Science Arts & Métiers (SAM)
is an open access repository that collects the work of Arts et Métiers Institute of
Technology researchers and makes it freely available over the web where possible.
This is an author-deposited version published in:
https://sam.ensam.eu
Handle ID: .
http://hdl.handle.net/10985/9958
To cite this version :
Amine AMMAR, Ali ZGHAL, Franck MOREL, Francisco CHINESTA - On the space-time
separated representation of integral linear viscoelastic models Comptes Rendus Mécanique
-Vol. 343, n°4, p.247-263 - 2015
Any correspondence concerning this service should be sent to the repository
Administrator :
archiveouverte@ensam.eu
Contents lists available atScienceDirect
Comptes Rendus Mecanique
www.sciencedirect.com
On
the
space-time
separated
representation
of
integral
linear
viscoelastic
models
Représentation
séparée
espace-temps
pour
des
comportements
viscoélastiques
linaires
intégraux
Amine Ammar
a,
b,
∗
,
Ali Zghal
a,
Franck Morel
b, Francisco Chinesta
c aUMSSDT, ENSIT, Université de Tunis, 5, avenue Taha-Hussien, Montfleury 1008, Tunis, TunisiabArts et Métiers ParisTech, 2, bd du Ronceray, BP 93525, 49035 Angers cedex 01, France cGEM, UMR CNRS–Centrale Nantes, 1, rue de la Noe, BP 92101, 44321 Nantes cedex 3, France
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Article history:
Received1December2014 Accepted10February2015 Availableonline9March2015
Keywords: PGD Viscoelasticity Integro-differentialmodels Fatigue Mots-clés : PGD Viscoélasticité Modèleintegro-differentiel Fatigue
Theanalysisofmaterialsmechanicalbehaviorinvolvesmanycomputationalchallenges.In thiswork,weare addressing thetransientsimulation ofthemechanical behavior when thetimeofinterestismuchlargerthanthecharacteristictimeofthemechanicalresponse. Thissituation isencountered inmanyapplications,as for exampleinthe simulation of materialsaging,orinstructuralanalysiswhensmall-amplitudeoscillatoryloadsareapplied duringalongperiod,asitoccursforexamplewhencharacterizingviscoelasticbehaviors bycalculatingthecomplexmodulusorwhenaddressingfatiguesimulations.Moreover,in the case ofviscoelasticbehaviors,the constitutiveequationis manytimesexpressedin anintegralformavoidingthenecessityofusinginternalvariables,factthatresultsinan integro-differentialmodel.Inordertoefficientlysimulatesuchamodel,weexploreinthis worktheuseofaspace-timeseparatedrepresentation.
©2015Académiedessciences.PublishedbyElsevierMassonSAS.All rights reserved.
r
é
s
u
m
é
L’analyseducomportementmécaniquedesmatériauxentraîne denombreusesdifficultés dupoint devuenumérique.Danscetravail,nousallonsnousfocalisersurl’uned’entre elles, celle associée à la simulation transitoire du comportement mécanique quand l’intervalletemporeld’intérêtestsubstantiellementpluslongqueletempscaractéristique associé à la réponse mécanique. Cette situation est fréquemment retrouvée dans la caractérisation rhéologique des matériaux viscoélastiques (pour la détermination du modulecomplexe)ainsiquequandons’attaqueàlasimulationdelafatigue.Deplus,dans lecasdesmatriauxviscoélastiques,lecomportementestgénéralementdécritparuneloi decomportementintégralequiévitelebesoind’utiliserdesvariablesinternes,donnantlieu
*
Correspondingauthor.E-mail addresses:Amine.Ammar@ensam.eu(A. Ammar),Ali.Zghal@gmail.com(A. Zghal),Franck.Morel@ensam.eu(F. Morel), Francisco.Chinesta@ec-nantes.fr(F. Chinesta).
http://dx.doi.org/10.1016/j.crme.2015.02.002
àunmodèlemécaniqueintegro-différentiel.Pourunerésolutionefficace,nousanalysons icil’utilisationd’unereprésentationséparéeenespace-temps.
1. Introduction
The presentwork focuseson theefficient treatment ofmodels involvingtransient fieldsthat must be solved inlarge time intervals usingvery small time steps. In thiscontext, if one uses standard incremental time-discretizations, in the generalcase(modelsinvolvingtime-dependentparameters,non-linearmodels,etc.),onemustsolveatleastalinearsystem ateachtimestep.Whenthetimestepbecomestoosmallasaconsequenceofthestabilityrequirements,andthesimulation timeintervalislargeenough,standardincrementalsimulationsbecomeinefficient.Theymustbereplacedwithothermore efficienttechniques.
Model order reduction—MOR—techniques consider reduced bases on which the solution is projected. As such bases
involve ingeneralfew functions, compared withthe standard approximationbases inwhich an interpolation function is attachedtoeachnodeofamesh,onemustconsiderareduceddiscretemodelwhosesolutioncanbeinmanycasessolved inrealtime.
TherearethreemainstrategiesbasedonMOR.Thefirstone concernstheso-calledProperOrthogonalDecomposition— POD—that proceedsby extractingthe mostsignificant functionsinvolved in the model’s solution.For that purpose,the high-fidelity modelissolved by usinga standard discretizationtechnique anddifferentsnapshotsare extracted(solution atdifferenttimes).Then,by applyingtheproperorthogonaldecomposition,themostsignificant modesare identifiedand then usedforprojectingthesolutionto“similar” problems.Bysimilar problems,we understandmodelsinvolvingslightly different parameters, boundary conditions, geometries, than the onesinvolved in the original model that served for ex-tracting thereducedbasis.Thereis anextensiveliterature regardingthisissue.The interestedreaderscan referto[1–12]
andthe numerousreferencestherein.The extractionofthereducedbasis isthe keypoint whenusing POD-basedmodel orderreduction,aswellasitsadaptivitywhenaddressingscenariosfarfromtheonesconsideredintheconstructionofthe reducedbasis[13,14].
Anotherfamilyofmodelorderreductiontechniqueswelladaptedtothesolutiontoparametricmodelsliesintheused ofreducedbases—RB—constructed bycombiningagreedyalgorithmandan apriorierrorindicatordrivingtheexploration of the parametric space. Thus, RB techniques need for some amount off-line work, but then the reduced basis can be usedonlineforsolving differentmodelswithaperfectcontrolofthesolutionaccuracybecauseoftheavailabilityoferror bounds. Whentheerrorisinadmissible,thereducedbasis canbeenriched byinvokingagainthesamegreedyalgorithm. Theinterestedreaderscanreferto[15–18]andthereferencestherein.
Manyyearsago,P.Ladevezeproposed theuseofspace-timeseparatedrepresentations,attheheartofthethirdkindof MOR strategieshereaddressedandthat wascoinedasProperGeneralizedDecomposition—PGD.Heintroducedthe space-time separatedrepresentationas one ofthe mainbricks composing the LATINmethod,a powerful nonlinearsolver. The interestedreadercanreferto[19–24]andthevaluablereferencestherein.
Whenusingspace-timeseparatedrepresentations,theapproximationofatransientfieldu
(
x,
t)
,x∈
⊂
R
D, D=
1,
2,
3 andt∈
I = (
0,
T ]
⊂
R
,isexpressedas u(
x,
t)
≈
N i=1 Xi(
x)
·
Ti(
t)
(1)The constructor of such a separated representation consists of a double iteration loop: the first associated withthe calculation of each term
(
Xn(
x)
·
Tn(
t))
,∀
n∈ [
1,
· · · ,
N]
, of the finite sum (1), and the other for solving the nonlinear problemrelatedtothecalculationofeachcoupleoffunctions( Xn(
x)
andTn(
t)
)becausebothbeingunknowntheproblem results nonlinear.The numericalalgorithmwas deeplyreported inourformerworks, butforthesake ofcompleteness it hasbeensummarizedinAppendix A.Anadditionaladvantageofseparatedrepresentationsisthattheycanbeappliedtothesolutiontoproblemsdefinedin highly dimensionalspaces becausetheyallow circumventingtheso-calledcurse ofdimensionality.Thus,we appliedsuch kindofseparatedrepresentations forsolvingmodels involvingmanyconformationalcoordinates encounteredinquantum chemistry,kinetictheorydescriptionsofmaterialsorcellsignalingprocesses[25–29].Moreover,weproposedaddingmodel parametersasextra-coordinatesforconstructingparametricsolutionsthatcanbeseenascomputationalvademecumsfrom whichwecanperform,inrealtime,optimization,inverseanalysis,andsimulation-basedcontrol[30–35].
Theinterestedreadercanalsorefertotherecentreviews[36–39]andthereferencestherein.
1.1. Non-incrementalversusincrementaltimeintegrations
It isusefultoreflectonthe considerabledifferencebetweentheabove PGDstrategy andtraditional,incrementaltime integrationschemes.
Indeed, the PGD allows for a non-incremental solution to time-dependent problems. Let
Q
n denote the number of non-lineariterationsrequiredtocomputethenewterm Xn(
x)
·
Tn(
t)
atenrichmentstepn.Then,theentirePGDprocedure toobtainthe N-termapproximation(1)involvesthesolutiontoatotalofQ
= (Q
1+ · · · +
Q
N)
decoupled,boundaryand initialvalueproblems.TheBVPsaredefinedoverthespacedomain,andtheircomputationalcomplexityscaleswiththe mesh used to discretize them.The IVPsare defined over the time interval
I
, andtheir complexity is usually negligible comparedtothatoftheBVPs,evenwhenextremelysmalltimestepsareusedfortheirdiscretization.Thisisvastly differentfroma standard,incremental solutionprocedure.If P is thetotalnumberoftime steps forthe completesimulation,i.e.P
=
T /
t,anincrementalprocedureinvolvesthesolutiontoaBVPinateachtimestep,i.e.a totalof P BVPs.Thiscanbe averylarge numberindeed,asthetimestep
t mustbe chosensmallenoughtoguarantee thestabilityofthenumericalscheme.
NumericalexperimentswiththePGDshowthatthe
Q
nsrarelyexceedten,whileN isafewtens.Thus,thecomplexity of the complete PGD solution is a few hundreds of BVP solutions in. This is in many applications several orders of magnitudelessthanthetotalofP BVPsthatmustbesolvedusingastandardincrementalprocedure.
Thisandotherrelatedadvantagesinusingspace-timeseparatedrepresentationswereanalyzedin[26,40,41]and[42].
1.2. Separatingthephysicalspace
Sometimes,thedomain
,assumedtobethree-dimensional,canbefullyorpartiallyseparated,andconsequentlyitcan beexpressedas
=
x×
y×
z or=
xy×
z,respectively.Thefirstdecompositionisrelatedtohexahedraldomains, whereasthesecondoneisrelatedtoplate,beamsorextrudeddomains.Bothwerewidelyconsideredin[43,37,44–47].We considerbelowtheapproximationsrelatedtobothscenarios.(i) Thespatialdomain
ispartiallyseparable.InthiscaseEq.(1)canberewrittenas:
u
(
x,
z,
t)
=
N i=1 Xi(
x)
·
Zi(
z)
·
Ti(
t)
(2) wherex= (
x,
y)
∈
xy,z∈
zandt∈
I
.Thus,iterationp ofthealternatingdirectionstrategyatagivenenrichmentstepn consistsinthefollowingthreetasks, employingthenotationintroducedinAppendix A:
(a) solvein
xy atwo-dimensionalBVPtoobtainthefunction Xnp, (b) solvein
z aone-dimensionalBVPtoobtainthefunction Znp, (c) solvein
I
aone-dimensionalIVPtoobtainthefunction Tnp.WecanrepeatourdiscussionregardingthecomplexityofthisPGDnon-incrementalstrategyversusstandard incremen-talschemes.Clearly,what willdominatethecostofthePGDprocedureisthetotalof
Q
two-dimensionalBVPstobe solvedinxy.TheBVPsin
zandIVPsin
I
beingone-dimensional,theircomplexityiscomparativelynegligible.Thus, thecomputationalcostofthePGDsimulationwillbeordersofmagnitudesmallerthanthatofastandardincremental procedure,whichrequiresthesolutiontoathree-dimensionalBVPateachtimestep.(ii) Thespatialdomain
x isfullyseparable.Inthiscase,Eq.(1)canberewrittenas:
u
(
x,
y,
z,
t)
=
N
i=1
Xi
(
x)
·
Yi(
y)
·
Zi(
z)
·
Ti(
t)
(3) Iteration p ofthealternatingdirectionstrategyatagivenenrichmentstepn consistsinthefollowingfourtasks: (a) solveinx aone-dimensionalBVPtoobtainthefunction Xnp,
(b) solvein
y aone-dimensionalBVPtoobtainthefunctionYnp, (c) solvein
zaone-dimensionalBVPtoobtainthefunction Znp, (d) solvein
I
aone-dimensionalIVPtoobtainthefunctionTnp.Thecostsavings providedbythePGDarepotentially phenomenalwhenthespatialdomainisfullyseparable. Indeed, the complexity ofthe PGDsimulation now scales with the one-dimensional meshes used to solve the BVPs in
x,
y and
z, regardless ofthe time step used in thesolution to the decoupled IVPsin
I
.The computational cost is thus orders of magnitude smaller than that of a standard incremental procedure, which requires the solution to a three-dimensionalBVPateachtimestep.Evenwhenthedomainisnotfullyseparable,afullyseparatedrepresentationcouldbeconsideredbyusingappropriate geometricalmappingsorbyimmersingthenon-separabledomainintoafullyseparableone.Theinterestedreadercanrefer to[48]and[49].
AfterthisshortintroductioninSection 2,we define theintegro-differential viscoelasticmodelwithin thesmall trans-formations frameworkwhosespacediscretization willbe carriedout inSection 3.InSection 4,thespace-time separated representationwill be introduced andits construction will be consideredin detail inSection 5. Finally,in Section 6,we
addresssomenumericalexamplesforverifyingtheproposedstrategyandtoproveitsabilitytoaddressefficientlycomplex scenarios.
2. Linearviscoelasticintegralmodel
The mechanicalmodelisdefinedinthe domain
whoseboundary
∂
≡
is decomposedintoDand
N inwhich
velocitiesandtractionsareprescribedrespectively.
Weconsiderthestandardmomentumbalanceequationneglectingtheinertiaandmassterms:
∇ · =
0 (4)where
isthestandardCauchy’sstresstensor. Theboundaryconditionswrite:
v
(
x∈
D,
t∈
I)
=
vg(
x∈
D,
t∈
I)
(
x∈
N,
t∈
I)
·
n(
x∈
N)
=
tg(
x∈
N,
t∈
I)
(5) where n isthe unitoutwards vectordefinedon theboundary
N,vg the prescribedvelocities on
Dandtg the applied tractions on
N.It was assumedthat the mechanicalproblemis linearimplyingboth alinear constitutivelawandsmall
displacements andstrains.Thus,weassumethatdomain
remainsunchangedallalongthetimeandthenunaffectedby thekinematicsinducedbytheappliedboundaryconditions.
The weak formrelatedtothemomentum balanceateach time t consistsin lookingforthevelocity field v
∈
V
,withV =
v(
x,
t)
∈
H
1()
3,
v(
x∈
D,
t∈
I) =
vg(
x∈
D,
t∈
I)
suchthat D∗:
dx=
N v∗·
tgdx,∀
v∗∈
V
∗ (6) withV
∗=
v∗(
x,
t)
∈
H
1()
3,
v∗(
x∈
D,
t∈
I) =
0 .InEq.(6),D istheusualrateofstraintensorandweuse
insteadoftheusual
σ
notationforthestresstensorbecause inwhatfollowsσ
willrefertothevectorformofthestresstensor.Usingvectornotations,integral(6)writes
d∗·
σ
dx=
N v∗·
tgdx (7)whered isthevectorformoftherateofstraintensorD.
Theconstitutiveequationhereconsideredconsistsofthestandardviscoelasticintegralform
=
t −∞λ(
t−
τ)
Tr(
D(τ
))
·
I dτ
+
t −∞ 2μ(t−
τ)
D(τ
)
dτ (8)whereTr
()
referstothetraceoperatorandλ
andμ
aretwomemoryfunctions.Even if here we only address the simplest viscoelastic constitutive model, all the developments can be extended to generalizedviscoelasticmodelsinvolvingseveralrelaxationtimes.
Byusingvectornotations,theconstitutiveequationcanbewrittenas
σ
=
t−∞
C
(
t−
τ
)
·
d(τ
)
dτ (9)beingd thevectorformofthestrainratetensorD.Inplanestrain,with
σ
=
⎛
⎝
1122
12
⎞
⎠
(10) and d=
⎛
⎝
DD1122 2D12⎞
⎠
(11)theexpressionofC
(
t−
τ
)
writes: C(
t−
τ
)
= λ(
t−
τ
)
⎛
⎝
1 1 01 1 0 0 0 0⎞
⎠ +
μ(
t−
τ
)
⎛
⎝
2 0 00 2 0 0 0 1⎞
⎠ = λ(
t−
τ
)
Gλ+
μ(
t−
τ
)
Gμ (12)Thevectorformofthestrainratetensorreads:
d
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
∂
∂
x 0 0∂
∂
y∂
∂
y∂
∂
x⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
·
vx vy (13)wherevx andvyarethevelocityvectorcomponents:vT
= (
vx,
vy)
. 3. SpacediscretizationWecanassumeastandardfiniteelementapproximationofthevelocityfield,involvingamesh
M
consistinginN
nodes with coordinates Xi, i=
1,
2,
· · · ,
N
.Thus, if Ni(
x)
denotes the shape function relatedto node Xi, that by construction verifiestheKroeneckerdeltapropertyNi(
Xj)
= δ
i j,thevelocityfieldvanbewrittenas⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
vx=
N i=1 Ni(
x)
vx(
Xi)
=
NT·
Vx vy=
N i=1 Ni(
x)
vy(
Xi)
=
NT·
Vy (14)whereVx andVyarethevectorsthatcontainthenodalvelocitycomponentsvx
(
Xi)
andvy(
Xi)
(i=
1,
2,
· · · ,
N
)respectively andN thevectorcontainingthedifferentshapefunctions.Thisapproximationcanbewritteninamorecompactformaccordingto:
v
=
NT 0T 0T NT·
Vx Vy=
M·
V (15)where0T istherowvectorofsize
N
withnullentries. Thus,thevectorformoftherateofstraind reads:d
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
∂
NT∂
x 0 T 0T∂
N T∂
y∂
NT∂
y∂
NT∂
x⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
·
Vx Vy=
B·
V (16)Now,comingbacktotheweakform(7),itsleftmemberresults
d∗·
σ
dx=
V∗T(
t)
·
⎧
⎨
⎩
t −∞λ(
t−
τ
)
Kλ+
μ(
t−
τ)
Kμ·
V(τ)
dτ⎫
⎬
⎭
(17) with⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
Kλ=
BT·
Gλ·
B dx Kμ=
BT·
Gμ·
B dx (18)Ontheotherhand,theright-hand-sidememberofEq.(6)writes:
N v∗(
x,
t)
·
tg(
x,
t)
dx=
V∗T·
⎧
⎪
⎨
⎪
⎩
N MT·
M dx⎫
⎪
⎬
⎪
⎭
·
f(
t)
=
V ∗T(
t)
·
F(
t)
(19)Thusfinally,afterdiscretizinginspace,theproblemreads:
V∗T
(
t)
·
⎧
⎨
⎩
t −∞λ(
t−
τ)
Kλ+
μ(
t−
τ)
Kμ·
V(τ
)
dτ⎫
⎬
⎭ =
V∗T(
t)
·
F(
t)
(20)whichleadstothelinearsystem: t
−∞λ(
t−
τ)
Kλ+
μ(
t−
τ)
Kμ·
V(τ
)
dτ=
F(
t)
(21)complementedwiththeDirichletboundaryconditionsapplyingon
D.
Eq.(21)canberewrittenas
Kλ
·
t −∞λ(
t−
τ)
V(τ
)
dτ+
Kμ·
t −∞μ(
t−
τ)
V(τ
)
dτ=
F(
t)
(22)4. Space-timeseparatedrepresentation
Now,weconsiderEq.(22)andassumethatboththeappliedtractionF
(
t)
andthevelocityfieldV(τ
)
canbewrittenin aseparatedform,respectively:F
(
t)
≈
NF i=1 SiS
i(
t)
(23) and V(
t)
≈
NV i=1 XiX
i(
t)
(24)Thus,Eq.(22)results: NV
i=1⎧
⎨
⎩
Kλ·
Xi·
t −∞λ(
t−
τ
)X
i(τ
)
dτ+
Kμ·
Xi·
t −∞μ(
t−
τ
)X
i(τ
)
dτ⎫
⎬
⎭ =
NF i=1 SiS
i(
t)
(25)The time integralscan be approximatedby using an adequatenumerical quadrature.Ifwe assume that F
(
t)
andV(
t)
vanishatt≤
0,andconsiderdiscretetimestn=
nt,thenwecanwrite:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
t1 0 g(
t)
dt≈
g(
t1)
t t2 0 g(
t)
dt≈
g(
t1)
t+
g(
t2)
t..
.
tn 0 g(
t)
dt≈
i=n i=1 g(
ti)
t (26)⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
t1 0λ(
t1−
τ
)X
i(τ
)
dτ≈ λ(
t0)X
i(
t1)
t t2 0λ(
t2−
τ
)X
i(τ
)
dτ≈ λ(
t0)X
i(
t2)
t+ λ(
t1)X
i(
t1)
t..
.
tn 0λ(
tn−
τ
)X
i(τ
)
dτ≈
n j=1λ(
tn−
tj)X
i(
tj)
t..
.
(27)whosematrixformreads:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
t1 0λ(
t1−
τ
)X
i(τ
)
dτ t2 0λ(
t2−
τ
)X
i(τ
)
dτ..
.
tP 0λ(
tP−
τ
)X
i(τ
)
dτ⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
t⎛
⎜
⎜
⎜
⎝
λ(
t0)
0· · ·
0λ(
t1)
λ(
t0)
· · ·
0..
.
..
.
. .
.
..
.
λ(
tP) λ(
tP−1)
· · · λ(
t0)
⎞
⎟
⎟
⎟
⎠
·
⎛
⎜
⎜
⎜
⎝
X
i(
t1)
X
i(
t2)
..
.
X
i(
tP)
⎞
⎟
⎟
⎟
⎠
=
t Lλ· Xi
(28) withPt
=
T
.Consideringnowtheintegralinvolvingthememoryfunction
μ
(
t−
τ
)
andusingthesamequadrature,itresults:⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
t1 0μ(
t1−
τ
)X
i(τ
)
dτ t2 0μ(
t2−
τ
)X
i(τ
)
dτ..
.
tP 0μ(
tP−
τ
)X
i(τ)
dτ⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
t⎛
⎜
⎜
⎜
⎝
μ(
t0)
0· · ·
0μ(
t1)
μ(
t0)
· · ·
0..
.
..
.
. .
.
..
.
μ(
tP)
μ(
tP−1)
· · ·
μ(
t0)
⎞
⎟
⎟
⎟
⎠
·
⎛
⎜
⎜
⎜
⎝
X
i(
t1)
X
i(
t2)
..
.
X
i(
tP)
⎞
⎟
⎟
⎟
⎠
=
t Lμ· Xi
(29)Forevanescentmemory,functions
λ(
tm)
andμ
(
tm)
vanishuptoacertainvaluen,andconsequentlyonlym diagonalsof LλandLμ mustbecomputed.5. Separatedrepresentationconstructor
Weconsiderthepreviousdiscreteform(25) NV
i=1⎧
⎨
⎩
Kλ·
Xi·
t −∞λ(
t−
τ)X
i(τ)
dτ+
Kμ·
Xi·
t −∞μ(
t−
τ
)X
i(τ)
dτ⎫
⎬
⎭ =
NF i=1 SiS
i(
t)
(30)andassume thatatpresentiteration we alreadycomputedtheq
−
1 firstterms ofthefinitesum(24),withq−
1<
NV,leadingtothe
(
q−
1)
-approximate:Vq−1
(
t)
=
q−1
i=1Xi
X
i(
t)
(31)Vq
(
t)
=
q
i=1
Xi
X
i(
t)
=
Vq−1+
XqX
q(
t)
(32)Now,inordertoapplytherationaledescribedinAppendix A,weconsiderthetestfunction
V∗q
=
X∗X
q(
t)
+
XqX
∗(
t)
(33)andfrom(25)theextendedweakform: T
0 X∗X
q(
t)
+
XqX
∗(
t)
·
⎧
⎨
⎩
q i=1⎧
⎨
⎩
Kλ·
Xi·
t −∞λ(
t−
τ)X
i(τ)
dτ+
Kμ·
Xi·
t −∞μ(
t−
τ
)X
i(τ
)
dτ⎫
⎬
⎭
−
NF i=1 SiS
i(
t)
⎫
⎬
⎭
dt=
0 (34)thatcanberewrittenundertheform: T
0 X∗X
q(
t)
+
XqX
∗(
t)
·
⎧
⎨
⎩
Kλ·
Xq·
t −∞λ(
t−
τ
)X
q(τ
)
dτ+
Kμ·
Xq·
t −∞μ(
t−
τ
)X
q(τ
)
dτ⎫
⎬
⎭
dt= −
T 0 X∗X
q(
t)
+
XqX
∗(
t)
·
⎧
⎨
⎩
q−1 i=1⎧
⎨
⎩
Kλ·
Xi·
t −∞λ(
t−
τ
)X
i(τ)
dτ+
Kμ·
Xi·
t −∞μ(
t−
τ
)X
i(τ
)
dτ⎫
⎬
⎭
−
NF i=1 SiS
i(
t)
⎫
⎬
⎭
dt (35)thatcontainstheunknownfieldsintheleft-hand-sidememberandtheknown(alreadycomputed)fieldsinthe right-hand-sideone.
Now, asdescribed in Appendix A,for computingthe coupleof unknown functionsXq and
X
q(
t)
, we are considering againan alternateddirectionsfixedpoint strategythat computedXq by assumingX
q(
t)
known(it israndomlychosen at the beginning ofthe process),andthen updatingX
q(
t)
from thejustcalculated Xq. The process continue untilreaching convergence,thatis,thefixedpoint.Inwhatfollowwearedevelopingbothsteps.
5.1. CalculationofXq
WhencalculatingXq,
X
q(
t)
isassumedknown(X
∗(
t)
=
0 inEq.(35)),andwithitallfunctionsdependingontime.Thus, alltimeintegralscanbeperformed,leadingtoalinearproblemforcalculatingtheunknownvectorXq.ThefirstintegralinEq.(35)concerns T
0X
q(
t)
⎧
⎨
⎩
t −∞λ(
t−
τ
)X
q(τ
)
dτ⎫
⎬
⎭
dt (36)thatusingthenotationpreviouslyintroducedresults
α
qλ=
T 0X
q(
t)
⎧
⎨
⎩
t −∞λ(
t−
τ
)X
q(τ)
dτ⎫
⎬
⎭
dt=
t2X
Tq·
Lλ· Xq
(37)Similarly,wecandefine:
α
qμ=
T 0X
q(
t)
⎧
⎨
⎩
t −∞μ(
t−
τ
)X
q(τ
)
dτ⎫
⎬
⎭
dt=
t2X
Tq·
Lμ· Xq
,
(38)α
λ q,i=
T 0X
q(
t)
⎧
⎨
⎩
t −∞λ(
t−
τ)X
i(τ
)
dτ⎫
⎬
⎭
dt=
t2X
Tq·
Lλ· Xi
,
(39)α
μq,i=
T 0X
q(
t)
⎧
⎨
⎩
t −∞μ(
t−
τ
)X
i(τ
)
dτ⎫
⎬
⎭
dt=
t2X
Tq·
Lμ· Xi
(40)∀
i∈ [
1,
2,
· · · ,
q−
1]
;andβ
q,i=
T 0X
q(
t)
·
S
i(
t)
dt (41)∀
i∈ [
1,
2,
· · · ,
NF]
;fromwithEq.(35)reducedto:X∗
·
α
λ qKλ·
Xq+
α
qμKμ·
Xq=
X∗·
⎧
⎨
⎩
q−1 i=1α
λ q,iKλ·
Xi+
α
qμ,iKμ·
Xi−
NF i=1β
q,iSi⎫
⎬
⎭
(42)oritsassociatedlinearsystem
α
λ qKλ·
Xq+
α
qμKμ·
Xq=
⎧
⎨
⎩
q−1 i=1α
λ q,iKλ·
Xi+
α
qμ,iKμ·
Xi−
NF i=1β
q,iSi⎫
⎬
⎭
(43)thatcanbesolvedforcalculatingXq
α
qλKλ+
α
qμKμ·
Xq=
⎧
⎨
⎩
q−1 i=1α
λq,iKλ·
Xi+
α
μq,iKμ·
Xi−
NF i=1β
q,iSi⎫
⎬
⎭
(44) or Xq=
α
qλKλ+
α
qμKμ−1·
⎧
⎨
⎩
q−1 i=1α
qλ,iKλ·
Xi+
α
qμ,iKμ·
Xi−
NF i=1β
q,iSi⎫
⎬
⎭
(45) 5.2. CalculationofX
q(
t)
Whencalculating
X
q(
t)
,Xqisassumedknown(X∗=
0 inEq.(35)).Thus,allmatrixproductsinEq.(35)canbecalculated, fromwhichthenextscalarsresult:γ
λ q=
XTq·
Kλ·
Xq (46)γ
qμ=
XqT·
Kμ·
Xq (47)γ
λ q,i=
XTq·
Kλ·
Xi (48)γ
qμ,i=
XTq·
Kμ·
Xi (49)∀
i∈ [
1,
2,
· · · ,
q−
1]
;andδ
q,i=
XTq·
Si (50)∀
i∈ [
1,
2,
· · · ,
NF]
.Byusingpreviousnotation,Eq.(35)reducesto: T
0X
∗(
t)
⎧
⎨
⎩
γ
qλ t −∞λ(
t−
τ)X
q(τ)
dτ+
γ
qμ t −∞μ(
t−
τ)X
q(τ
)
dτ⎫
⎬
⎭
dt= −
T 0X
∗(
t)
⎧
⎨
⎩
q−1 i=1⎧
⎨
⎩
γ
qλ,i t −∞λ(
t−
τ
)X
i(τ)
dτ+
γ
qμ,i t −∞μ(
t−
τ
)X
i(τ
)
dτ⎫
⎬
⎭ −
NF i=1δ
q,iS
i(
t)
⎫
⎬
⎭
dt (51) ort2
X
∗T·
γ
qλLλ+
γ
qμLμ· Xq
= −X
∗T·
⎧
⎨
⎩
t2 q−1 i=1
γ
qλ,iLλ+
γ
qμ,iLμ· Xi
−
t NF i=1δ
q,iSi(
t)
⎫
⎬
⎭
(52)where
S
i isthevectorthatcontainsthevalueofS
i(
t)
attimesn·
t,n∈ [
1,
2,
· · · ,
P]
.Thusthestrongformrelatedto(52) resultsγ
λ qLλ+
γ
qμLμ· Xq
=
⎧
⎨
⎩
q−1 i=1γ
λ q,iLλ+
γ
qμ,iLμ· Xi
−
1t NF i=1
δ
q,iSi(
t)
⎫
⎬
⎭
(53)fromwhichitfinallyresults:
Xq
=
γ
qλLλ+
γ
qμLμ−1·
⎧
⎨
⎩
q−1 i=1γ
qλ,iLλ+
γ
qμ,iLμ· Xi
−
1t NF i=1
δ
q,iSi(
t)
⎫
⎬
⎭
(54)5.3. Separatedrepresentationconstructoroverwiew
– Assumingatiterationq
≥
1 vectorsXi andX
i(
t)
,i∈ [
1,
2,
· · · ,
q−
1]
,known– while
Vq−1(
t)
−
Vq−2(
t)
>
calculateVq
(
t)
=
Vq−1(
t)
+
XqX
q(
t)
by solving until reaching the fixed point thetwo problemsbelow:– calculateXqfromEq.(45)
Xq
=
α
qλKλ+
α
qμKμ−1·
⎧
⎨
⎩
q−1 i=1α
λq,iKλ·
Xi+
α
qμ,iKμ·
Xi−
NF i=1β
q,iSi⎫
⎬
⎭
(55)– calculate
X
q(
t)
fromEq.(54)Xq
=
γ
qλLλ+
γ
qμLμ−1·
⎧
⎨
⎩
q−1 i=1γ
qλ,iLλ+
γ
qμ,iLμ· Xi
−
1t NF i=1
δ
q,iSi(
t)
⎫
⎬
⎭
(56) 6. NumericalresultsIn thissection, we are first verifyingthe proposed strategy by solving a quite simpleproblemand thenaddressing a
more complex problemclose to the one found in assembled systems involving elastomers. As we are considering here
linearbehaviors,itisexpectedthatafteracertaintimetheresponsebecomessteadyharmonic,withacertainphaseangle withrespecttotheappliedload. Thus,simulationsinthelinearcasedonot needtocovertheentirelife period,butonly thetransientregime.
6.1. Strategyverification
For strategy verification, we consider the plane deformation quasi-incompressible viscoelastic model in
= (
0,
L)
×
(
0,
H)
,withL=
1 and H=
1;andI = (
0,
T ]
,withT =
0.
25 (allunitsinthemetricsystem).Aharmonictractionisappliedtotheupperboundary y
=
H givenby tg(
x,
y=
H,
t)
= (
sin(ω
t),
0)
T,withω
=
2π
.The lateralsidesarefree,thatistg(
x=
0,
y,
t)
=
tg(
x=
L,
y,
t)
=
0.Onthelowerboundary,thedisplacementandvelocitiesare enforcedtozero,thatisv(
x,
y=
0,
t)
=
0.WeconsideredtheviscoelasticlawgivenbytheMaxwell’smodel(assumingsmalldisplacementsandstrains)
θ
ddt
+ =
2 Gθ
D (57)whereG denotestheshearmodulusand
θ
therelaxationtime. TheintegralcounterpartoftheMaxwellmodel(57)reads:(
t)
=
t
0
2 G e−t−θτD d
τ
(58)Usingthenotationintroducedintheprevioussectionsweconsider:
λ(
t)
=
e−θtμ(
t)
=
2 G e−tθ(59) Inthenumericaltestscarriedout,weconsidered
largeenoughforensuringthemodelincompressibilityand2G
=
0.
3356.Usingthestrategydescribedintheprevioussectionwecomputedthevelocityfieldrelatedtotheappliedload,andthe displacementwasobtainedbyintegratingthecalculatedvelocity.ForMaxwell’smodelitiswellknownthatthetangentof thephaseangle(anglebetweentheappliedloadandtheresultingdisplacement),tan
(ϕ)
,isrelatedto therelaxationtime andtheappliedfrequencyfrom:tan(ϕ)
=
1ω
θ
(60)Thus,itfollowsfromEq.(60)thattheknowledgeofthephaseangle
ϕ
allowsidentifyingtherelaxationtimeθ
.Tocheck it,wesolvedthejust-presentedmodelforthreedifferentvaluesoftherelaxationtime:θ
1=
0.
05,θ
2=
21π ,andθ
3=
2.Bysolving thethreeviscoelasticproblems,weobtainedthethreeassociated displacementfieldsui
(
x,
t)
, i=
1,
2,
3.Now,the post-treatment ofthe obtainedresults allows calculating thethree phase anglesϕi
, i=
1,
2,
3 and from them thethree relaxationtimesthatwereinperfectagreementwiththeonesthatwerechosenforperformingthecalculation.6.2. Analysisofarigid–viscoelasticjoining
Inthepresentanalysis,weconsideragainasquaredomain
= (−
L,
L)
× (−
H,
H)
,withL=
3 and H=
3,containinga circularholeH(C,
R)
centeredatC =
0 andofradius R=
1.ThesystemwasanalyzedinthetimeintervalI = (
0,
T ]
,withT =
20.Thevelocitywas prescribedonthe domainboundary≡ ∂
,consistingoftheexternalboundarye andofthe internalone(holeboundary)
i
≡ ∂H
,=
e∪
i:vg
(
x∈ e
,
t)
= (
sin(0.1πt2),
0)Tvg
(
x∈ i
,
t)
=
0(61) Thebehaviorlawwasgivenby
⎧
⎪
⎪
⎨
⎪
⎪
⎩
λ(
t)
=
a1e− t b1+
a2e− t b2μ(
t)
= ϒ
c1e− t c1+
c2e−d2t (62) witha1=
a2=
c1=
c2=
1,b1=
5,b2=
0.
1,d1=
10,d2=
0.
5,=
(1+ν)(Eν1−2ν),ϒ
=
2(1E+ν),E=
1 andν
=
0.
3.ThesolutionV
(
t)
involvesonlysixmodesfortheprescribedprecision(Xi,X
i),i=
1,
· · · ,
6,whosefourmostsignificant are depicted inFig. 1.The time-associated functionsX
i(
t)
, i=
1,
· · · ,
4,are depicted inFig. 2. InFig. 3 theapplied dis-placementandtheassociatedtractionarerepresented.From thisfigure,itcan benoticedthat whenthefrequencyofthe applieddisplacementincreases,thetensionamplitudedecreasesandthephaseangleincreases,asexpectedforviscoelastic behaviors.7. Conclusions
In this work, we extended the domain of applicability of space-time separated representations to integro-differential modelsdescribingviscoelasticbehaviors. Theadvantagesinusingsuchdecompositionfollowfromthefactthat spaceand time arediscretized independentlyandthenafine resolutionofbothdiscretizations canbe considered,withoutaffecting the globalefficiency ofthe coupled model.Depending on the analyzed case, thespeeding up can reach some orders of magnitude.
Hereweusedthemostdirectformulationthatonlyinvolveskinematicdegreesoffreedom(velocities);however,amixed formulation(stress velocity)asinthe LATINmethod(see [19]) couldbe envisaged inorderto separate thegloballinear problemfromthelocalonethatdependsonthehistorydespiteofitslinearity.
Another appealing possibility in using such kind of separated representations is the fact of introducing some model parameterasextra-coordinateinordertocalculateageneralparametricsolutiontothetransientinteger-differentialmodel. Thispossibility,andtheconsiderationofnonlinearviscoelasticbehaviors,constitutesomeworkinprogress.
Acknowledgements
FranciscoChinestaacknowledgesthesupportoftheInstitutUniversitairedeFrance (IUF).
Appendix A. Space-timeseparatedrepresentationconstructor
Forthesakeofsimplicity,weconsiderheretheone-dimensionalproblemofcomputingthefieldu
(
x,
t)
governedby∂
u∂
t−
k∂
2uFig. 1. (Color online.) Four most significant modes Xi: (top-left) X1; (top-right) X2; (down-left) X3and (down-right) X4.
Fig. 2. (Color online.) Four most significant modesXi(t), i=1,· · · ,4.
inthespace-timedomain
=
x×
t= (
0,
L)
× (
0,
τ
]
.Thediffusivityk andsourceterm f areassumedtobeconstant.We specifyhomogeneousinitialandboundaryconditions,i.e.u(
x,
t=
0)
=
u(
x=
0,
t)
=
u(
x=
L,
t)
=
0.Morecomplexscenarios wereaddressedin[50].Theweightedresidualformof(63)reads
x×t u∗∂
u∂
t−
k∂
2u∂
x2−
f dx dt=
0 (64)Fig. 3. (Color online.) Applied displacement (blue curve) versus its associated tension (green curve).
OurobjectiveistoobtainaPGDapproximatesolutionintheseparatedform
u
(
x,
t)
≈
N
i=1
Xi
(
x)
·
Ti(
t)
(65)We doso by computingeach termof theexpansion at each stepof an enrichmentprocess, until asuitable stopping criterionismet.
A.1. Progressiveconstructionoftheseparatedrepresentation
Atenrichmentstepn,then
−
1 firsttermsofthePGDapproximation(65)areknown:un−1
(
x,
t)
=
n−1
i=1Xi
(
x)
·
Ti(
t)
(66)WenowwishtocomputethenexttermXn
(
x)
·
Tn(
t)
togettheenrichedPGDsolutionun
(
x,
t)
=
un−1(
x,
t)
+
Xn(
x)
·
Tn(
t)
=
n−1 i=1Xi
(
x)
·
Ti(
t)
+
Xn(
x)
·
Tn(
t)
(67) Onemustthussolvea non-linearproblemfortheunknownfunctions Xn(
x)
andTn(
t)
bymeansofasuitable iterative scheme.Werelyonthesimplebutrobustalternatingdirectionscheme.Atenrichmentstepn,thePGDapproximationun,p obtainedatiteration p isgivenby
un,p
(
x,
t)
=
un−1(
x,
t)
+
Xnp(
x)
·
Tnp(
t)
(68) Startingfroman arbitraryinitialguessTn0(
t)
,thealternating directionstrategycomputes Xnp(
x)
fromTp−1
n
(
t)
,andthenTnp
(
t)
fromX pn
(
x)
.Thesenon-lineariterationsproceeduntilreachingafixedpointwithinauser-specifiedtolerance,i.e. Xnp
(
x)
·
Ynp(
y)
−
Xnp−1(
x)
·
Ynp−1(
y)
<
(69) where
·
isasuitablenorm.Theenrichmentstepn thusendswiththeassignments Xn
(
x)
←
Xnp(
x)
andTn(
t)
←
Tnp(
t)
.The enrichmentprocess itself stopswhen an appropriate measure oferror
E(
n)
becomes smallenough, i.e.E(
n)
<
˜
. Onecanapplythestoppingcriteriadiscussedin[51,52].Letuslookatoneparticularalternatingdirectioniterationatagivenenrichmentstep.
A.2. Alternatingdirectionstrategy
– Calculating Xnp
(
x)
fromT p−1n
(
t)
.Atthisstage,theapproximationisgivenby
un
(
x,
t)
=
n−1
i=1Xi
(
x)
·
Ti(
t)
+
Xnp(
x)
·
Tnp−1(
t)
(70)whereallfunctionsbut Xnp
(
x)
areknown.Thesimplestchoicefortheweightfunctionu∗ in(64)is
u∗
(
x,
t)
=
Xn∗(
x)
·
Tnp−1(
t)
(71) whichamountstoconsideraGalerkinformulationofthediffusionproblem.Introducing(70)and(71)into(64),weobtain
x×t Xn∗·
Tnp−1·
Xnp·
dTnp−1 dt−
k d2Xnp dx2·
T p−1 n dx dt= −
x×t Xn∗·
Tnp−1·
n−1 i=1 Xi·
dTi dt−
k d2Xi dx2·
Ti dx dt+
x×t X∗n·
Tnp−1·
f dx dt (72)Asallfunctionsoftimet areknown,wecanevaluatethefollowingintegrals:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
α
x=
t Tnp−1(
t)
2 dtβ
x=
t Tnp−1(
t)
·
dTnp−1(
t)
dt dtγ
ix=
t Tnp−1(
t)
·
Ti(
t)
dtδ
xi=
t Tnp−1(
t)
·
dTi(
t)
dt dtξ
x=
t Tnp−1(
t)
·
f dt (73)Eq.(72)thentakestheform
x Xn∗·
−
k·
α
x·
d 2Xp n dx2+ β
x·
Xp n dx=
x Xn∗·
n−1 i=1 k·
γ
ix·
d 2X i dx2− δ
x i·
Xi dx+
x Xn∗· ξ
xdx (74)Thisdefinesaone-dimensional boundaryvalueproblem(BVP),whichisreadilysolved bymeansofastandard finite-elementmethodtoobtainan approximationofthefunction Xnp.Asanotheroption,one cangobacktotheassociated strongform
−
k·
α
x·
d2X p n dx2+ β
x·
Xp n=
n−1 i=1 k·
γ
ix·
d2Xi dx2− δ
x i·
Xi+ ξ
x (75)andthensolveitusinganysuitable numericalmethod,such asfinitedifferencesforexample.Thestrongform(75)is asecond-orderdifferentialequation forXnp duetothefactthattheoriginaldiffusionequation(63)involvesa second-orderx-derivativeoftheunknownfield u.
The homogeneousDirichletboundary conditions Xnp
(
x=
0)
=
X pn
(
x=
L)
=
0 arereadilyspecified witheitherweakor strongformulations.– Calculating Tnp
(
t)
fromthejust-computed Xnp(
x)
.The proceduremirrorswhatwehavejustdone.Itsufficestoexchangetherolesplayedbytherelevantfunctionsofx