differ only slightly. Furthermore,
the stress norms in response to shear in the load cases ˜
D4and ˜
D5are smaller than to shear in the load
cases ˜
D6. Again, this is caused by the strong alignment of the fibers in the coordinate direction e1, which
increases the flow resistance stronger in the e1-e2-plane than in other shear planes [51]. For the largest
considered shear rate of ˙γ= 105s−1, the stress norms are 5.5 MPa, 6.0 MPa, and 8.4 MPa for the load
cases ˜
D4,˜
D5, and ˜
D6, respectively. Overall, the predicted ℓ2-norm of the effective stress response ¯
σDM N
of the FDMN agrees closely with the effective stress ¯
σFFT computed via FFT-based homogenization for
all investigated load directions ˜
Di, i ∈ {1, ..., 6}, and shear rates ˙γ∈S˙γ, see Figure 8(a).
Not only the ℓ2-norm of the effective stress response is predicted well by the FDMN, but the direction
as well. This is evident from the relatively small online errors eon(¯
D)shown in 8(b). Depending on the
load case, the error eon increases up to a point in the interval [102,103] s−1, before decreasing again for
24
higher shear rates. The largest error occurs for the elongational load case ˜
D3with 4.31%, and is closely
followed by 4.08% for the elongational load case ˜
D1. For the shear load cases ˜
D4,˜
D5, and ˜
D6, the
largest errors are 3.05%, 2.78%, and 3.23%, respectively, while the largest error for the elongational load
case ˜
D2is 2.2%. The largest error for the shear load case ˜
D6occurs at a shear rate of ˙γ= 102s−1,
whereas the largest errors for all other load cases occur at a shear rate of ˙γ= 5 ·102s−1. This is
directly tied to the Cross-type constitutive equation (4.4), which has two Newtonian plateaus for low
and high shear rates and a non-linear transition in between. Consequently, for large shear rates ˙γ→ ∞
the Matrix behavior is Newtonian and the homogenization function of the microstructure approaches the
linear homogenization function. Since the FDMN approximates the non-linear homogenization function
the microstructure to first order [54], the prediction quality depends on the degree to which the matrix
behaves non-linearly. This is in line with previous observations regarding the modelling of shear-thinning
fiber suspensions [17, 51].
4.4 Computational Speedup
Sampling total Sampling per orientation (mean) Training
CPU Hours 24415 788 0.08
Wall-clock time (h) 305 9.8 0.08
Table 2: CPU hours and wall-clock time for the training of one FDMN and the sampling of effective
viscosities ¯
Vsusing FFT-based homogenization.
FFT (h) DMN (ms) Speedup
Min 2.08 600 11785
Mean 2.61 630 14828
Max 3.07 650 17225
Table 3: Speedups and wall-clock times for the evaluation of a single load case using FFT-based homog-
enization compared to an FDMN.
For sampling the linear homogenization functions of the 31 considered fiber orientation states λ∈ST,
we used a workstation with two AMD EPYC 7552 48-Core processors and 1024 GB DRAM. We relied
on a single thread per FFT-based computation and ran 80 computations in parallel. In total, the com-
putations took 24415 CPU hours and a wall-clock time of 305 hours, see Table 2. Averaging over all 31
considered orientations λ, this leads to 788 CPU Hours and 9.8h wall-clock time per orientation. For the
training and the online evaluation of the FDMNs, as well as the non-linear FFT-based computations,
we used a workstation with two AMD EPYC 9534 64-Core Processors and 1024 GB DRAM, and ran
all computations on a single thread. The wall-clock time to evaluate all six load cases ˜
Di, i = 1, ..., 6,
and thirteen shear rates ˙γ∈S˙γin series for a single fiber orientation state λis listed in Table 3. The
wall-clock times for the FFT-based computations range from 2.08h to 3.07h with the mean wall-clock
time over all considered orientation states λbeing 2.61h. In contrast, the online evaluation of the FDMNs
takes between 600ms and 630ms, with a mean of 650ms. Consequently, the speedup factors range between
11785 and 17225 with a mean of 14828. This considerable speedup is achieved by investing computational
resources into the sampling of the linear homogenization functions of the microstructures. It is straight-
forward to judge whether this investment is sensible for the considered setup. For one microstructure the
25
generation of training data and the training of a single FDMN takes 9.88h wall-clock time on average,
while the non-linear FFT-based computations take 2.61h wall-clock time on average. Therefore, assuming
the non-linear computations run on one thread as in the presented setup, it would take four non-linear
computation to offset the initial investment into an FDMN on average. Given that component scale
simulations routinely require thousands or millions of microscale computations [46,59], the sampling and
training effort is offset easily in engineering problems.
4.5 Comparison with machine learning aided analytical models
In this section, we compare the FDMN based approach to predicting the effective behavior of fiber sus-
pensions with another machine learning approach suggested by Sterr et al. [51]. The authors presented
four different analytical constitutive equations for the effective viscosity of shear-thinning fiber suspen-
sions, and identified the model parameters using supervised machine learning. The model parameters
were learned from simulation data obtained with FFT-based computational techniques for the same
material as considered in this article: rigid fibers suspended in a Cross-type matrix material with pa-
rameters as shown in Table 1. However, the FDMN approach and the analytical approach follow very
different paradigms. The FDMNs are trained to approximate the non-linear homogenization function of
the microstructure by learning the respective linear homogenization function without prior knowledge
of the actual constitutive equations. In contrast, the parameters of the analytical models were learned
from the non-linear stress response of the suspension, and the analytical models incorporate knowledge
about the expected material behavior. Even though the two approaches follow different strategies, they
achieve comparable prediction accuracy. For the load cases ¯
D˙γand fiber orientation states λconsidered
in this article, the largest validation error of the FDMNs is 4.31% and occurs for the fiber orientation
state λ= (0.8055,0.1389)T. For the same load cases ¯
D˙γand the same set of fiber orientation states λ,
three of the four analytical models by Sterr et al. [51] achieve a maximum prediction error of 5.00%, which
occurs for the fiber orientation state λ= (0.8334,0.0833)T. The fourth model did not compare favorably
with the other three, because the built-in assumptions of stress-strain rate superposition and orientation
averaging did not hold well for the considered suspension [51]. Consequently, the FDMNs achieved a
slightly higher prediction accuracy for the type of fiber suspension considered in this article. Also, within
the constraints of a first order approximation, FDMNs generalize to different constitutive equations while
the presented analytical models do not, reducing the required modelling effort for FDMNs. However, the
FDMNs utilized in this article use 1150 free parameters, whereas the analytical models use between 11
and 49 parameters.
In terms of computational cost, generating the training data and training an FDMN for a single mi-
crostructure took 9.88h wall-clock time on average, see Table 2. The parameters of the analytical mod-
els for a single microstructure were identified based on six non-linear FFT-based simulations, which
took 2.61h wall-clock time on average. This leads to an average total wall-clock time of 15.70h if all
required simulations are run in series on one thread, and 0.04 h are allocated to identify the model pa-
rameters. However, with the considered setup, the non-linear FFT-based simulations could be run in
parallel, such that the analytical model parameters can be obtained more quickly than an FDMN. In
summary, FDMNs offer a higher degree of accuracy and flexibility regarding the considered constitutive
equations, but require more computational resources and a larger amount of parameters. Furthermore,
FDMNs are inherently thermodynamically consistent [54, §3.1], and inherit stress-strain rate monotonic-
ity from their phases [54, App. C]. In contrast, both thermodynamic consistency and stress-strain rate
monotonicity must be ensured manually in constitutive models.
26
5 Conclusions
In this article, we extended the direct DMN architecture to the Flexible DMN (FDMN) architecture for
the treatment of fiber suspensions with infinite material contrast and shear-thinning matrix behavior.
To do so, we derived linear homogenization functions for two-phase layered emulsions that are governed
by Stokes flow and consist of linearly viscous phases. More specifically, we utilized results by Kabel et
al. [70] and Milton [69] on the homogenization of laminates to derive closed form analytical expressions
for the effective properties of such layered emulsions. Because rank-one layered materials are ill-suited
as DMN building blocks in case of infinite material contrast, we investigated the effective properties of
coated layered materials (CLMs). We leveraged the linear homogenization functions of rank-one layered
materials and investigated under which conditions the effective behavior of CLMs is non-singular if the
core material is rigid. The conditions depend on the physical constraints of the employed materials, and
involve the rank of the CLM and the relative orientation of the layering directions. In the relevant case
for incompressible fiber suspensions, a CLM consists of an incompressible coating material and a rigid
core material. Then, the effective viscosity of a rank-3CLM is non-singular if the three layering directions
are mutually non-orthogonal and mutually non-collinear.
Using the derived homogenization functions for layered materials and CLMs, we extended the (direct)
DMN architecture to the FDMN architecture by replacing the lowest layer of rank-one laminates in a
(direct) DMN with non-singular CLMs, and the other rank-one laminates with rank-one layered materials
capable of treating fluids. For the offline training of an FDMN, we presented a strategy where the relative
angles of the CLM layering directions are fixed to reduce the amount of free parameters and guarantee
non-singular CLMs. Furthermore, we modified the online evaluation strategy of direct DMNs to account
for incompressible phases. We leveraged the FDMN architecture to predict the non-linear effective be-
havior of fiber suspensions with a Cross-type matrix material. Compared to direct numerical simulations
with FFT-based computational techniques, the FDMNs achieved validation errors below 4.31% for a
variety of 31 fiber orientation states, six different load cases, and a wide range of shear rates relevant to
engineering processes. If the time required to generate the training data and train the FDMNs is not
considered, the FDMNs achieved an average speed up factor of 14828 as compared to FFT-based sim-
ulations. Additionally, FDMNs achieve higher accuracy than another machine learning based approach
by Sterr et al. [51] for the same composite material. However, this competitive accuracy improvement
comes at the cost of an increased computational effort to obtain the training data for the FDMNs.
In future work, the application and extensions of the FDMN architecture to problems involving more
than two phases, where multiple phases may be singular, could be explored. This could enable the FDMN
based treatment of three-phase systems containing a fluid, fibers, and gas, which occur in engineering
processes like injection molding [92] and flotation de-inking [93]. Furthermore, problems involving solids
with defects could be explored as well. To account for the locally varying microstructure in component
scale simulations, fiber orientation interpolation schemes [46,57,63] could be combined with the FDMN
architecture to increase versatility, and enable new concurrent two-scale simulations involving defects and
rigid inclusions.
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.
27
Data availability
The data that support the findings of this study are available from the corresponding author upon
reasonable request.
Acknowledgements
The research documented in this manuscript was funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation), project number 255730231, within the International Research Train-
ing Group “Integrated engineering of continuous-discontinuous long fiber-reinforced polymer structures“
(GRK 2078/2). The support by the German Research Foundation (DFG) is gratefully acknowledged.
Author contributions
The present study was conceptualized by B. Sterr, M. Schneider and T. Böhlke. The presented FDMN
architecture was derived by B. Sterr, S. Gajek, M. Schneider, and T. Böhlke. B. Sterr, S. Gajek, and
M. Schneider implemented and validated the software. B. Sterr performed the simulations, analyzed
and visualized the data, and drafted the manuscript. The original manuscript draft was extensively
reviewed and edited by B. Sterr, A. Hrymak, M. Schneider, and T. Böhlke. Resources were provided
by M. Schneider and T. Böhlke. The research project was supervised by A. Hrymak, M. Schneider and
T. Böhlke.
28
Appendix
A Coated layered materials with singular core
A.1 Singularity condition for coated layered materials with singular core
In the particular cases that the core material m1of a CLM is rigid or represents a void, the dual or
the primal material properties K1or M1approach zero, respectively. Thus, more generally, we let the
material tensor A1approach zero, and rewrite equation (2.42) as
lim
A1→0
¯
A=A2
I+ (1 −f2) −I+f2
R
X
r=1
cr
L
A2(nr)A2!−1
.(5.1)
In the following, we study under which conditions the effective material properties ¯
Aof a rank-RCLM are
singular. The effective material properties ¯
Aare singular precisely if there exists an effective tensor ¯
g∈
{Sym(3),Sym0(3)}, such that
¯
A[¯
g] = 0.(5.2)
Using the definition (5.1), equation (5.2) may be equivalently rewritten in the form
R
X
r=1
cr
L
A2(nr)A2![¯
g] = ¯
g.(5.3)
Equation (5.3) is satisfied precisely if ¯
glies in the intersection of all subspaces EA
r, such that
¯
g∈
R
\
r=1 EA
r,(5.4)
or, equivalently,
L
I(nr)[¯
g] = ¯
g,∀r∈ {1, ..., R}.(5.5)
We prove this as follows. First, if ¯
glies in the section of all subspaces such that (5.4) is satisfied, it
follows from equation (5.5) and the definition of the operator
L
A2(nr)(2.41) that
R
X
r=1
cr
L
A2(nr)A2![¯
g] = R
X
r=1
cr
L
A2(nr)A2
L
I(nr)[¯
g]!=
R
X
r=1
cr¯
g=¯
g,(5.6)
and hence equation (5.3) is satisfied. Second we wish to prove that the converse is also true, i.e., if
equation (5.3) is satisfied, equation (5.4) is satisfied as well. Let TA2denote a convex combination of the
projectors
L
A2(nr)A2, i.e.,
TA2= R
X
r=1
cr
L
A2(nr)A2!,(5.7)
with
cr>0and
R
X
r=1
cr= 1.(5.8)
Then, fixed points of the operator TA2lie in the set [94, Lemma 1.4]
E=
R
\
r=1 EA
r.(5.9)
29
Therefore, any eigenvector of the operator TA2corresponding to the eigenvalue one must lie in the
intersection of the subspaces EA
r, which we denote with E. In particular, if equation (5.3) is satisfied, the
element ¯
glies in E, i.e.,
TA2[¯
g] = ¯
g,=⇒¯
g∈ E (5.10)
and equation (5.4) is satisfied as well.
With the result (5.10) at hand, we observe that a necessary and sufficient condition for the regularity of
a rank-R CLM is that the space Eis trivial, i.e.,
E={0}.(5.11)
We refer to equation (5.11) as non-singularity condition in the following. In the context of linear elastic
and linearly viscous materials, we derived the non-singularity condition (5.11) for the spaces EA
r, such
that
EA
r⊂Sym(3),∀r∈ {1, ..., R}or EA
r⊂Sym0(3),∀r∈ {1, ..., R}.(5.12)
However, the presented proof extends to other problems that can be formulated in the form of equa-
tion (5.1), such as thermoelastic or piecoelectricity problems [69, §9]. With the goal of using CLMs as
building blocks in a DMN architecture, we are interested in whether there are particular choices of the
layering directions nrand the rank Rof the CLM for which the effective properties of a CLM are always
non-singular, i.e., the non-singularity condition (5.11) is satisfied. For incompressible suspensions of rigid
fibers, we study this question for CLMs with a rigid core material m1and an incompressible coating
material m2.
A.2 Coated layered materials with incompressible coating and rigid core
For applications involving rigid inclusions in incompressible media, we consider CLMs with an incom-
pressible coating material m2and a rigid core material m1. Therefore, the spaces EA
r⊂Sym0(3)have the
form
EA
r={B∈Sym0(3)|B·(nr⊗sa)=0,∀a∈R3},(5.13)
with the dimensions of the spaces EA
rand Sym0(3)given by
dim(EA
r)=3,and dim(Sym0(3)) = 5.(5.14)
Because the dimension dim(EA
1+EA
2)is bounded by the dimension of the space Sym0(3), such that
dim(EA
1+EA
2)≤5,(5.15)
it follows from the dimension of the intersection of the two spaces EA
1and EA
2
dim(EA
1∩ EA
2) = dim(EA
1) + dim(EA
2)−dim(EA
1+EA
2),(5.16)
that
dim(EA
1∩ EA
2)≥3+3−5=1.(5.17)
In other words, the intersection of the spaces EA
1and EA
2is at least one dimensional. Consequently,
two layering directions are not sufficient to satisfy the condition (5.11) Besides the required number of
layerings, we are also interested in whether there are restrictions on the angles between the layering
directions. Suppose that one layering direction is orthogonal to the other two. We discuss the case that
the direction n3is orthogonal to the normals n1and n2. The other cases work similarly via permuting
the indices. Then it holds that
n3·n1= 0,and n3·n2= 0,(5.18)
30
and the intersection EA
1∩ EA
2∩ EA
3does not only contain the zero element 0. Hence, the condition (5.11)
is violated. To show this, we consider the alternative description of the spaces EA
r
EA
r={B∈Sym0(3)|Bnr=αnrfor some α∈R}(5.19)
which is equivalent to equation (5.13) and characterizes the space EA
rvia all tensors B∈ EA
rthat have
the layering direction nras an eigenvector. Then the tensor
B=n1⊗n1+m2⊗m2−2n3⊗n3,(5.20)
where the layering direction m2is constructed by orthogonalization, such that
m2=n2−(n1·n2)n1,(5.21)
is an element of the intersection EA
1∩ EA
2∩ EA
3. This is true because it holds that
tr(B)=0,Bn1=n1,Bn2=n2,Bn3=n3,(5.22)
and therefore condition (5.11) is violated. In other words, the effective properties ¯
Aof a rank-3CLM
with incompressible coating and rigid core, where one layering direction n3is orthogonal to the other
two layering directions n1and n2, is always singular. Particularly, if all three layering directions are
mutually orthogonal, the effective properties ¯
Aof a rank-3CLM are singular.
In contrast, let the three layering directions be mutually non-orthogonal and mutually non-collinear, i.e.,
0<|n1·n2|<1,0<|n1·n3|<1,0<|n2·n3|<1.(5.23)
We consider an element Bin the intersection EA
1∩ EA
2∩ EA
3and wish to show that the tensor Bvan-
ishes, such that the non-singularity condition (5.11) is satisfied. Because the tensor Bis symmetric, its
eigenvectors corresponding to distinct eigenvalues are orthogonal. Thus, if the eigenvalues were distinct,
the layering directions would need to be orthogonal. However, this contradicts our assumption (5.23).
Therefore, the tensor Bhas a single eigenvalue αwith the corresponding eigenspace R3, i.e., the tensor B
attains the form
B=αI, α ∈R.(5.24)
Additionally, because the trace of the tensor Bvanishes, i.e., tr(B)=0, it follows that the eigenvalue α
is zero. Consequently, the tensor Bmust vanish, the non-singularity condition (5.11) is satisfied, and
the effective properties ¯
Aof a rank-3CLM with an incompressible coating material m2and a rigid
core material m1are non-singular, if the three layering directions are mutually non-orthogonal and non-
collinear.
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