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Giacomo QUARANTA, Clara ARGERICH MARTIN, Rubén IBÁÑEZ, Jean Louis DUVAL, Elias
CUETO, Francisco CHINESTA - From linear to nonlinear PGD-based parametric structural
dynamics - Comptes Rendus - Mecanique - Vol. 347, n°5, p.445-454 - 2019
Any correspondence concerning this service should be sent to the repository
From
linear
to
nonlinear
PGD-based
parametric
structural
dynamics
Giacomo Quaranta
a,
Clara Argerich Martin
b,
Ruben Ibañez
c,
Jean
Louis Duval
a,
Elias Cueto
d,
Francisco Chinesta
c,
∗
aESIGROUP,99,ruedesSolets,94513Rungiscedex,France
bGeM,ÉcolecentraledeNantes,1,ruedelaNoe,44321Nantescedex3,France cESIGROUPChair,ENSAMParisTech,151,boulevarddel’Hôpital,75013Paris,France
dAragonInstituteofEngineeringResearch,UniversidaddeZaragoza,EdificioBetancourt,MariadeLuna,s.n.,50018Zaragoza,Spain
a
b
s
t
r
a
c
t
Keywords:
Linearandnonlinearparametricdynamics ProperGeneralizedDecomposition Harmonicanalysis
Modalanalysis Lowfrequencydomain
The present paper analyzes different integration schemes of solid dynamics in the frequency domain involving the so-called Proper Generalized Decomposition – PGD. The lastframework assumesfor the solution aparametric dependency with respect to frequency. Thisprocedure allowed introducingotherparametric dependences relatedto loading,geometry,andmaterialproperties.However,inthesecases, affinedecompositions are required for an efficient computation of separated representations. A possibilityfor circumventing such difficulty consists in combining modal and harmonic analysis for defining an hybrid integration scheme. Moreover, such a procedure, as proved in the present work, can be easily generalized to address nonlinear parametric dynamics, as wellastosolveproblemswithnon-symmetricstiffnessmatrices,alwaysoperatinginthe domainoflowfrequencies.
1. Introduction
Governingequationsinsoliddynamicsareusuallyformulatedeitherinthetimeorinthefrequencydomain.Theformer is preferredwhencalculating transient responses, whereas thefrequency approachis an appealingchoice forcalculating forcedresponses.Bothapproacheshavebeenextensivelyusedanddescribedinmanyclassicalbooksas,forinstance,[1].
Timedescriptions are used inboth the linear andthe nonlinear cases,being speciallyefficient when combined with modal analysis. The latter allows expressing the solution in a series of decoupled ordinary differential equations. Other worksconsideredadvanced space–timeseparatedrepresentations[2–4],foraddressing transientdynamics[5–7]. Medium frequencies were efficientlyattainedwithin the VariationalTheoryofComplex Rays—VTCR—proposed by P. Ladeveze and intensively andsuccessfully used (the interested reader can refer to [5] and the numerousreferences therein). Recently, a PGD-based dynamical integrator that takes asa parameter the field of initial conditions—conveniently expressed in a reducedbasis—hasalsobeendeveloped[8,9].
*
Correspondingauthor.E-mailaddresses:giacomo.quaranta@esi-group.com(G. Quaranta),clara.argerich-martin@ec-nantes.fr(C. Argerich Martin),
Ruben.Ibanez-Pinillo@eleves.ec-nantes.fr(R. Ibañez),Jean-Louis.Duval@esi-group.com(J.L. Duval),ecueto@unizar.es(E. Cueto),Francisco.Chinesta@ensam.eu
As discussed in[10–12], problemsbecome abit morecomplex inthe caseofparametrizeddynamics, andmore con-cretely whenthoseparametersdependonfrequency.Inthiscase,frequency-basedmodelingseemsmoreappropriatethan itstimecounterpart,assoonasthefunctionalformsexpressingtheparametricdependenceonfrequencyiscompatiblewith theuseofaspace-frequency-parameterseparatedrepresentation[13–15].
We revisit in next section, for a sake of completeness, the case of linear dynamics and the harmonic-modal hybrid approachdevelopedin[16].
2. Classicallineardynamicsinthetimeandfrequencydomains
Thegeneral,semi-discretizedformofthelinearsoliddynamicsequations,writes
Md
2U
(
t)
dt2+
CdU
(
t)
dt
+
K U(
t)
=
F(
t)
(1)whereM,C andK arerespectivelythemass,damping,andstiffnessmatrices.Massandstiffnessmatricesareusuallypositive definite. U representsthe vectorthat contains thenodaldisplacements andF the nodalexcitations(forces). Theproblem boundaryconditionswereincorporatedwhenthespacediscretizationleading toEq. (1) wasperformed. Thisequation can beobtainedthroughanysuitablemesh-baseddiscretizationtechniquelike,forinstance,theFiniteElementMethod,andcan besolvedassoonasthedisplacementsandvelocitiesarespecifiedattheinitialtime.
BymovingtothefrequencydomainthroughtheFouriertransform
F(•)
—denotingFˆ
(
ω
)
=
F(
F(
t))
andUˆ
(
ω
)
=
F(
U(
t))
—, itresults−
ω
2M+
iω
C+
Kˆ
U(
ω
)
= ˆ
F(
ω
)
(2)Ifdampingvanishes,i.e.C
=
0 (ifit isnot thecase,itisusuallyassumedtobeproportional, C=
a0M+
a1K),andonefocusesonthefreeresponseofthesystem,F
ˆ
(
ω
)
=
0,Eq. (2) reducesto:KU
ˆ
=
ω
2MUˆ
(3)This defines an eigenproblem whose result is givenby the eigenmodes Pi and the associated eigenfrequencies
ω
2i. The inversetransformallowscomingbacktothetimedomain,U(
t)
=
F
−1( ˆ
U(
ω
))
.2.1. Thehybridharmonic-modalapproach
When dampingisneglected, C
=
0 (or whenproportionaldampingisconsideredC=
a0M+
a1K),thesingle-parameter (frequency)dynamicequationreads−
ω
2M+
KUˆ
(
ω
)
= ˆ
F(
ω
)
(4)WenowconsidermatrixP diagonalizingmatricesM andK.Inotherwords,
PM P
= M
PK P= K
where
M
i j=
miiδ
i j andK
i j=
kiiδ
i j.Here,δ
i j representstheKroenecker’sdelta,i.e.M
andK
becomediagonalwithentriesmiiandkii,respectively.
Such a choice impliesthat the systemis no longerdescribed in terms ofits nodaldegrees offreedom, butrather in termsofthemodalcontent.Bothareformallyrelatedthroughthelineartransformation
ˆ
U
(
ω
)
=
Pξ (
ω
)
(5)Thus,thedynamicalproblemreducesto[16]
−
ω
2M + K
ξ (
ω
)
=
PFˆ
(
ω
)
= ˆ
f(
ω
)
(6)whichresultsinasystemofNn decoupledalgebraicequations(Nn beingthesizeofmatricesM andK)
−
ω
2mii+
kiiξ
i(
ω
)
= ˆ
fi(
ω
),
i=
1,
2, . . . ,
Nn (7)fromwhichitresults
ξ
i(
ω
)
=
ˆ
fi(
ω
)
−
ω
2m ii+
kii,
i=
1,
2, . . . ,
Nn (8)Thus,thespace-frequencyseparatedrepresentationreads
ˆ
U(
ω
)
=
Nn i=1 Ziξ
i(
ω
)
whereZiisthei-columnofmatrixP.
TheobtentionofU
ˆ
(
ω
)
allowsustocomebacktothetimedomainU(
t)
byapplyinganinverseFouriertransform,U
(
t)
=
F
−1( ˆ
U(
ω
))
Remark.Asitisusualwhenusingharmonicanalysis,theforcedtimeresponsecanberecoveredatanytime,exceptduring thetransientregime.
Itisimportanttohighlight—thiswillbecruciallaterwhenaddressingnonlineardynamics—thateachterm
ξ
i(
ω
)
involves transformednodalforces,thatis,nodalforcesaffectedbythetransformationmatrix P.Thus,Eq. (9) representsacanonical space-frequency-loadingseparatedrepresentation,whichonlymakesuseofaproportionaldampingassumption.2.2. Extensiontodynamicsinvolvingnon-symmetricstiffnessmatrices
Above,matricesM andK wereconsideredsymmetric,makingitpossibletodiagonalizethemaccordingtoEq. (13).Some dynamicalsystems,whichwillbediscussedlater,likenetworkscomposedofHelmholtzresonators,leadtonon-symmetric stiffness matrices KNS.Therationaleconsidered abovecanbeextendedaccordinglybyjustdefiningthesymmetricmatrix KS insuchawaythat
KS
=
KNS+
KNS (10)andrewritingthedynamicalsystemaccordingto
Md 2U
(
t)
dt2+
KNS+
KNS−
KNS U(
t)
=
F(
t)
(11) or,equivalently, Md 2U(
t)
dt2+
KSU(
t)
=
F(
t)
+
K NSU(
t)
(12)Remark.ThedecompositionofKNS initssymmetricandskew-symmetriccomponentscanalsobeconsidered. Thus,theleft-handsideofthepreviousequationcanbediagonalizedaccordingto
PMP= M
PKSP= K
(13) whichleadsto−
ω
2M + K
ξ (
ω
)
=
PFˆ
(
ω
)
=ˆ
f(
ω
)
(14)where,inthepresentcase,F
ˆ
(
ω
)
istheFouriertransformoftheright-handsideofEq. (12),i.e.ˆ
F
(
ω
)
=
F
(
F(
t)
+
KNSU(
t))
(15) Thistermwillbelinearized,intheframeworkofaniterativescheme,bysimplyevaluatingU(
t)
atthepreviousconverged iteration.ThetimesolutioncanbeobtainedbyapplyingtheinversetransformtothefrequencysolutionU
ˆ
(
ω
)
=
Pξ (
ω
)
,according to3. Nonlineardynamics
Inthenonlinearcase,thegeneralsemi-discretizedequilibriumequationwrites
Md
2U
(
t)
dt2+
CdU
(
t)
dt
+
KU(
t)
−
H(
U)
=
F(
t)
wherethenonlinearcontributionisgroupedinthevectorH
(
U)
andagainaproportionaldampingisassumed.The simplestlinearization of thisequation consists in an explicit evaluation ofthe non-linear term H
(
U)
atthe last converged iteration. For the sake of notational simplicity, it will be denoted by U−(
t)
. Thus, as soon as the nonlinear contributionisassumedknown,itcanbemovedtotheright-hand-sidemember,i.e.Md 2U
(
t)
dt2+
C dU(
t)
dt+
KU(
t)
=
H(
U −(
t))
+
F(
t)
(16)AnobviouspossibilityconsistsincomputingtheFouriertransformoftheright-handmemberofEq. (16),
ˆ
F
(
ω
)
=
F
(
H(
U−(
t))
+
F(
t))
and then toproceed exactly inthe samewayas inthe linearcase. However, in orderto take benefitfrommodel order reduction,inwhatfollowswepresentanalternativebutequivalentformulation,moreadaptedtotheuseofreducedbases.
ByinvokingthelinearityofFouriertransform,wewrite
ˆ
F
(
ω
)
=
F
H(
U−(
t))
+
F(
t)
=
F
H(
U−(
t))
+
F
(
F(
t))
= ˆ
FH(
ω
)
+ ˆ
FF(
ω
)
which could be expressed using a piecewise linearapproximation basis Nl
(
ω
)
(like the usually considered one in linear finiteelementanalyses)ˆ
FH(
ω
)
=
L l=1ˆ
FH(
ω
l)
Nl(
ω
)
orˆ
fH(
ω
)
=
PFˆ
H(
ω
)
Inturn,ˆ
FF(
ω
)
=
L l=1ˆ
FF(
ω
l)
Nl(
ω
)
orˆ
fF(
ω
)
=
PFˆ
F(
ω
)
eachonecontributingtothesolution
ξ (
ω
)
accordingtoξ (
ω
)
= ξ
H(
ω
)
+ ξ
F(
ω
)
(17) withξ
iH(
ω
)
=
L l=1ˆ
fiH(
ω
l)
Nl(
ω
)
−
ω
2m ii+
kii,
i=
1,
2,
· · · ,
Nn (18)ξ
iF(
ω
)
=
L l=1ˆ
fiF(
ω
l)
Nl(
ω
)
−
ω
2m ii+
kii,
i=
1,
2,
· · · ,
Nn (19)Thus,itfinallyresults
ˆ
U
(
ω
)
=
Pξ (
ω
)
= ˆ
UH(
ω
)
+ ˆ
UF(
ω
)
=
Pξ
H(
ω
)
+
Pξ
F(
ω
)
beingitstime-dependentcounterpart
Fig. 1. Simple case study.
Itisimportanttohighlightthattheinversetransforminvolvesterms
T
il,i=
1,
. . . ,
Nnandl=
1,
. . . ,
L
,whoseformreadsT
il≡
F
−1Nl
(
ω
)
−
ω
2mii
+
kiithatcanbecomputedofflineandstoredinmemory.
Our numerical experiments reveal that this offline–online procedure does not allow for significant computing time savings when one considers a standard piecewise linearapproximation bases. However, by considering reducedbases to approximatefunctionsF
ˆ
H(
ω
)
andFˆ
F(
ω
)
,thenumberofintegralstobeperformeddrasticallyreducesandtheoffline–onlineprocedurebecomesvaluable.Thesereducedbasisarecomputedofflineafteranadequatetrainingstage.Moreover,the com-putationalcostfortheevaluationofthenonlinearterminthetime domainH
(
U−(
t))
canbe drasticallyreducedbyusing empiricalinterpolationtechniques[17–19].As soonasreduced basesare available inthe frequencydomain,their counterpart inthe time domainis easily com-putable,andfromit,directFouriertransformcouldbealsocomputedoffline.Theuseofstrategiesbasedontheusereduced basesconstitutesaworkinprogress.
4. Numericalresults
4.1. Nonlineardynamics
Inthissection,weusetheproposedstrategytoaddressthe1DdynamicsofarodoflengthL,crosssection A,clampedat itsleftboundaryandsubjectedtoanaxialloadappliedonitsrightboundaryasdepictedinFig.1,subjectedtohomogenous displacementandvelocityinitialconditions.
Ifalinearelasticbehaviorisassumed,therelationbetweenthestress
σ
andthestrainε
readsσ
=
Eε
where E istheYoung modulus.Ineverysimulation,weused L
=
1 m, E=
2·
109 N/m2,andρ
=
8000 kg/m3,whereρ
isthedensity.Whendampingvanishesorwhenproportionaldampingisassumed,themechanicalresponseiscomputedfrom thediscretesystem
Md
2U
(
t)
dt2
+
KU(
t)
=
F(
t)
(20)whoseexpressioninthefrequencydomainisgivenbyEq. (4).However,whenanonlinearelasticbehaviorisassumed,the stress–strainrelationreads
σ
=
C
(
ε
)
where
C
isanon-linearfunctionofε
.Inthenumericaltesthereaddressed,itisassumedthatσ
=
E(
ε
+
cε
3)
(21)wherec canbeconsideredasaparameter(obviously,whenc
=
0,thenonlinearcasereducestothelinearcase). RecallingthelinearizationdescribedinSection3,themechanicalresponseiscomputedfromthediscretesystemMd
2U
(
t)
dt2+
CdU
(
t)
dt
+
KU(
t)
=
c H(
t)
+
F(
t)
(22)whereH
(
t)
accountsforthenonlinearcontribution.Thus,usingthenotationintroducedintheprevioussection,thefrequencyandtimedomainsolutionsread
ˆ
U
(
ω
;
c)
= ˆ
UF(
ω
)
+
cUˆ
H(
ω
)
(23) thatcanbeseenasaparametricsolutioninvolvingboththefrequencyω
andtheparameterc controllingthenonlinearity, withitstimecounterpartexpressedfromFig. 2. Displacement at the rod right border for two different values of c: (top) c=0, which corresponds to the linear case and (bottom) c=220.
Table 1
HybridvsNewmarkmethods,T=4 s.
Hybrid Newmark Nn=100 1.34 s 5.97 s Nn=250 3.26 s 14.29 s Nn=500 7.19 s 39.08 s Nn=750 11.70 s 68.12 s Nn=1000 16.33 s 100.75 s Table 2
HybridvsNewmarkmethod,Nn=100.
Hybrid Newmark
T=4 s: 1.34 s 5.97 s
T=8 s: 2.44 s 12.20 s
T=20 s: 6.32 s 29.75 s
T=40 s: 13.54 s 59.30 s
Both formulations,theonedefinedinthetime domainandtheharmonic-modal hybridformulationsaresolved inthe timeinterval I
= [
0,
T]
withT=
4 s,thefirstbyusingastandardNewmarktime-stepping(withtimestept
=
10−2s).Fig.2comparesbothsolutionsfortwodifferentvaluesoftheparameterc:c
=
0,whichcorresponds tothelinearcase andc=
220 forthesimpleharmonicloadingF(
t)
=
1010sin(
2π
t)
.Thecomputedresultsagreeperfectly,butwhenusingthe hybrid solver,significantcomputingtime savingsare noticed.ThesearesummarizedinTables 1and2.Moreover, Table1 alsoreflectsthefactthatcomputingtimesavingremainsalmostindependentoftheconsideredmeshsize.Finally,weconsideramorecomplexloadingscenario,asdepictedinFig.3.Itcontainsaricherfrequencyspectrum,with
c
=
0 andc=
4000.ThesolutionswhenusingtheNewmark(t
=
10−3 s)versustheharmonic-modalhybridschemesareFig. 3. Loading containing a richer frequency spectrum.
Fig. 4. Displacement at the rod right border for two different values of c: (top) c=0, that corresponds to the linear case and (bottom) c=4000.
Table 3
HybridvsNewmarkmethod,T=4 s.
Hybrid Newmark Nn=100: 2.49 s 9.68 s Nn=250: 6.90 s 23.38 s Nn=500: 14.28 s 48.26 s Nn=750: 21.90 s 69.72 s Nn=1000: 30.13 s 93.11 s
Table 4
HybridvsNewmarkmethod,Nx=100.
Hybrid Newmark
T=4 s: 2.49 s 9.68 s
T=8 s: 4.95 s 21.32 s
T=20 s: 13.57 s 59.48 s
T=40 s: 28.29 s 122.23 s
Fig. 5. Case of study for non-symmetric dynamics.
4.2. Non-symmetricstiffness
Inthissection,weconsideranetworkofHelmholtzresonators,depictedinFig.5.Here,theoscillationamplitude ofthe airineach pipecontributestothecompressionoftheairfillingthecavitiesconnectingdifferenttubes.Thatcompression results in a pressure variation that generates an extra force, acting atthe entrance section of each tube. In each tube, the inertia contribution involves the second time-derivative of the oscillation amplitude. The whole model results in a quitestandard dynamicalsystemwiththeonlyexception ofcontaining anon-symmetricstiffnessmatrix.Inthat case,the modal-harmonic hybridmethodisappliedasdescribedinSection 2,whichproceedsbysymmetrizingtheproblembefore applyingthehybridstrategy.
AscanbenoticedfromFig.5,thegeometryofthesystemfollowsafractalstructure.Therefore,thegeometrical charac-teristics ofsubsequentgenerationswillbedeterminedbythecharacteristicsofthefirstgeneration:thetubecrosssection,
A1,thetubelengthl1,andthecavityvolume, V1.Subsequentgenerationsarecharacterizedbyagivenreductionfactor,in
ourcaseof 2.Standardthermodynamicalairpropertiesareconsidered. ThemodelofasingleHelmholtzresonatorreads
ρ
A Ld 2U(
t)
dt2
+
γ
p A2V U
(
t)
=
f(
t)
and,byassemblingthecontributionofthoseinvolvedinthenetworkofFig.5,itresults
Md 2U
(
t)
dt2+
K U(
t)
=
F(
t)
where Mii=
ρ
AiLi Ki j=
γ
P0AiAj 1 Vi j,
for i=
jwith j referringtoatubeconnectedtooneofthecavitiesassociatedwithtubei,withvolumeVi j,and
Kii
= −
γ
P0A2i Ni j=1 1 Vi jFig. 6. Displacement of the air filling the first tube.
The discrete systeminvolves the dynamicsof the airdisplacement in each tubeand its only specificityconcerns the factofhavinganon-symmetricstiffnessmatrix.Fig.6comparesthesolutionobtainedbyusingthemodal-harmonichybrid techniquesdescribed in Section 2andthe referencesolutionobtainedby a standard time-steppingintegration.A perfect agreementbetweenbothsolutionscanbenoticed.Theconvergenceisevaluatedfrom
=
||
Ui(
t)
−
Ui−1(
t)
||
||
Ui(
t)
||
wherethe superscripti refersto theiteration within thehybridstrategy. The convergenceisattainedvery fastafterfew iterations,threebeingingeneralenoughforreducingtheerrorbymorethanthreeordersofmagnitude.
5. Conclusions
Inthisnote,weproposedanextension ofthehybridmethodologycombiningharmonicandmodalanalysesfortreating nonlinearparametricdynamics.
The main contribution of the present work is the derivation of a parametric solution in the frequency space. Apart fromits naturaldependenceon frequency,the above-developedmethod alsoaccountsforother modelparameters. More importantly, itexplicitly decouples the dependenceon the amplitude of nodalloading. Thislast fact makes possiblethe solutionofnonlinearmodelscombinedwithsimplelinearizations.
Preliminary numerical results evidence the potentialities of the proposed technique, while proving its computational efficiency.
Acknowledgements
This work has been supported by the Spanish Ministry of Economy and Competitiveness through Grants Nos. DPI2017-85139-C2-1-RandDPI2015-72365-EXP,aswellasbytheRegionalGovernmentofAragonandtheEuropeanSocial Fund,researchgroupT2417R.ThisprojecthasalsoreceivedfundingfromtheEuropeanUnion’sHorizon2020researchand innovationprogrammeundertheMarieSklodowska-CuriegrantagreementNo. 675919.
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