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An efficient modeling procedure is proposed for viscoelastic (VE) solids subjected to large numbers of loading cycles. While the Laplace-Carson transform (LCT) is often used to solve VE creep or relaxation problems, the originality here is an efficient extension of the approach to a plethora of cycles, based on some key ingredients. The time history of the cyclic loading is decomposed into transient and periodic signals, leading to two subproblems. Each one is transformed into a finite number of linear elastic analyses in the L-C domain. A method to choose the number and positioning of the L-C domain sampling points for each one of the two subproblems is detailed. Specific LCT inversion methods are applied to each subproblem in order to reconstruct the displacement, strain and stress fields in the time domain. For the transient subproblem, Schapery's collocation method based on exponential basis functions is used, while a new LCT inversion method is proposed for the periodic subproblem based on sinusoidal basis functions and a Newton-Gauss algorithm. After a verification on well-known 1D functions, the accuracy of the proposed method is assessed on two structural problems with large numbers of cycles. Comparison with reference finite element analyses conducted directly in the time domain shows that the proposed methodology provides excellent predictions, both at local scale (displacement, strain and stress components at various points) and macroscale (global energy indicator). The important speedup factor (e.g., 32 for 10k cycles) will increase significantly with the number of cycles, enabling the proposed method to be extended to high cycle fatigue of thermoplastic polymer structures in future work. KEYWORDS Laplace-Carson numerical inversion, high cycle simulation, periodic basis extension, viscoelastic structure.
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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)eptdt, (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
L1
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 et/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 + k
.(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
t0f(t) = lim
p+f?(p),(5)
141
B= lim
t+
f(t)
t= lim
p0pf?(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 + Cet/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 M3xN, 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 Rtol = 106). 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
Giet/gi,(22)
228
K(t) = K+
J
X
j=1
Kjet/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) =
uT=,uT=,
ui(x, t) = ud
i(x, t), on u,
Ti(x, t) = Td
i(x, t), on T,
(24)
8
where uand Tare 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) =
uT=,uT=,
u?
i(x, p) = ud?
i(x, p), on u,
T?
i(x, p) = Td?
i(x, p), on T,
(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),xu.(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 et/
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
VZ
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)WLCT1(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 T. 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
24
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.
25
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