Thermal and thermo-mechanical solution of laminated composite beam based on a variables separation for arbitrary volume heat source locations

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Thermal and thermo-mechanical solution of laminated composite beam based on a variables separation for

arbitrary volume heat source locations

Philippe Vidal, Laurent Gallimard, Isabelle Ranc, Olivier Polit

To cite this version:

Philippe Vidal, Laurent Gallimard, Isabelle Ranc, Olivier Polit. Thermal and thermo-mechanical solu- tion of laminated composite beam based on a variables separation for arbitrary volume heat source lo- cations. Applied Mathematical Modelling, Elsevier, 2017, 46, pp.98-115. �10.1016/j.apm.2017.01.064�.

�hal-01676516�

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Thermal and thermo-mechanical solution of laminated

composite beam based on a variables separation for arbitrary volume heat source locations

P. Vidal

^∗

, L. Gallimard , I. Ranc , O. Polit

UPL, Univ Paris Nanterre, Laboratoire Energétique, Mécanique, Electromagnétisme, 50 rue de Sèvres - Ville d’Avray, 92410 France

a b s t r a c t

Inthiswork, amethod tocompute explicitthermal solutions for laminatedand sand- wichbeamswitharbitraryheatsourcelocationisdeveloped.Thetemperatureiswritten asasumofseparatedfunctionsoftheaxialcoordinatex,thetransversecoordinatezand thevolumetricheatsourcelocationx₀.Thederived non-linearproblemimpliesanitera- tiveprocessinwhichthree1Dproblemsaresolvedsuccessivelyateachiteration.Inthe thicknessdirection,afourth-orderexpansionineachlayerisconsidered.Fortheaxialde- scription,classicalFiniteElementmethodisused.Thepresentedapproachisassessedon variouslaminatedandsandwichbeamsandcomparisonswithreferencesolutionswitha ﬁxedheatsourcelocationareproposed.Basedontheaccurateresultsofthethermalanal- ysis,thermo-mechanicalresponseisalsoaddressedusingalsoaseparatedrepresentation.

1. Introduction

Composite andsandwich. structures areincreasingly used inindustrialapplications owingto their specificmechanical properties. Theycan be also submitted to multi-physics environments such as thermal effects, andin particular due to theinternal volumeheat source.Someapplicationscan bementioned.Inthe thermallyactivated composite(smartmate- rial), thedeformationofthestructurecan becontrolled byactingontheinternalheat source[1].Itrequiresto knowthe thermo-mechanicalresponseofthestructureforvariouslocalizedthermalsollicitations.Theidentificationofthedamageof compositecanbe viewedasanotherillustration.Infact,thedeterminationofthepositionoftheheatsourceallows some researcherstolocalizethedamagedregion,asin[2]forcarbonfibrereinforcedlaminates.Inall thesescases,theclassical way consistsin performing differentcomputations witha fixed value ofthe heat source position. The presentapproach basedonthe variablesseparationaimsatbuildingtheexplicitthermalsolutionwithrespecttothislocationavoidingthe costofnumerouscomputations.Then,thermo-mechanicalanalysiscanbeperformed.

Inopenliterature,variousmechanicalmodelsdedicatedtocompositeorsandwichbeamstakeintoaccountthethermal effects withafixed value of theheat sourcelocation. Theycan be classifiedintotwo familiesasEquivalent Single Layer (ESL)orLayerwise(LW)models.IntheESLmodels,thenumberofunknownsremainlowasitisindependentofthenum- ber oflayers. Unfortunately, the physicalrequirements such asthe shearstress continuity atthe layer interfacesare not fulfilled.Inthisfamily,differenttypesofmodelscanbedistinguished:theclassicallaminatetheory[3],thefirstordershear

∗ Corresponding author.

E-mail address: philippe.vidal@u-paris10.fr (P. Vidal).

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deformationtheory ([4]witha linearvariationoftemperaturethrough thethickness),andhigherordertheories [5,6]for thermalstressesanalysisofbeamsorplates.ThelimitationsoftheESLapproachcanbeovercomebytheLWmodels.The thermaleffectforbeamstructures hasbeenincludedin[7].But,thecostbecomesvery highasthenumberofunknowns increaseswiththenumberofplies.Itshouldbenotedthesubstantial worksdevelopedbyCarrera([8,9]).VariousESLand LWtheorieswithmixedanddisplacement-basedapproachesforassumeddistributionoftemperatures(uniform,linear,localized)are addressedintheframework oftheCarrera’s Uniﬁed Formulation(CUF).Different displacementorstress ﬁeld approximationsoverthecross-sectionareconsidered.Afamilyofhigher-orderbeamelementscanbededuced.

Analternativestrategyconsistsinconsideringaprioriphysicalrequirementsinthekinematics.Thus,reﬁnedmodelscan bebuilttoimprovethe accuracyofESLmodelsavoidingtheadditionalcomputational costofLWapproach.Inthisway,a zigzagtheoryforbeamshasbeencarriedoutbyKapuriain[10].Apiecewiselinearfunctionalongthethicknessisusedfor thetemperature. In[11],aglobal-local approach inconjunctionwitha Hermiteinterpolation ofthe temperaturethrough thethicknessisused.Foranoverviewaboutthethermalanalysis,see[5,12–14].

Until now, the so-calledProperGeneralized Decomposition (PGD) [15]has shownadvantageous characteristics to de- creasesigniﬁcativelythe computational costs intheframework of thereduction model.Indeed,it allows todecrease the dimensionalityofnumerousphysicalproblems.Forinstance,thecoordinatevariablesseparationinmulti-dimensionalPDEs hasbeenperformedin[15].Moreover,thisapproachhasbeencarriedout in[16,17]formechanicalapplicationsinvolving compositebeamsandplates. Concerningthethermalanalysis,the variablesseparation ofthespatialcoordinates xandz hasalreadybeenused in[18]forlaminates.Other parametrizedthermalproblemshave beenalsoconsidered in[19–21]. Notethat theintroductionoftheloadpositionasaparameterofthesolutionintheframeworkofmechanicalsimulations hasbeencarriedoutin[22].ForareviewaboutthePGDanditsﬁeldsofapplications,thereadercanreferto[23,24].

This work aims at modeling composite beam structures regardless of the volume heat source location for thermo- mechanicalanalysis.Thisparticularstudyhasnotbeenyetaddressedinthepreviousquoted works.Forthispurpose,the presentapproachisbasedontheseparatedrepresentationwherethetemperatureiswrittenasasumofproductsofunidi- mensionalpolynomialsofxcoordinate,zcoordinate,andalsothevolumethermalheatpositionx₀.Apiecewisefourth-order Lagrangepolynomialofzischosenasitisparticularlysuitabletomodelcompositestructures.Asfarasthevariationwith respecttotheaxial coordinateisconcerned,a1D three-node beamFinite Element(FE)isemployed. Thefunctionsofthe volumeheatsourcepositionarealsopiecewisequadratic.Finally,thededucednon-linearproblemislinearized.Thus,three 1D linearproblems aresolved alternatively, inwhich the numberofunknownsis much smallerthan ina Layerwise approach.Finally,thesolutiondependsexplicitlyontheloadposition.

Then,thethermo-mechanicalanalysiscanbeaddressedasaweakcouplingisconsidered.Thesolutionofthemechanical problem,alsowrittenunderaseparatedformasin[25],isbuiltby consideringthetemperaturedistribution asaload. As theseparatedform ofthetemperatureunknowns arechosen, the natureof themechanicalproblemremains unchanged.

Moreover,the approachallows usto reﬁneeasily the descriptionofthe temperaturevariation withoutincreasing signiﬁ- cantlythecomputationalcostwhichisimportanttoderiveanaccuratemechanicalresponse.

This article is organized as follows. Firstly, the thermo-mechanical formulation is recalled. The resolution of the parametrizedthermal problembased onthe variables separationis described. The particularassumption on thetemper- aturefield yieldsanon-linearproblemwhichissolvedby aniterativeprocess.Then,theFEdiscretizationisgiven.Finally, numericaltestsareprovided.Boththermalloadsandthermo-mechanicalcouplingareconsideredforlaminatedandsand- wichbeams.Thebehaviorofthemethodisillustratedanddiscussed.Itisassessedbycomparingwith2DFEMcomputations issuedfromacommercialfiniteelementsoftwarewithdifferentfixedheatsourcelocationsandsizes.Theresultsshowthe accuracyandtheefficiencyofthepresentedmethod.

2. Thermo-mechanicalproblemdescription

AlaminatedorsandwichbeamwithNClayers isdeﬁnedinadomain B=Bx×By×Bz=[0,L]×[−^b₂≤y≤₂^b]×[−^h₂≤ z≤ ^h2]expressedinaCartesiancoordinate(x,y,z).Thecross-sectionofthebeamisrectangular.handbaretheheightand thewidth,respectively.Thecentrallineofthebeamischosenasthexaxis.ItisshowninFig.1.

Inthiswork,aplanestressassumptionisused(σ12=σ23=σ22=0).

x

h/2

y

y z z

L

b

Fig. 1. The laminated beam and co-ordinate system.

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2.1. Heatconductionproblem

2.1.1. Thermalconstitutiveequation

Here, thetwo-dimensional linear heat conductionequations are considered.The constitutive equation is givenby the Fourierlaw

Q=

q1

q3

=−

λ

¯^grad(

θ

)^, ⁽¹⁾

where

λ

¯ ⁼

λ

¹ ⁰

0

λ

3

andgrad(

θ

)=

θ

,x

θ

,z

.

λ1,λ3 arethethermalconductivitycomponents,q₁,q₃aretheheatﬂuxcomponents.Theyareexpressedinthecoordinate systemx,z.Thederivativeofthetemperaturewithrespecttoxandzaredenotedθ^,^x^,θ^,^z^,respectively.

2.1.2. Theclassicalweakformoftheboundaryvalueproblem

Forδθ ∈δ⁽δ={θ∈H¹(B)/θ=0on∂B_θ}^),^thevariationalprincipleisgivenby:

ﬁndθ ∈^such^that

Bgrad(

δθ

)^T^Q(

θ

)^dB+

B

δθ

^rddB+

∂Bh

δθ

^hdd

∂

B=0,

∀ δθ

^∈

δ

⁽²⁾

wherer_dandh_daretheprescribedvolumeheatsourceandsurfaceheatfluxappliedon∂Bh.Thetemperatureθdisimposed on∂B_θ.îs^the^spaceôfâdmissibletemperatures,i.e.={θ∈H¹(B)/θ=θd on∂B_θ}^.

Inthepresentwork,theconvectionheattransferandtheprescribedsurfaceheatﬂuxarenotconsidered.

2.2. Thermo-mechanicalproblem

2.2.1. Constitutiverelation

Thereducedtwodimensionalconstitutivelawofthek^th layerwiththermo-mechanicalcouplingisgivenby

σ

⁽^k⁾=¯C⁽^k⁾

ε

⁽^k⁾−

α

¯⁽^k⁾

θ

⁽^k⁾

, (3)

whereθ⁽^k⁾=θ⁽^k⁾−θ0isthevariationoftemperature.

Thestressandstraintensoraresupposedtobewrittenas

σ

^T⁼^[

σ

¹¹

σ

³³

τ

¹³^]^,

ε

^T⁼^[

ε

¹¹

ε

³³

γ

¹³^]^.

Wehavealso

¯C⁽^k⁾=

⎡

⎣

C¯₁₁⁽^k⁾ C¯₁₃⁽^k⁾ 0 C¯₃₃⁽^k⁾ 0 symm C¯₅₅⁽^k⁾

⎤

⎦

,

whereC¯_{i j}⁽^k⁾arethereducedmoduliofthematerialforthek^thlayerissuedfromtheplanestressassumptions(σ22=0).They aregivenby

C¯_{i j}⁽^k⁾=C_{i j}⁽^k⁾−C_i⁽₂^k⁾C⁽_j2^k⁾/C₂₂⁽^k⁾. (4)

C_{i j}⁽^k⁾are orthotropic three-dimensional elastic moduli. And, we have ¯C₅₅⁽^k⁾=C₅₅⁽^k⁾. Thevectorofthethermalexpansioncoeﬃcientsisα¯⁽^k⁾^T =[α1⁽^k⁾α3⁽^k⁾0].

2.2.2. Theweakformoftheboundaryvalueproblem

The formulationofthe thermo-mechanicalproblemis writtenundera displacement-basedapproach forgeometrically linearelasticbeams.UsingtheabovenotationandEq.(3),thevariationalprincipleisgivenby:

Findu(M)∈Usuchthat

B

δε

^T^¯C⁽^k⁾

ε

dB=

∂VF

δ

^u·t dS+

B

δε

^T^¯C⁽^k⁾

α

^¯⁽^k⁾

θ

dB

∀ δ

^u∈

δ

U, (5) where t is the prescribed surface forces applied on ∂VF. U is the space of admissible displacements, i.e. U={û∈ (^H¹(B))²/u=u_don∂Bu}^.^We^haveâlsoδÛ={û∈(^H¹(B))²/u=0on∂Bu}^.^The^body^forcesâre^not^consideredⁱⁿÊq.⁽⁵⁾^.

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Fig. 2. Volume heat source applied on the laminated beam.

3. Applicationofthevariablesseparationtothethermalanalysisofbeamwithanyvolumeheatsourcelocations

TheProperGeneralizedDecomposition(PGD)hasbeenintroducedin[15].Itisbasedonanaprioriconstructionofsepa- ratedvariablesrepresentationofthesolution.Intheframeworkofthermalproblems,thesolutionwithseparatedcoordinate variableshasbeendevelopedin[19].Thefollowing sectionsarededicatedtothe introductionofthisapproachtobuild a parametricsolutionforcompositebeamsubmittedtoheatsourceswithanylocations.TheproblemdeﬁnedbyEqs.(1)and (2)isconsideredasaparametrizedproblemwheretheheat sourcelocation x₀isdeﬁnedinaboundedinterval[0,L].The thermalsolutionforapointMofcoordinate(x,z)ofthestructuredependsonthevaluesofthisparameterandisdenoted

θ⁽^x,^z,^x0)(seeFig.2).

Inthepresentwork,aweakthermo-mechanicalcouplingisaddressed.Oncethethermalproblemhasbeensolved,the mechanicalresponseisdeducedusingalsothevariableseparationfollowing[17].Itwillnotbedetailedhereafterforbrevity reason.OnlythecouplingtermisdetailedinAppendixA.

3.1. Theparametrizedproblem

Thetemperaturesolutionθ îs^builtâs^the^sumôf^N ^productsôf^functionsôfônlyône ^spatial^coordinate⁽^N∈Nisthe orderoftherepresentation)suchthat:

θ

(x,z,x0)= N

i=1

fⁱ(^x0)^Txⁱ(^x)^Tzⁱ(^z), (6)

whereT_zⁱ(^z)îs^definedônBzandT_xⁱ(^x),fⁱ(x₀) aredefinedonBx.AclassicalquadraticFEapproximationisusedinBx anda LWdescriptionischoseninBz asitisparticularlysuitableforthemodelingofcompositestructures.

Theexpressionofthetemperaturegradientiswrittenasfollows:

grad(

θ

)= N

i=1

fⁱ(^x0)

_T_i

x,x(^x)^Tzⁱ(^z) T_xⁱ(^x)^T_z,zⁱ (^z)

. (7)

Fora structuresubmitted to a localizedinternal heat source atx=x0 in one layer, theparametrized problemcan be formulatedas:

findθ ∈êxt⁽êxt={θ ∈H¹(B×[0,L])/θ =θdon∂B_θ×[0,L]}⁾^such^that:

a(

θ

^,

δθ

)=b(

δθ

),

∀ δθ

^∈

δ

^ext, (8)

with

a(

θ

^,

δθ

)⁼⁻

x0

Bgrad(

δθ

)^T^Q(

θ

)^d^Bd^x0

b(

δθ

)=

x0

B^zs

δθ

^rd

δ

(^x−x0)^dBzsdx0

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whereδ^is^the^DiracdistributionandBz_s representsthelocationwheretheinternalheatsourceisapplied(seeFig.2).

Theassumption on theexpression ofthe temperaturegivenin Eq.(6)impliesthe resolution ofa non-linearproblem whichisdescribedinthefollowingsection.

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3.2. Resolutionoftheparametrizedproblem

A greedy algorithm is used to solve Eq. (8). If we assume that the ﬁrst m functions(cf. Eq.(6)) have been already computed,thetrialfunctionfortheiterationm+1canbegivenby

θ

^m⁺¹(^x,z,x₀)=

θ

^m(^x,z,x₀)+f(^x0)^T^x(^x)^T^z(^z), (10) wheref,TxandTz aretheunknownfunctionsandθ^mîs^theâssociated^known^setsâtîteration^m^defined^by

θ

^m(x,z,x0)= m

i=1

fⁱ(^x0)^Txⁱ(^x)^Tzⁱ(^z). (11)

TheproblemtobesolvedgiveninEq.(8)canbewrittenas

a(^f^TxTz,

δθ

)=b(

δθ

)−a(

θ

^m,

δθ

). (12)

Thetestfunctionis

δ

(^f^TxTz)=

δ

^f^T^x^T^z⁺^f

δ

^T^x^T^z⁺^f^T^x

δ

^T^z^. ⁽¹³⁾

Introducing the test function deﬁned by Eq. (13) and the trial function deﬁned by Eq. (10) into the weak form Eq.(12)leadstothethreefollowingproblems:

• forthetestfunctionδ^Tx

a(^f^T^z^T^x,fTz

δ

^T^x)=b(^f^T^z

δ

^T^x)−a(

θ

^m,fTz

δ

^T^x),

∀ δ

^T^x, (14)

• forthetestfunctionδ^T^z

a(^f^TxTz,fTx

δ

^T^z)=b(^f^Tx

δ

^T^z)−a(

θ

^m^,^f^T^x

δ

^T^z),

∀ δ

^T^z^, ⁽¹⁵⁾

• forthetestfunctionδ^f

a(^TxTzf,TxTz

δ

^f)=b(^TxTz

δ

^f)−a(

θ

^m^,^T^x^T^z

δ

^f),

∀ δ

^f. ⁽¹⁶⁾

FromEqs.(14)–16,acouplednon-linearproblemtobesolvedisdeduced.Classically,aﬁxedpointmethodiscarriedout forsimplicityreason. Starting froman initial functionT˜_x⁽⁰⁾, T˜_z⁽⁰⁾, f˜⁽⁰⁾, we constructa sequenceT˜_x⁽^k⁾,T˜_z⁽^k⁾ and f˜⁽^k⁾ which satisfyEqs.14–16,respectively.Foreachproblem,onlyoneunknown 1Dfunctionhastobefound,thetwoother onesare assumedtobe known(fromtheprevious stepoftheﬁxed pointstrategy). So,theapproachleadstotheprocess givenin Algorithm1.

Algorithm1 Presentalgorithm.

forn=1toNmaxdo InitializeT˜_x⁽⁰⁾,T˜_z⁽⁰⁾, f˜⁽⁰⁾ fork=1tokmaxdo

Compute f˜⁽^k⁾fromEq.(16)(straightforwardresolution),T˜_x⁽^k⁻¹⁾,T˜_z⁽^k⁻¹⁾beingknown ComputeT˜_z⁽^k⁾fromEq.(15)(linearequationonBz), f˜⁽^k⁾andT˜_x⁽^k⁻¹⁾beingknown ComputeT˜_x⁽^k⁾fromEq.(14)(linearequationonBx), f˜⁽^k⁾andT˜_z⁽^k⁾beingknown Checkforconvergence

endfor

Set fⁿ⁺¹=f˜⁽^k⁾,T_zⁿ⁺¹=T˜_z⁽^k⁾,T_xⁿ⁺¹=T˜_x⁽^k⁾ Setθⁿ⁺¹=θⁿ+fⁿ⁺¹T_zⁿ⁺¹T_xⁿ⁺¹

Checkforconvergence endfor

Theﬁxedpointalgorithmisstoppedwhen

^f⁽^k⁺¹⁾^Tx⁽^k⁺¹⁾T_z⁽^k⁺¹⁾−f⁽^k⁾T_x⁽^k⁾T_z⁽^k⁾

B×Bx

^f⁽⁰⁾^Tx⁽⁰⁾T_z⁽⁰⁾

B×Bx

≤

ε

^, ⁽¹⁷⁾

where^AB×Bx=

Bx

Bz

BxA²dxdzdx₀1/2

andεîsâ^small^parameter^to^be^fixed^by^theûser.În^the^present^study,^we^set ε=10⁻⁶.

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3.3.FEdiscretization

Tosolvethebeamproblempreviouslyintroduced,adiscreterepresentationofthefunctions(Tx,Tz)ischosen.Aclassical finiteelementapproximationinBx andBz isused.Theelementaryvectorsofdegreeoffreedom (dof)associatedwiththe finiteelementmeshinBx andBz are denotedq^x_e andq^z_e,respectively.The temperaturefieldsandthe associatedgradient arecomputedfromthevaluesofq^x_e andq^z_e by

Txe=Nxq^x_e, Ex^e=Bxq^x_e,

Tze=Nzq^z_e, E_z^e=Bzq^z_e, (18)

where Ex^eT

=[T_x,x T_x] and Ez^eT

=[T_z T_z,z]. The matrices [N_x], [B_x], [N_z], [B_z] contain the interpolation functions, their derivativesandthejacobiancomponents.Theyarebuiltfollowingthechosendiscreterepresentation.

3.4.FiniteelementproblemtobesolvedonBx

Forthesakeofsimplicity,thefunctionsf˜⁽^k⁾andT˜_z⁽^k⁾whichareassumedtobeknown,willbedenoted f˜andT˜_z,andthe functionT˜_x⁽^k⁾tobecomputedwillbedenotedTx.ThegradientinEq.(14)canbewrittenas

grad(^T^˜zTx)=z(^T^˜z)Ex, (19)

with

z(^T^˜z)=

_˜

Tz 0 0 T˜z,z

. (20)

ThevariationalproblemdeﬁnedonBx fromEq.(14)is

Bx

δ

^Ex^T

λ

^x⁰^z(^f^˜,T˜z)Exdx=r_dQzs(^T^˜z)

Bx

f˜

δ

^T^x^dx⁻

Bx

δ

^Ex^TQx0z(^f^˜,T˜z,

θ

^m)dx, (21) with

λ

^x⁰^z(^f^˜,T˜z)=

Bx

f˜²dx0

Bz

z(^T^˜z)^T

λ

^¯z(^T^˜z)dz, (22)

Qx0z(^f^˜^,^T^˜^z^,

θ

^m)⁼

Bx

Bz

f˜^z(^T^˜^z)^T

λ

^¯^grad(

θ

^m)^d^z^d^x0, (23)

Qzs(^T^˜^z)⁼

Bzs

T˜zdz, (24)

r_d isconsideredasconstant.

NotethatEq.(23)cansplitintotwoindependentintegralsoverBx andBzasθ^mîs^writtenûnderâ^separated^form.

TheintroductionoftheﬁniteelementapproximationEq.(18)inthevariationalEq.(21)allowsustodeducethefollowing linearsystem:

x0z(^f^˜,T˜z)^q^x=Rx(^f^˜,T˜z,

θ

^m), (25) whereq^xisthenodaltemperaturesvectorassociatedwiththeFEmeshinBx,x₀z(^f^˜,T˜z)^theconductivitymatrixobtained bysummingtheelements’conductivitymatricesêx₀z(^f^˜,T˜z)ândRx(^f^˜,T˜z,θ^m)^theequilibriumresidualobtainedbysumming theelements’residualloadvectorsRêx(^f^˜,T˜z,θ^m)

^ex0z(^f^˜,T˜z)=

Le

B^T_x

λ

^x0z(^f^˜,T˜z)^Bxdx (26)

and

R^e_x(^f^˜^,^T^˜^z^,

θ

^m)⁼^rdQs(^T^˜^z)

Le

N^T_xf˜dx−

Le

B^T_xQx0z(^f^˜^,^T^˜^z^,

θ

^m)^dx. ⁽²⁷⁾

Le referstooneelementofthemeshinBx(Bx=∪Le).

3.5.FiniteelementproblemtobesolvedonBz

Forsimplicityreasonasin theprevious section, thefunctions f˜⁽^k⁾ andT˜_x⁽^k⁻¹⁾ whichis assumedtobe known,willbe denoted f˜,T˜x,andtheunknownfunctionT˜_z⁽^k⁾willbedenotedTz.ThegradientinEq.(15)becomes

grad(^T^˜xTz)=x(^T^˜x)Ez (28)

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with

x(^T^˜x)=

_˜

Tx,x 0 0 T˜x

. (29)

ThevariationalproblemdeﬁnedonBzfromEq.(15)is

Bz

δ

^Ez^T

λ

^x⁰^x(^f^˜,T˜x)Ezdx=r_dQx0xs(^f^˜,T˜x)

Bzs

δ

^T^z^dz⁻

Bz

δ

^Ez^TQx0x(^f^˜,T˜x,

θ

^m)^dx ⁽³⁰⁾

with

λ

x0x(^f^˜,T˜x)=

B^x

f˜²dx0

B^xx(^T^˜x)^T

λ

^¯x(^T^˜x)dx, (31)

Qx0x(^f^˜,T˜x,

θ

^m)=

Bx

f˜x(^T^˜x)^T

λ

^¯^grad(

θ

^m)^d^x^d^x0, (32)

Qx0xs(^f^˜,T˜x)=

B^x

f˜T˜xdx. (33)

AsinEq.(23),theintegralinEq.(32)cansplitforthesamereason.

TheintroductionoftheﬁniteelementdiscretizationEq.(18)inthevariationalEq.(30)leadstothelinearsystem

x0x(^f^˜,T˜x)^q^z=Rz(^f^˜,T˜x,

θ

^m), (34) whereq^zisthenodaltemperaturevectorassociatedwiththeFEmeshinBz,x₀x(^f^˜,T˜x)^theconductivitymatrixobtainedby summingtheelements’ conductivitymatricesêx₀x(^f^˜,T˜x)ândRz(^f^˜,T˜x,θ^m) ^theequilibriumresidualobtainedbysumming theelements’residualloadvectorsRêz(^f^˜,T˜x,θ^m)

^ex0x(^f^˜^,^T^˜^x)⁼

Lze

B^T_z

λ

^x0x(^f^˜^,^T^˜^x)^B^z^dz ⁽³⁵⁾

and

R^ez(^f^˜,T˜x,

θ

^m)=rdQx0xs(^f^˜,T˜x)

Lz^sourcee

N^T_zdz−

Lze

B^T_zQx0x(^f^˜,T˜x,

θ

^m)dz. (36) Lze referstooneelementofthemeshinBz (Bz=∪Lze).

3.6. DeterminationoffonBx

Aspreviously,theknownfunctionsT˜_x⁽^k⁻¹⁾,T˜_z⁽^k⁻¹⁾ andtheunknown function f˜⁽^k⁾aredenotedT˜x,T˜z andf,respectively.

TheintroductionoftheseparatedformEq.(10)inEq.(16)allowsustodeducethestraightforwardrelation:

Bgrad(^T^˜zT˜x)^T

λ

^¯^grad(^T^˜zT˜x)dB

f=r_dT˜x

Bzs

T˜zdz−

Bgrad(^T^˜zT˜x)^T

λ

^¯^grad(

θ

^m)^dB. (37) Remark:ForavolumeheatsourceappliedonagivenregiondenotedI_s,thecalculationofthetemperatureiscarriedout followingtwosteps.First,thecomputationofthefunctionsfⁱ,T_xⁱandT_zⁱinEq.(6)isperformed. Second,onlyaintegration overIsofthetermsfⁱintheexplicitsolutioniscomputedatthepost-processinglevel.

4. Numericalresults:thermalanalysis

Inthissection, thefollowing FEdiscretizationischosen:(i) a classical3-node quadraticFEforthethermalunknowns dependingonthex-axiscoordinate,(ii)aclassicalquadraticinterpolationforthefunctionf,(iii)afourth-orderlayer-wise descriptionforthetransversedirection.

Static thermaltestsare presentedevaluatingthe eﬃciencyandthepropertiesofthealgorithm andalsovalidatingour approach.Tothispurpose,a sandwichtestisaddressed.Theconvergenceofthemethod,theinﬂuenceofthelocation and sizeofthethermalloadarediscussed.Theapproachisassessedby comparingwith2DFEMsolutions usingacommercial code. Finally, the separatedvariables decompositionis compared witha common tensor decomposition to illustrate the performanceofthemethod.

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4.1. Localizedvolumetricheatsourceontheupperfaceofasandwichbeam

Asandwichbeam, givenin[10],submitted toa localizedheat sourceon theupperface isconsidered. The testis de- scribedasfollows:

geometry:Athree-layersandwichbeamwithgraphite-epoxyfacesandasoftcorewiththicknesse=0.1hisconsidered.

S= L h=10.

boundaryconditions:localizedvolumetricheatsourceover[xs_min⁽¹⁾,xs_max⁽¹⁾]intheupperfaceofthesandwichwithaconstant valuer_d=0.1W.m⁻³

zerotemperatureonthebottomsurfaceofthebeamsuchasθ(^x,z=−h/2)=0 materialproperties:Theorientationofallthepliesisγk=0,k=1,2,3.

fortheface:λL=1.5W.m⁻¹.K⁻¹,λT=0.5W.m⁻¹.K⁻¹

whereLreferstotheﬁberdirection,Treferstothetransversedirection.

forthecore:isotropicmaterialsuchasλ=3.0W.m⁻¹.K⁻¹ mesh:N_x=60(numberofelementsalongthewholebeam)

results: Fora ﬁxedheat sourcelocation, 2DFEM computationsusingAnsyswithavery reﬁnedmeshofPLANE25 ele- mentsisprovided(120elementsalongthelengthand70elementsalongthethickness).Resultsbasedonthevariables separationxandz,denotedVS-LD4,arealsoprovided.Theseresultsareconsideredasreferencesolutions.

4.1.1. heatsourceforthreelocations

First, the approach is assessed forthree different locations of the heat source over I₁=[0,10], I₂=[10,20] andI₃= [40,50].ForathermalloadoverI_i,threeerrorindicatorsbetweenareferencesolutionθ^ref ^(VS-LD4) ^and^a^separated^vari-

ablessolutionθⁿôfôrderⁿîsintroduced:

1. energyerrorindicator

ⁿ=100

aerr(

θ

ⁿ−

θ

^{re f},

θ

ⁿ−

θ

^{re f}) aerr(

θ

^{re f}^,

θ

^{re f}) withaerr(

θ

1,

θ

2)=

B^x

B^zgrad(

θ

1)^T

λ

^¯^grad(

θ

2) ^d^z^dx,

(38)

2. globalerrorindicator

n=100

Bx

Bz

(ⁿ−^{re f})² ^d^z^d^x

Bx

Bz

^{re f}² dzdx

, (39)

where standsforθ^,^q1andq₃ 3. errorindicatoronthemaximumvalue

n^θ^max=100

(

θ

maxⁿ −

θ

max^{re f})²

θ

max^{re f}2 , (40)

ThevariationoftheenergyerrorindicatorEq.(38)withrespect tothenumberoftriplets(^Txⁱ,T_zⁱ,fⁱ)îsrepresentedon Fig. 3for the three locations ofthe heat source.The trendof these three configurations is rather similar. The obtained curve can be divided into two parts withtwo different slopes, thefirst one being over 0to 15 triplets, and thesecond

Fig. 3. Energy error indicator with respect to the number of triplets for three locations of volume heat source.

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Fig. 4. Global error indicator (left) - θmaxerror indicator (right) for three locations of volume heat source.

Fig. 5. Global error indicator q 1(left) - q 3(right) for three locations of volume heat source.

Fig. 6. Distribution of the temperature along the beam axis for 3 locations of volume heat source at x = (xs ⁽_min¹⁾+ xs ⁽max¹⁾)/ 2 - 50 triplets.

one beyond 15triplets. For further investigations, Fig.4 showsthe errorrate onthe globaland maximumtemperatures versus thenumberof termsforthedifferentcases.Again, we cannoticethat the errorrateis greatlyreducedwithonly 15couples,then thenewcouples donotbringanysignificantcorrections. Nevertheless,asit canbehighlighted inFig.5, the axialheat flux q₁ is themostdifficult quantity toobtain withaccuracy, andadditionaltermsare neededto improve theestimationofq₁.50tripletsareneededtohaveanerrorrateoflessthan1%,whereas40tripletsaresufficientforthe maximumtemperaturewiththesameleveloferror.

Toillustratetheaccuracyoftheapproach,thedistributionsoftemperaturealongtheaxisandthethicknessofthebeam, andthevariationofthe transverseheatﬂux forthree locationsoftheheatsourceare showninFigs.6–8,respectively.It canbeinferredfromtheseﬁguresthattheparametrizedapproachisinverygoodagreementwiththe2Dreferencesolution

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Fig. 7. Distribution of the transverse heat ﬂux along the thickness for 3 locations of volume heat source at x = (xs _min⁽¹⁾+ xs ⁽max¹⁾)/ 2 - 50 triplets.

Fig. 8. Distribution of the temperature across the thickness for 3 locations of volume heat source - 50 triplets.

Fig. 9. Distribution of the temperature for 3 locations of volume heat source - 50 triplets.

computedwithAnsysregardlessofthepositionofthevolumetricthermalload,fromtheedgetothemiddleofthestructure.

Thelocalizedtemperatureriseiswell-captured(seeFigs.6and9).Asfarasthetransverseheatﬂuxisconcerned,thefree boundary conditionon the top ofthe beam andthe continuity conditionat the layers interface are fullﬁled (cf. Fig. 7).

Thehighvariationinthe upperlayerisalsowell-represented. Finally,itcanbe noticedthat theseparatedrepresentation approach(x andz coordinates), denoted VS-LD4, witha ﬁxedlocation of heat sourceleadsto very accurate results,and couldbeconsideredasareferencesolution.

4.1.2. Inﬂuenceofthesizeoftheheatsourceregion

Tofurtherassessthemethod,localizedvolumeheatsourceoveraverysmallregionisconsidered withthreelocations (I₁^L=[0,5/3],I₂^L=[10,35/3], I₃^L=[40,125/3]). The variationof theerrorrateis showninFig. 10.It can beinferred from thisfigurethattheconvergencerateisslowerthanthepreviouscaseasitcouldbeexpected.Indeed,thisloadcaseimplies a highvariation of thetemperature along the beamaxis around the heat sourceregion as it can be seen inFig. 11.34 couples are needed to obtain a global error rate n^θ<1%. Nevertheless, 55 couples are required to obtain the maximal temperaturewithaccuracy(about2%)duetothelocalizationeffectonthedistribution.Forillustration,thefiftyfirsttriplets T_xⁱ,T_zⁱandfⁱaregiveninFigs.12–14.ThevariationsofT_xⁱandfⁱarerathersimilar.Theseresultswereexpectedconsidering Eq.(37)becausetheyhaveshorterwavelengthswhenthenumberofthetripletincreases.ForthefunctionT_zⁱ,thevariation alongthe thickness(Fig.13) seems to be identicalbeyondthe 10thone.Nevertheless, itis not thecaseasheat flux are correctedbytheseadditionalterms.

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Fig. 10. Global error rate (left) - θmaxerror rate (right) for three very localized regions of volume heat source ( I ^L₁, I ^L₂, I ^L₃).

Fig. 11. Distribution of the temperature along the beam axis for 3 locations of volume heat source - 55 triplets.

Fig. 12. T _xⁱ(x ).

Finally,itcanbeconcludedthattheaccuracyofthemethodisverygoodwhencomparedwiththereferencesolution.

4.2. ComparisonwithCANDECOMP-PARAFACapproach

Toevaluate theeﬃciencyofthedecompositionalgorithm developedinthe previoussection, theresultsare compared withthecanonicaldecompositionandparallelfactor(CANDECOMP-PARAFAC,denotedHOSVD-PC)[26,27]whichisacom- mon tensordecomposition.Forthat, an estimateofthe bestrank-R CandecompParafacmodelof atensorXis computed using an alternating least-squares (ALS) algorithm available in [28].The process consists infactorizing agiven N^th order tensor Xinto asum ofcomponentrank-one tensors (set of NcomponentmatricesAⁿ withAⁿ=[aⁿ₁,...,aⁿ_R],Aⁿ∈R^Iⁿ^×R,

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Fig. 13. T zⁱ(z).

Fig. 14. f ⁱ( x 0).

(ⁿ=1,...,N)⁾^that^is X≈

R

r=1

a¹_r◦a²_r...a^N_r =Xsuchthat min

X

^X⁻^X

²F

whereF istheFrobeniusnormand“◦” denotesouterproduct.

Inourcase, we haveN=3 anda¹_r,a²_r, a³_r corresponds to T_x^r,T_z^r,f^r respectively.The tensorXiscomputed usingthe solutionXobtainedwith55triplets.Thetestinvolvesavolumeheatsourceappliedononlythesizeofoneelement(xs∈ [40,125/3],namely1.6%ofthelengthofthebeam).Tocomparewiththepresentapproach,twotypesoferrorrate,aglobal oneEq.(39)andalocaloneEq.(40),aregiveninFig.15.Itcanbeobservedthattheresultsarerathersimilar.Itshowsthe performanceofthepresentdecompositionprocess.

Then,theenergycontributionofeachtriplettothetotalsolutionisalsocomparedinFig.16.Forthepresentapproach, 55% and 87% of the total energyare included in the first 8 and 25 triplets, respectively. The figure illustrates well the behavior of the greedy type algorithm. On the contrary, for the HOSVD-PC approach, it is important to notice that the decompositionprovided foragivennumberof termscannot be truncated.This numbermustbe chosen a priori.We see clearlythat thereisno hierarchyinthedifferenttermsofthesummation(seeFig.16right),whichisadrawback.Infact, toobtainthisfigure,itisneededtocomputeallthetermseachtimeforafixedvalueofR.

5. Numericalresults:thermo-mechanicalanalysis

Thenumericalmethod describedin[17] tosolve themechanicalproblemisused asithasshowninteresting features intermsofaccuracyandeﬃciency.Again,afourth-orderexpansionforthez-functions isconsidered.Asthepresentstudy isrestrictedtoaweakthermo-mechanical coupling,oncethethermalproblemissolvedfollowingthepresentedapproach, theresolution ofthemechanical oneis limitedto thecomputationofthe second termofthe righthandside ofEq.(5). Itis importantto note that thisintegral can be alsodivided into2 parts,one over Bx and theother one over Bz asthe temperatureiswrittenunderaseparatedform. Thus,the keypoint ofthemethodremains available.Thiscomputationis giveninAppendixA.