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1

Modeling and simulation of viscoelastic solids2

under large numbers of loading cycles3

Darith Anthony Huna, Mohamed Haddada, Issam Doghria

4

aUniversit´e catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering, Louvain-La-Neuve5

B-1348, Belgium.6

ARTICLE HISTORY7

Compiled January 30, 20228

ABSTRACT9

An eﬃcient modeling procedure is proposed for viscoelastic (VE) solids subjected to large numbers of10

loading cycles. While the Laplace-Carson transform (LCT) is often used to solve VE creep or relaxation11

problems, the originality here is an eﬃcient extension of the approach to a plethora of cycles, based on some12

key ingredients. The time history of the cyclic loading is decomposed into transient and periodic signals,13

leading to two subproblems. Each one is transformed into a ﬁnite number of linear elastic analyses in the14

L-C domain. A method to choose the number and positioning of the L-C domain sampling points for each15

one of the two subproblems is detailed. Speciﬁc LCT inversion methods are applied to each subproblem16

in order to reconstruct the displacement, strain and stress ﬁelds in the time domain. For the transient17

subproblem, Schapery’s collocation method based on exponential basis functions is used, while a new18

LCT inversion method is proposed for the periodic subproblem based on sinusoidal basis functions and a19

Newton-Gauss algorithm. After a veriﬁcation on well-known 1D functions, the accuracy of the proposed20

method is assessed on two structural problems with large numbers of cycles. Comparison with reference21

ﬁnite element analyses conducted directly in the time domain shows that the proposed methodology22

provides excellent predictions, both at local scale (displacement, strain and stress components at various23

points) and macroscale (global energy indicator). The important speedup factor (e.g., 32 for 10k cycles)24

will increase signiﬁcantly with the number of cycles, enabling the proposed method to be extended to25

high cycle fatigue of thermoplastic polymer structures in future work.26

KEYWORDS27

Laplace-Carson numerical inversion, high cycle simulation, periodic basis extension, viscoelastic28

structure.29

1. Introduction30

Numerical simulation of high cycle fatigue for structures whose material behavior remains inelastic31

and dissipative through the cyclic history is an important challenge, in particular for thermoplastic32

polymer structures. The latter are increasingly used especially with the emergence and improvement33

of diﬀerent additive manufacturing technologies [1, 2], which allow the user to design the geometry34

and/or the mechanical performance of the structure. However, in the long-term and under repeated35

loading these structures weaken and may even fail. Proposing a model that can accurately predict36

these phenomena while taking into account the intrinsic behavior of the medium represents a great37

interest for mechanical engineering. In the literature, many constitutive models and numerical algorithms38

about the thermomechanical behavior of these thermoplastic polymer materials have been developed. A39

ViscoElastic-ViscoPlastic model (VE-VP) [3, 4] has been often selected because it allows a ﬂexibility of40

responses in both the ViscoElastic (VE) and ViscoPlastic (VP) regimes. Extended to the framework of41

large numbers of cycles, Haouala and Doghri [5] propose for single volume elements under imposed strain42

history a method of time homogenization allowing a strong gain of computation time (∼94%) for a small43

error on the stress response (≤4%). The natural extension of this method would be an adaptation to a44

structural problem. Indeed, a classical ﬁnite element solution would require a prohibitive computation45

CONTACT Issam Doghri. Email: issam.doghri@uclouvain.be

Mechanics of Advanced Materials and Structures (in press).

time due ﬁrst to the large numbers of loading cycles and also to the solvers used to numerically manage46

the material non-linearities.47

Figure 1.: Multiscale in space and time strategy proposed for structures subjected to large numbers of

cycles and made of heterogeneous VE-VP thermoplastic polymer materials (see [6]).

To overcome this issue, a multiscale strategy in space and time is proposed, which is illustrated in48

Fig. 1. As proposed by Krairi et al. in [6], high cycle fatigue of thermoplastic polymer structures can be49

explained at microlevel by supposing that each representative volume element is made of a VE matrix50

material containing a small volume fraction (∼2%) of VE-VP weak spots. We also consider a small (∼51

3%) porosity distribution due to manufacturing process (e.g., selective laser sintering, [1, 2]). Under the52

hypothesis of small deformations, the idea is then in the present work to postulate that the structure will53

follow globally a VE deformation throughout the cyclic loading and that the plastiﬁcation leading to the54

weakening of the structure occurs at microscale in some local regions, aﬀecting ﬁnally only marginally55

the macroscopic structural response. In order to implement and validate the proposed strategy, there56

are two major issues which need to be addressed, and each one represents an original contribution. The57

ﬁrst problem to overcome, located at the macroscopic scale, would be to propose a method to solve a58

structural VE problem at a low computational cost. Once the deformation history is obtained, the next59

step would be to couple this structural deformation history with the local time-homogenization method60

[5] applied on the real complex microstructure (modeled as VE matrix, porosity and weak spots) and to61

validate the whole multiscale strategy on a full reference structural computation. In this paper we focus62

on the ﬁrst problem, the study aiming to accelerate the simulation time of VE structures under large63

numbers of cycles.64

65

The cost of solving a VE time problem is dependent on the number of cycles and requires time66

integration schemes that must respect approximations [7] to converge to the solution. This cost becomes67

prohibitive for a problem with large numbers of cycles. In the literature, diﬀerent numerical methods allow68

to reduce the calculation time. An extended PGD (Proper General Decomposition) method is presented69

in [8] for a VE material, under creep or cyclic loadings. Basically this method deﬁnes an approximate70

solution constructed on an enriched basis function. In this application, the response is compared to a71

reference and allows a time saving of 5 and an error lower than 3%. The jump cycle method is used72

in [9] for a more general material (VE-VP-damage). This method works in two steps, the ﬁrst one73

consists in calculating a solution for a certain number of cycles and then extrapolating it temporally74

on a neighboring interval. The operation is then repeated until the complete history time interval is75

covered. In that work, the ﬁnal solution is then obtained by skipping 60% of the cycles for an error76

around 5%. In the present article, in order to reduce the computational time for VE structures subjected77

to large numbers of cycles, we use another method based on the Laplace-Carson transform (LCT) and78

its numerical inversion. We develop some key ingredients which enable to combine an important speedup79

factor (much higher than with existing methods) with a very accurate reconstruction of the strain and80

stress ﬁelds in realistic structural problems.81

2

The LCT of a function fis deﬁned as follows :82

Lp{f(t)}=f?(p) = pZ∞

0

f(t)e−ptdt, (1)

where t7→ f(t) is a scalar or tensorial function deﬁned in the original space and p7→ f?(p) is in the L-C83

space.84

Basically, this LCT allows to change an integral problem with a convolution product in the original space,85

into a simple product in the L-C space, facilitating the algebraic and/or numerical resolution. This LCT86

is often used in engineering, e.g, in the ﬁeld of electrical engineering [10, 11] or in mechanics of materials87

with VE behavior [12, 13, 14]; in these two ﬁelds the original space is time. This transformation calling88

the correspondence principle of Lee-Mandel [15] provides a faster solution in the L-C domain. However,89

the inversion of the response in the time domain (Bromwich formula) :90

L−1

t{f?(p)}=f(t).(2)

is not always applicable in practice. During this step the accuracy depends directly on the response91

function, which for simple cases (well-known algebraic functions) can be solved analytically. For more92

complex functions f, numerical approximation inversion methods are then employed. Among all these93

numerical methods, we can reference two main families :94

(1) General inversion algorithms : Abate and Whitt provide in [16] a large bibliography of these95

numerical methods which use general function bases, e.g. : Fourier series originally coming from96

[17, 18], Gaver’s functional [19, 20], or Bromwich’s integral [21].97

(2) Speciﬁc inversion algorithms : where the inversion basis is ”designed” by the shape of the fsolution98

known a priori. These methods are particularly adapted to the linear VE problem which gives by99

construction of a constitutive model and/or boundary conditions an indication on the shape of100

the responses. For creep/relaxation type mechanical responses, by ﬁxing certain parameters of101

their models [22, 23] proposed a direct inversion method under an exponential projection basis,102

particularly adapted to VE behavior where the time-evolution of material properties is described103

with decaying exponentials (Prony series). Indirect methods like [14, 24, 25], use optimization104

algorithms to reduce the error between the solution and the approximate function in the L-C105

domain. One of the advantages is a better accuracy compared to the direct method against an106

additional time cost due to the convergence of the optimizer.107

In both cases, these two families of algorithms continuously deﬁne the solution function fin the time108

domain on the basis of the discretized solution function f?in the L-C domain.109

The method proposed in this paper belongs to the second family and applies to VE structural mechanics110

problems in the framework of large numbers of loading cycles. The main focus and the originality with111

respect to the existing literature is the accurate reconstruction in time domain of all strain components in112

any Gauss point of the structural FE mesh. This is achieved thanks to some key ingredients : (i) structural113

VE problem decomposition into two subproblems, transient and periodic; (ii) use of exponential basis114

functions and Schapery’s collocation method for the transient subproblem; (iii) use of sinusoidal basis115

functions and a Gauss-Newton algorithm for the periodic subproblem; (iv) a rigorous procedure to116

determine the adequate number and the positioning of the sampling points in the L-C domain for117

each subproblem; (v) solving a limited number of structural elastic problems in the L-C domain; (vi)118

computationally eﬃcient and accurate reconstruction of displacement, strain and stress ﬁelds in the119

time domain. The paper is organized as follows: In Section 2, two LCT inversion methods are presented120

in a general setting and applied on well-known 1D periodic functions (mono and multi-harmonic). In121

Section 3, the proposed computational procedure for VE structures subjected to large numbers of loading122

cycles is developed. The methodology is applied on on two diﬀerent structures. A comparison between123

results and computational cost obtained by the proposed method and reference calculations in time124

domain is also shown. Conclusions are drawn is Section 4. Three appendices (A, B and C) deal with the125

inﬂuence of the number of sampling points, the sampling method and the computational time comparison.126

3

2. LCT inversion methods for VE problems127

In this section, we present two LCT inversion methods. The ﬁrst is Schapery’s collocation method used128

when the time function is a sum of exponentials. The shortcomings of the method are illustrated. The129

second method is proposed for periodic time functions expressed as a sum of sinusoids.130

2.1. Schapery’s collocation method131

The collocation method proposed by Schapery [22] for numerical LCT inversion is still often used in VE132

mechanical problems. In this framework the author deﬁned the solution fof the problem in time tas a133

series of exponentials such that :134

f(t) = A+Bt +

N

X

k=1

bk(1 −e−t/θk),(3)

This decomposition is parametrized by the set (bk,θk), A and B are shift parameters. This decomposition135

is adapted to represent both the classical phenomena of creep (θk<0) and relaxation (θk>0) related136

to a VE behavior.137

This decomposition in time of the fsolution has its analogous form f?in the L-C domain by successive138

transformation of each term such that :139

f?(p) = A+B

p+

N

X

k=1

bk

1

1 + pθk

.(4)

The A and B parameters are evaluated with the limit values in time or (in practice) in L-C domain as :140

A= lim

t→0f(t) = lim

p→+∞f?(p),(5)

141

B= lim

t→+∞

f(t)

t= lim

p→0pf?(p).(6)

The LCT solution function f?is then estimated on several discrete points pi(i∈ {1, ..., M},Mthe142

number of discrete points) allowing the identiﬁcation of the set (bk,θk) and the reconstruction of a143

reference function f(t) in the time domain. Based on the discretization of Eq. 4, and using the constraint144

:θi= 1/pi, the system is reduced to computing bkparameters only. The coeﬃcients are identiﬁed by145

solving the following linear system :146

b1

b2

b3

.

.

.

bN

=

1

1 + p1/p1

1

1 + p1/p2

1

1 + p1/p3

. . . 1

1 + p1/pN

1

1 + p2/p1

1

1 + p2/p2

1

1 + p2/p3

. . . 1

1 + p2/pN

1

1 + p3/p1

1

1 + p3/p2

1

1 + p3/p3

. . . 1

1 + p3/pN

.

.

..

.

..

.

.....

.

.

1

1 + pN/p1

1

1 + pN/p2

1

1 + pN/p3

. . . 1

1 + pN/pN

−1

f?(p1)−(A+B/p1)

f?(p2)−(A+B/p2)

f?(p3)−(A+B/p3)

.

.

.

f?(pN)−(A+B/pN)

.(7)

It should be noted that the order Nof the series and the number Mof discrete points piare dependent147

on each other (N=M) to ensure a well-posed system. The advantage of this method lies in a direct148

numerical solution and an easy implementation. Theoretically, once the system is solved and all param-149

eters bkare identiﬁed, f(t) can be reconstructed at any time. However, in practice, the accuracy of the150

reconstruction strongly depends on the number of sample points pi, as illustrated in Fig. 2.151

4

10-2 100102104

0.2

0.4

0.6

0.8

1

1.2

1.4

(a) Creep test.

10-2 100102104

0.6

0.8

1

1.2

1.4

1.6

1.8

(b) Relaxation test.

Figure 2.: Eﬀect of number of collocation points (under-sampling and over-sampling) on the accuracy of

Schapery’s method.

In this illustrative example the reference time function e

f(t) to be recovered is :152

e

f(t) = 1 + Ce−t/θ,(8)

representing the shape of the response under relaxation (C= 1, see Fig. 2b) or creep test (C=−1,153

see Fig. 2a) and θ= 0.02s, the relaxation time. In these illustrations, a low number of points (N =154

10) does not allow to capture enough information to represent the solution and an oversampling (N =155

100) increases the size of the system and its singularity, thus adding parasitic oscillations to the original156

solution in time, as reported in [13]. In practice and in this example, a short preliminary study was157

conducted and a choice of N= 25 points piequally positioned on a logarithmic scale allows to recover158

the solution correctly.159

2.2. Proposed LCT inversion method for sinusoidal functions160

Schapery used an exponential projection basis to approximately reconstruct a time function, particularly161

well adapted for creep and relaxation response. However, for a cyclic structural simulation, our goal is162

to accurately reconstruct the time histories of the strain components εij(x, t) at various positions xof163

a structure subjected to cyclic loading. In this context, it is more pertinent to consider a decomposition164

built on a sinusoidal basis, which is described in time as :165

f(t) =

N

X

k=1

aksin(ωkt+φk),(9)

with the set Ak= (ak,ωk,φk) of parameters to be identiﬁed, which are respectively the amplitude, the166

pulsation and the shift of the approached function. The analogous form in L-C domain of this projection167

basis is deﬁned as :168

f?(p) =

N

X

k=1 ak

p2

ω2

k+p2sin(φk) + ak

ωkp

ω2

k+p2cos(φk).(10)

The solution function f?is then estimated on several discrete points piand an algorithm allows the169

identiﬁcation of the set Akand the reconstruction in the time domain.170

For this example, an optimization algorithm was selected, as in [14, 24, 25], namely the Gauss-Newton171

algorithm. This choice was motivated by the use of a tangent operator (additional constraint on the ﬁrst172

5

derivatives) which can be deﬁned analytically on the basis of projection in the L-C domain and allowing173

the convergence as well as the elimination of spurious higher modes generated by oversampling. It is an174

iterative method, well-known and general. The algorithm is explained and applied in the present case on175

the periodic projection basis and the (3xN)-parameters are identiﬁed using this following vector form :176

a

ω

φ

new

(3×N,1)

=

a

ω

φ

old

(3×N,1)

+

∆a

∆ω

∆φ

(3×N,1)

,(11)

or :177

{A}new ={A}old +{∆A},(12)

where {A}=(a,ω,φ)Tis a (3xN) size vector containing respectively all the set of Akparameters, and178

∆A is the correction vector of the Newton-Gauss algorithm deﬁned classically as :179

{∆A}(3×N,1)=[−JTJ]−1JT(3×N,M)

{r}(M×1)(13)

where {r}is the residual error vector containing ri=e

f?(pi)−f?(pi,A) component, [J] is the (M,3xN)180

Jacobian matrix (not necessarily square) written as :181

[J](M,3xN)=[Ja](M,N),[Jω](M,N ),[Jφ](M,N )(14)

and deﬁned as the derivative of the residuum error vector and parameters182

[Ja]ik =−∂f ?

i

∂ak

,[Jω]ik =−∂f ?

i

∂ωk

,[Jφ]ik =−∂f ?

i

∂φk

.(15)

Where f?

i=f?(pi;A) is the discretized f?with Acoeﬃcients to identiﬁed and e

f?the function to183

recover. It should be noted that on the basis of the sinusoidal decomposition used, the system requires184

a discrete number of points M≤3xN, the oversampling does not aﬀect the accuracy of the projected185

solution.186

187

A ﬁrst illustration of this LCT inversion method is proposed on a unimodal periodic signal (N=1) in188

Fig. 3b, where the reference solution to be recovered (in black curve) is written as :189

e

f(t) = asin(ωt +φ) (16)

and its LCT analogous form as :190

e

f?(p) = ap2

ω2+p2sin(φ) + aωp

ω2+p2cos(φ),(17)

with (a= 1, ω= 2, φ=π/6).191

This reference function e

f(in black) is compared ﬁrst to Schapery’s collocation method (in blue),192

where the numerical approached solution and the exponential basis used have been described in Eq. 3193

& 4, with N = 60 discrete points (pi), and then the proposed method (in red), where the reconstructed194

solution is approached by a sinus basis and computed by an optimization algorithm described in Eq. 13195

& 14, with the starting coeﬃcients (a(0)

1= 0.8×a,ω(0)

1= 1 ×ω,φ(0)

1= 2 ×φ), for N = 30 discrete points.196

Fig. 3b shows that in order to reconstruct the sinusoidal function of Eq. 16, the proposed method197

needs only 30 sampling points to yield a very accurate reconstruction. However, on the same example,198

Schapery’s collocation method needs twice as many points (60) and nevertheless loses accuracy after two199

cycles. An extended study of the accuracy of the two methods in the reconstruction of the sinusoidal200

6

10-4 10-2 10010210 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(a) Convergence of the residue and the iterative ap-

proached solutions associated with the reference solu-

tion in L-C domain.

012345678910

-1.5

-1

-0.5

0

0.5

1

1.5

(b) Comparison of two approximate reconstructions

with the reference solution in time domain.

Figure 3.: Accuracy of the proposed LCT inversion for a sinusoidal function.

reference solution depending on the number of sampling points is provided in Appendix A.201

With this parametrization, the iterative convergence of the LCT (see Eq. 10) with the Avector to be202

identiﬁed in L-C domain is shown in Fig. 3a. We have deﬁned the residuum indicator as :203

R= max(||{¯r}||),(18)

with ¯ri=ri/e

f?(pi). The accuracy of the Newton-Gauss algorithm is ensured by a ﬁxed tolerance tol204

(if R≤tol = 10−6). In this illustrative example ite = 4 iterations were enough to converge and we205

can observe that the decomposition on sinus basis function is more adapted to recover the reference206

sinusoidal solution function.207

208

The algorithm can also be applied to periodic signals of higher order, as in the following example,209

where the reference function to reconstruct is deﬁned as :210

e

f(t) =

3

X

k=1 eaksin(eωkt+e

φk),(19)

with (ea1,ea2,ea3) = (1,2,2), (eω1,eω2,eω3) = (2,15,3) and (e

φ1,e

φ2,e

φ3)=(π/6, π/4, π/3).211

The convergence of the algorithm is illustrated in Fig. 4a, which shows a negligible residuum after 7212

iterations. A comparison in time between the reference and the reconstructed function is illustrated in213

Fig. 4b. We observe that the superposed complex periodic signal (N=3) is well recovered.214

215

3. Computation of VE structures under large numbers of cycles216

In this section, we propose a computational procedure for VE structures subjected to large numbers of217

loading cycles.218

3.1. Problem statements in time and L-C domains219

We recall the equations associated with the problem in time and its equivalent in the L-C domain.220

Under the assumption of a quasi-static problem, without ageing, under small perturbations and in221

7

1234567

10-8

10-6

10-4

10-2

100

102

(a) Residuum of the Newton-Gauss algorithm to re-

construct the approached solution.

01234567

-5

-4

-3

-2

-1

0

1

2

3

4

5

(b) Comparison between the reference and recon-

structed function in time.

Figure 4.: Accuracy of the proposed LCT inversion for high order sinusoidal function (Eq. 19).

linear viscoelasticity, the evolution equations in time tare written as follows:222

223

•Field equations :224

P(t) =

∇.σ(x, t) + ρF(x, t)=0,

ε(x, t) = 1

2∇(u(x, t)) + ∇T(u(x, t)),

σ(x, t) = Zt

−∞

C(x, t −τ) : ˙ε(x, τ )dτ,

(20)

where C(t) is the fourth-order isotropic relaxation tensor :225

C(t)=2G(t)Idev + 3K(t)Ivol,(21)

the set (Idev,Ivol) are respectively the fourth-order deviatoric and volumetric projection tensors and226

(G, K) are the shear and bulk relaxation moduli expressed here as Prony series :227

G(t) = G∞+

I

X

i=1

Gie−t/gi,(22)

228

K(t) = K∞+

J

X

j=1

Kje−t/kj.(23)

Here (G∞, K∞) is the set of long-term moduli, (gi, kj) the set of relaxation times, and (Gi, Kj) the set229

of weights for the shear and bulk moduli.230

231

•Boundary conditions :232

B(t) =

∂Ωu∩∂ΩT=∅,∂Ωu∪∂ΩT=∂Ω,

ui(x, t) = ud

i(x, t), on ∂Ωu,

Ti(x, t) = Td

i(x, t), on ∂ΩT,

(24)

8

where ∂Ωuand ∂ΩTare respectively the regions where the boundary conditions in displacement ud

iand233

the stress vector Td

iare applied.234

235

The mechanical time problem can be transformed via the LCT into a linear elastic problem in the236

L-C domain (Correspondence principle [15, 26]), such as :237

238

•Field equations :239

P?(p) =

∇.σ?(x, p) + ρF?(x, p)=0,

ε?(x, p) = 1

2∇(u?(x, p)) + ∇T(u?(x, p)),

σ?(x, p) = C?(x, p) : ε?(x, p),

(25)

240

241

•Boundary conditions :242

B?(p) =

∂Ωu∩∂ΩT=∅,∂Ωu∪∂ΩT=∂Ω,

u?

i(x, p) = ud?

i(x, p), on ∂Ωu,

T?

i(x, p) = Td?

i(x, p), on ∂ΩT,

(26)

3.2. A ﬁrst structural application243

The structural application considered in this Section is illustrated in Fig. 5. The 2D plane strain conﬁg-244

uration (meshed with 10 344 linear triangular elements and 5356 nodes) has been chosen to accelerate245

the computation and obtain the direct reference solution under 10k cycles with a standard computer246

(6600U/i7, 2.60GHz/4CPU, 16GB/RAM). The reference solution computed in time domain will be247

compared to the solution reconstructed with the proposed method.248

Figure 5.: VE structure under cyclic loading, 3D geometry and 2D plane strain model.

3.2.1. Signal decomposition and split computational procedure249

In this application the system is loaded by a prescribed cyclic displacement (see the black curve in250

Fig. 6a), following the function :251

ud

1(x, t) = eud

1(x, t) + bud

1(x, t),∀x∈∂Ωu.(27)

9

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

(a) Loading functions decomposition in time domain.

10-4 10-2 10010210 4106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Loading functions split in L-C domain.

Figure 6.: Decomposition of the loading signal into Heaviside and periodic parts. (a) Time domain, (b)

L-C domain.

This decomposition of the loading signal is used to apply the superposition principle to cumulate the252

transient response due to the bud

1function (see the blue curves in Fig. 6a in time and in Fig. 6b in L-C253

domain), expressed as :254

bud

1(t) = b

A(1 −e−t/

b

θ),(28)

and the periodic response according to the sinus loading eud

i(see the red curves in Fig. 6a in time and in255

6b in L-C domain), such as :256

eud

1(t) = e

Asin(eωt +e

φ),(29)

where b

θ << eω(in mathematical sense), to approximate a Heaviside function by an exponential function257

in order to use a continuous function in L-C domain.258

259

Each subproblem (transient/periodic) needs a particular LCT inversion treatment :260

•transient subproblem will use the Schapery LCT inversion (see Sec. 2.1),261

•periodic subproblem will employ the proposed LCT inversion method based on the sinus basis262

decomposition (see Sec.2.2).263

In the next part we will describe step by step the proposed procedure (illustrated in Fig. 7) and264

compare the reconstructed solutions with the reference. The simulations are based on the parameters265

reported in Tab. 1 used in [27] for semi-crystalline polyamide polymer.266

The sampling points piin the L-C domain have to be chosen. Indeed they aﬀect directly the accuracy of267

the solution and their choice depends on the problem formulation and its parameters. A methodology is268

proposed here and as the cyclic problem is separated into two sub-problems, each case can have a diﬀerent269

discretization. For both, we have ﬁxed 30 sampling points distributed according to the evolution of each270

set of parameters : ( b

K?,b

G?,bu?) illustrated in Fig. 8a and ( e

K?,e

G?,eu?) in Fig. 8b, which are respectively271

attached to each sub-problem. The methodology here is to select some points : (p(1)

i, p(2)

i) which are272

suﬃcient to represent for each case the three evolutions simultaneously.273

The shear and bulk relaxation moduli in time deﬁned by Prony series in Eq. 22,23 are expressed274

10

Figure 7.: FE simulation strategy of VE structure under cyclic (displacement bd=udor force bd=Td)

loading. Left : direct time domain computation to obtain reference solution. Right : proposed procedure

in L-C domain followed by split numerical inversion for LCT.

VE parameters

Initial shear modulus G0= 1074 MPa Initial bulk modulus K0= 3222 MPa

Gi(MPa) gi(s) Kj(MPa) kj(s)

158 0.021 472 0.007

80 0.378 242 0.126

37 0.648 111 0.216

Loading parameters

bud

1b

A= 0.05 m b

θ= 0.01

eud

1e

A= 0.01 m eω= 4πrad/s e

φ= 0

Table 1.: VE problem parameters [27].

according to their LCTs as :275

G?(p) = G∞+

I

X

i=1

Gi

p

p+ 1/gi

.(30)

11

10-10 10-5 10010510 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) Transient problem.

10-5 100105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Periodic problem.

Figure 8.: Selection of the discretization points in L-C domain for (a) transient problem, (b) periodic

problem.

and276

K?(p) = K∞+

J

X

j=1

Kj

p

p+ 1/kj

,(31)

The LCT of the transient part (Eq. 28) of the displacement is :277

bu?

d(p) = b

Ap

p+ 1/b

θ,(32)

and the LCT of the periodic part (Eq. 29) is :278

eu?

d(p) = e

Ap2

eω2+p2sin e

φ+e

Apeω

eω2+p2cos e

φ, (33)

Fig. 8 shows how the sampling is placed in the L-C domain. To illustrate all the parameters of each279

sub-problem on the same graphic, we apply on these parameters the shifted and normalized function280

deﬁned as :281

P?

N=P?−min(P?)

max(P?−min(P?)),(34)

where P?is applied on (K?, G?) moduli and for the two prescribed displacements (bu?

d,eu?

d).282

283

We give hereafter guidelines regarding the assignment of sampling points, and more information is284

provided in Appendix B.285

•Minimum and maximum points containing for each sub-problem, the spectra in the L-C domain of286

the shear and bulk moduli and imposed displacement. In particular for the transient response, we287

have to identify the A and B parameters deﬁned as the limits in zero and inﬁnity (see Eq. 5 and288

6) for the LCT inversion using the Schapery’s collocation method. In pratice, as the parameters289

are deﬁned as a priori functions, it is straightforward to select numerically the two bounds.290

•Number of sampling points. For the transient response, as explained in Section. 2.1 and illus-291

trated in Fig. 2, the Schapery’s collocation method needs a number of points respecting over-292

and undersampling constraints. For the sinusoidal response, the method proposed and described293

12

in Section. 2.2 requires only a minimal number of points. In practice, this selection on sampling294

requires a short preliminary study as explained in Section 2 introducing the Schapery’s collocation295

method and the proposed method.296

•Positioning of the sampling between two bounds, highlighted on the regions of strong curvature.297

Indeed, the sampling in piof these functions is crucial to ensure an accurate LCT inversion, as298

suggested in [13]. In practice the application of a curvilinear log-linear distribution algorithm for299

sampling is appropriate.300

3.2.2. Application to the ﬁrst loading cycles301

In this part, we consider the ﬁrst ten cycles. The VE problem in time (Eq. 20 & 24) fully parametrized302

with Tab. 1 is solved ﬁrst, using implicit dynamic time scheme algorithm for N-cycles (t∈[0; 5]s,303

∆t= 0.02s, Nc=∼10 cycles), under ud

1(Heaviside+periodic) displacement boundary condition (here304

we imposed : Td= 0) as illustrated in Fig. 6a. The response constitutes the reference solution to assess305

the proposed method. Then the VE problem in L-C domain (P?,B?) (see Eq. 25 & 26) is solved for the306

displacement boundary condition (see Eq. 32 & 33) split in two (bud,eud). The methodology is illustrated307

in Fig. 7. The full comparison (large numbers of loading cycles) will be shown in the last part, here308

a preliminary validation for the VE problem in time is conducted, allowing to compare each evolution309

(periodic/transient) independently as well as the ﬁrst cycles. Numerically, the method employed to310

recover the periodic response function needs a starting point for the iterative resolution in each Gauss311

point of the structure. It was found that the choice of the starting point selected on the parameters312

of the imposed displacement function (a(0)

1(γ) = e

A,ω(0)

1(γ) = eω,φ(0)

1(γ) = e

φ, where γrepresents the313

displacement, strain, or stress solution) yields a fast convergence of the Newton-Gauss algorithm.314

(a) Nodal points selected. (b) Gauss points selected.

Figure 9.: Points of interest selected for future comparisons between reference and proposed methods,

(a) nodal points P17→3, (b) Gauss points T17→3.

315

Figure. 9 shows some points of interest chosen in the structure for comparison between reference316

solution and proposed method; there are 3 nodal points (P17→3, see Fig. 9a) and 3 Gauss points (T17→3,317

see Fig. 9b).318

The x- and y-displacements at points P17→3are plotted in Fig. 10 for the budHeaviside part of the319

imposed displacement and in Fig. 11 for the eudperiodic part.320

Figures. 10 and 11 show that for both problems, the recovered time evolutions of the x- and y-321

displacements at points P17→3obtained with the proposed LCT inversion methods match perfectly the322

13

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

5

10

15

20

25

30

35

40

45

50

(a) x-displacement

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

(b) y-displacement

Figure 10.: Displacements at 3 points of interest in Fig. 9a, computed with proposed method and reference

solution (Heaviside boundary condition).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-15

-10

-5

0

5

10

15

(a) x-displacement

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

(b) y-displacement

Figure 11.: Displacements at 3 points of interest in Fig. 9a, computed with proposed method and reference

solution (sinus boundary condition).

reference values obtained with direct Finite Element Analysis solved in the time.323

3.2.3. Application to a large number of loading cycles324

In this part, we compare the proposed method’s results with the reference full calculation for a large325

number of cycles (Nc= 10k, t∈[0; 5000] s, ∆t= 0.02s). For this purpose, each component of the strain326

and stress tensors in points of interest (T17→3in Fig. 9b) is compared with the reference response in327

Figs. 12-14.328

14

024

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

100 102 104 10001002 1004 3000 3002 3004 8400 8402 8404

024

-2

0

2

4

6

8

10

100 102 104 1000 1002 1004 3000 3002 3004 8400 8402 8404

Figure 12.: VE structure under a large number of loading cycles. Comparison of strain (ε11) and stress

(σ11) components at 3 points of interest of Fig. 9b computed with reference solution and with proposed

method.

024

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

100 102 104 10001002 1004 3000 3002 3004 8400 8402 8404

024

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

100 102 104 1000 1002 1004 3000 3002 3004 8400 8402 8404

Figure 13.: VE structure under a large number of loading cycles. Comparison of strain (ε22) and stress

(σ22) components at 3 points of interest of Fig. 9b computed with reference solution and with proposed

method.

The ﬁgures show that all strain and stress components are extremely well recovered.329

330

A second point of comparison based on a global energy indicator for the entire structure and providing331

a global estimation of the accuracy of the proposed method was established. The global energy indicator332

as a function of time is deﬁned here as :333

W(t) = 1

VΩZΩ

σ(x, t) : ε(x, t)dV. (35)

The comparison of W(t) obtained with the proposed method and with the reference solution is illus-334

15

024

-0.2

-0.1

0

0.1

0.2

0.3

0.4

100 102 104 1000 1002 1004 3000 3002 3004 8400 8402 8404

024

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

100 102 104 1000 1002 1004 3000 3002 3004 8400 8402 8404

Figure 14.: VE structure under a large number of loading cycles. Comparison of strain (ε12) and stress

(σ12) components at 3 points of interest of Fig. 9b computed with reference solution and with proposed

method.

trated in Fig. 15a, which shows that W(t) is qualitatively well reproduced for various cycles.335

024

0

20

40

60

80

100

120

100 102 104 1000 1002 1004 3000 3002 3004 8400 8402 8404

(a) Time (cycle) evolution of a global energy indicator.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0

1

2

3

4

5

6

7

8

9

10

(b) Time (cycle) evolution of the L2-error on W(t).

Figure 15.: Global accuracy evaluation of proposed method against reference solution for the VE structure

of Fig. 5.

To have a quantitative estimation, the L-2 relative error on W(t) :336

337

ErrL2(W(t)) = v

u

u

tRt

0[WRef(t0)−WLCT−1(t0)]2dt0

Rt

0[WRef(t0)]2dt0(36)

338

339

was calculated. The results are plotted in Fig. 15b.340

It is seen that the error on the global energy indicator drops rapidly, then it increases weakly with341

the number of cycles before stabilizing. At 10k cycles, the error remains lower than 2%.342

16

3.2.4. Computational gain343

The numerical costs of the reference calculation in time and the reconstructed solutions using the344

proposed method based on LCT inversion are detailed in Appendix C, Table C1, including the CPU345

time used at each calculation step, and following the methodology scheme in Fig. 7. As for the346

simulation tools, Abaqus software was used for all FE calculations, while the numerical LCT inversions347

to reconstruct temporal responses and the post-processing were performed in Matlab.348

349

To summarize the numerical study, the method that is proposed here and applied to the structure350

depicted in Figs. 5 and 9, for Nc=10k cycles, generates a relative error of L-2 norm less than 2 % in the351

global energy indicator for a factor gain in time of GT= 32, which represents Tref = 96 hours for the352

reference calculation against Tpm = 3 hours for the proposed method and its post-processing. It should353

be noted that the proposed method has a ﬁxed computational cost of 2h30 and is independent of the354

number of cycles whereas the reference time calculation is a linear function of the number of cycles. Thus,355

without taking into account the post-processing, we can estimate by extrapolation with the number of356

cycles a gain factor of time between the two calculations such as :357

GT=Tref

Tpm

,(37)

where T is the computational time, respectively of the reference (ref) and the proposed method (pm).358

359

The evolution of the computational gain is illustrated in Fig. 16 and shows a great time performance360

for very large numbers of cycles. As discussed in the introduction in Section. 1, the computational gain361

will play a key role in future work.362

1001021041061081010

10-2

100

102

104

106

108

Figure 16.: Computational time gain factor.

17

3.3. A second structural application363

In this Section, the proposed method is applied to a VE arc subjected to a cyclic force (see Fig. 17),364

applied to an extremity of the geometry ∂ΩT. The cyclic time history of the applied force and its365

decomposition are form-identical to those of the applied boundary displacement in the previous structural366

problem of Section 3.2; see Eqs. 27 to 29, with ( b

A= 1.106Pa, e

A= 1.105Pa). The force is applied367

vertically Td

1= 0. The structure is discretized with 13 860 linear triangular elements and 7 174 nodes.368

Figure 17.: VE arc under cyclic loading, 2D plane strain model and points of interest selected for future

comparisons between reference solution and proposed method; 3 Gauss points (T17→3).

024

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

100 102 104 10001002 1004 2500 2502 2504 4900 4902 4904

024

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

100 102 104 1000 1002 1004 2500 2502 2504 4900 4902 4904

Figure 18.: VE arc under a large number of loading cycles. Comparison of strain (ε11) and stress (σ11)

components at 3 points of interest of Fig. 17 computed with reference solution and with proposed method.

024

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

100 102 104 10001002 1004 2500 2502 2504 4900 4902 4904

024

-1.5

-1

-0.5

0

0.5

1

1.5

100 102 104 1000 1002 1004 2500 2502 2504 4900 4902 4904

Figure 19.: VE arc under a large number of loading cycles. Comparison of strain (ε22) and stress (σ22)

components at 3 points of interest of Fig. 17 computed with reference solution and with proposed method.

18

024

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

100 102 104 10001002 1004 2500 2502 2504 4900 4902 4904

024

-1

-0.5

0

0.5

1

1.5

2

2.5

3

100 102 104 1000 1002 1004 2500 2502 2504 4900 4902 4904

Figure 20.: VE arc under a large number of loading cycles. Comparison of strain (ε12) and stress (σ12)

components at 3 points of interest of Fig. 17 computed with reference solution and with proposed method.

As in the previous application, a comparative study was conducted, ﬁrst locally on three points369

(see Fig. 17), and on the entire structure with the global energy indicator (see.Eq. 35) over 5k cycles370

(t∈[0; 2500] s). The local comparisons are shown in Figs. 18,19,20 for strain and stress components and371

global values in Figs. 21a,21b.372

024

0

2

4

6

8

10

12

14

16

18

100 102 104 1000 1002 1004 2500 2502 2504 4900 4902 4904

(a) Time (cycle) evolution of a global energy indicator.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

(b) Time (cycle) evolution of the L2-error on W(t).

Figure 21.: Global accuracy evaluation of proposed method against reference solution for the VE arc of

Fig. 17.

Again, and similarly to the ﬁrst structural application, all the local and global results show a good373

qualitative and quantitative reproduction of the reference solution with an error on the global energy374

indicator lower than 1%. The CPU time speedup factor of the proposed method as compared to the375

reference solution is about 16.376

19

4. Conclusions377

In this paper, a computational time reduction method for solving VE structural problems under large378

numbers of loading cycles was proposed. The strategy is based on an extension of the LCT method379

and built on a decomposition of the problem into a ﬁrst transient part that can be treated numerically380

by Schapery’s collocation method, and a second sinusoidal periodic part using a new LCT inversion381

method. This proposed method deﬁnes the periodic solution on a sinusoidal basis in the L-C domain and382

uses a Newton-Gauss algorithm enabling the LCT inversion in the time domain. The modeling and the383

numerical implementation of these two parts were ﬁrst applied on the case of well-known 1D functions.384

Then, the accuracy of the proposed method was assessed on two structural problems under large385

numbers of cycles (10 k and 5 k cycles, respectively). A methodology as well as the comparison with386

reference solutions have been provided, both at local scale (displacement, strain and stress components387

at diﬀerent points) and macroscale (global energy indicator). The results show very accurate results at388

local and global levels, a L-2 relative error on the global energy indicator less than 2 %. The CPU time389

speedup factor of the proposed method as compared to the reference solution is about 33 for the 10k390

cycles application.391

392

The proposed method allows to obtain a total strain history at any location in a VE structure subjected393

to a large number of loading cycles. Consequently, a multiscale model combining VE computations at394

macroscale of the structure and complex microstructures and constitutive models at the microscale(VE395

matrix with porosity and VE-VP weak spots), as discussed in the introduction (Section 1) is now being396

developed. The (very) low computational cost of the procedure proposed in the present article will play397

a fundamental role in rendering the multiscale model aﬀordable for the simulation of high cycle fatigue398

of polymer solids and structures.399

Declaration of competing interest400

The authors declare that they have no known competing ﬁnancial interests or personal relationships that401

could have appeared to inﬂuence the work reported in this paper.402

Acknowledgments403

The present work was conducted and funded within the framework of the European Union’s Horizon 2020404

research and innovation programme for the project ”Multi-scale Optimisation for Additive Manufacturing405

of fatigue resistant shock-absorbing MetaMaterials (MOAMMM)”, grant agreement No. 862015, of the406

H2020-EU.1.2.1. - FET OpenProgramme.407

20

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22

Appendix A. Dependency of the LCT inversion methods on the number of sampling472

points473

012345678910

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(a) Reconstruction of the sinusoidal function of Eq. 16 with the proposed

method. Dependence on the number of sampling points.

012345678910

-1.5

-1

-0.5

0

0.5

1

1.5

(b) Reconstruction of the sinusoidal function of Eq. 16 with Schapery’s

collocation method. Dependence on the number of sampling points.

Figure A1.: Dependence of the two reconstruction methods on the number of sampling points for the

sinusoidal function of Eq. 16.

23

Appendix B. Sampling method474

In this appendix, e

K?is a well-known function in the L-C domain, which is illustrated in Fig. B1a as475

a function of the L-C variable p. The ﬁrst task is to ﬁnd the minimum and maximum values of the476

sampling points, pmin and pmax. There are two issues here. First, the [pmin, pmax] interval must be able477

to encompass the entire range of e

K?(p). Second, the interval should not be unnecessarily large as to478

not waste computing time on too many sampling points where e

K?(p) is a horizontal line. In order to479

achieve these two objectives, the following procedure is adopted.480

481

Initially, far apart estimates of pmin and pmax are found. Then, diﬀerent ﬁlters are applied in order to482

ﬁnd closer estimates. The ﬁrst considers the area under the e

K?(p) curve. The second ﬁlter is based on483

normalized values of the ﬁrst derivative of e

K?(p). And if e

K?(p) is not the L-C transform of a sinusoidal484

time function, then a third ﬁlter is applied, based on the second derivative of e

K?(p).

10-10 10-5 10010510 10

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3 109

(a) Selection of the bounds and the range of K?.

10-5 1001051010

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Sampling on the selected range.

Figure B1.: Numerical sampling selection.

485

Once the values of pmin and pmax have been found, the sampling points are chosen according to the486

following method. First, e

K?(p) is normalized as explained in Eq. 34. We thus obtain e

K?

N(p) illustrated487

in Fig. B1b. Next, a curvilinear sampling algorithm is applied. The main idea behind it is to divide488

the area under the curve e

K?

N(p) between pmin and pmax into a ﬁnite number of subdomains (typically,489

around 30) having the same area.490

491

A ﬁnal issue concerns the fact that when solving a structural VE problem under cyclic loadings, there492

are several known functions e

K?(p). Indeed, we have the L-C transforms of the time dependent bulk and493

shear moduli, and those of the boundary conditions corresponding to each one of the two structural494

sub-problems (transient and periodic). For the example problem studied in Section. 3.2, these are given495

by equations 30 to 33. The above procedure is applied to each of these known functions. Next, for each496

one of the two structural sub-problems considered independently :497

(i) the smallest values among the pmin values is found;498

(ii) the largest value among the pmax values is determined;499

(iii) all the sampling points are arranged in ascending order;500

(iv) the corresponding elastic problems are solved in the L-C domain for each sampling point.501

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Appendix C. Computational time comparison502

Reference solution Dependence on the number of cycles Final time ∼4 days

VE solver : Total time ∼3.5 days

Time domain Cumulative time = 300 k s

time for one tiincrement time ←0.6 s

time for one cycle (∆t= 0.02s 7→ 25 steps/cycle) time ←30 s

time for 10k cycles time ←time x 10k

Others : Post-treatment (data extraction) Total time <6h

Proposed method Independence from the number of cycles Final time <3h

Elastic (linear) solver : Total time = 1 min

L-C domain Cumulative time = 1 min

time for one discrete pipoint time ←0.8 s

for Npi= 30 discrete points time ←time x 30

for 2 boundary displacement functions time ←time x 2

LCT inversion : Total time <2h30

Schapery’s collocation Cumulative time = 30 min

method Calculation time ←0.027 s

for Ne(∼10k) element time ←time x Ne

for components of ﬁelds (u,ε) time ←time x 6

LCT inversion Cumulative time <1h30

for sinusoidal function Calculation for one iteration time ←0.083 s

for Nng <10 (Newton-Gauss) iteration time ←time x 10

for Ne(∼10k) elements time ←time x Ne

for components of ﬁelds (u,ε) time ←time x 6

Others : Post-treatment Total time <30 min

Table C1.: Computational time details for the structural VE problem of Figs. 5 and 9.

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