A posteriori global error estimator based on the error in the constitutive relation for reduced basis approximation of parametrized linear elastic problems

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A posteriori global error estimator based on the error in the constitutive relation for reduced basis approximation

of parametrized linear elastic problems

Laurent Gallimard, David Ryckelynck

To cite this version:

Laurent Gallimard, David Ryckelynck. A posteriori global error estimator based on the error in the constitutive relation for reduced basis approximation of parametrized linear elastic problems.

Applied Mathematical Modelling, Elsevier, 2016, 40, pp.4271-4284. �10.1016/j.apm.2015.11.016�. �hal-

01305736�

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A posteriori global error estimator based on the error in the constitutive relation for re duce d basis approximation of parametrized linear elastic problems

L. Gallimard

^a^,^∗

, D. Ryckelynck

^b

aLaboratoire Energétique Mécanique Electromagnétisme EA4416, Université Paris Ouest Nanterre La Défense, 50 rue de Sèvres Ville d’Avray 92410, France

bMINES ParisTech, PSL, research university, Centre des Matériaux, CNRS UMR 7633, France

Keywords:

Finite element analysis Model reduction Error bounds Global error estimator Reduced basis error indicator Finite element error indicator

a b s t r a c t

Inthispaperweintroduceaposteriorierrorestimatorbasedontheconceptoferrorinthe constitutiverelationtoverifyparametricmodelscomputedwithareducedbasisapproxima- tion.Wedevelopaglobalerrorestimatorwhichleadstoanupperboundfortheexacterror andtakesintoaccountalltheerrorsources:theerrorduetothereducedbasisapproximation aswellastheerrorduetotheﬁniteelementapproximation.Weproposeanerrorindicatorto measurethequalityofthereducedbasisapproximationandwededuceanerrorindicatoron theﬁniteelementapproximation.

1. Introduction

FiniteElementMethodisacommontoolusedtoanalyzeanddesignparametrizedmechanicalsystems.However,whena largesetofparametersneedstobeintroducedinthemodelthecomputationaleffortincreasesdrasticallyandmanyauthorshave recentlyshowninterestindevelopingmodelreductionmethodsthatexploitthefactthattheresponseofcomplexmodelscan oftenbeapproximatedbytheprojectionoftheinitialmodelonalow-dimensionalreducedbasis[1–4].Reducedbasismethods aimatspeedingupthecomputationaltimeforcomplexnumericalmodels.Theyarebasedonanoffline/onlinecomputational strategywhichconsistindetermininginafirststepasetofsnapshotsorareducedbasis(offlinecomputations)thatwillbe abletorepresentaccuratelythesolutionsfortheproblemstudied.Differenttechniquesareusedtogeneratethisbasis,themore commonlyfoundintheliteraturearetheproperorthogonaldecompositionandthegreedysamplingapproach[2,5].Inbothcase, thenumberoftermsinthereducedbasisisassumedtobeverysmallcomparedtothenumberofdegreeoffreedomofthefinite elementcomputation.Then,theapproximatesolutionsoftheparametrizedproblemarecomputedviaperformingaGalerkin projectionontothereducedbasisspace(onlinecomputations).

However,theaccuracyoftheobtainedsolutionsdependsonthequalityofthemeshusedaswellasonthequalityofthe chosenreducedbasis.Ifwedenotebyμ^the^vector^ofparameters,theglobalerroregisdeﬁnedforanyμ^by

eg

( μ )

=u

( μ )

−urb

( μ )

, ⁽¹⁾

∗ Corresponding author. Tel.: +33 140974865.

E-mail address: laurent.gallimard@u-paris10.fr (L. Gallimard).

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whereu(μ⁾îs^theêxact^solutionôf^theparametrizedproblemandu_rb(μ⁾îsîts^reduced^basisapproximation.Thisglobalerror canbesplitintotwoparts:

eg

( μ )

=u

( μ )

−u_h

( μ )

+u_h

( μ )

−u_rb

( μ )

=e_h

( μ )

+e_rb

( μ )

⁽²⁾

whereu_h(μ⁾îs^the^finiteêlement^solutionôf^theparametrizedproblemande_rb(μ)=u_h(μ)−u_rb(μ)^(resp.êh(μ)=u(μ)− u_h(μ)⁾îs^theêrror^due^to^the^projectionôf^the^finiteêlement^solutionⁱⁿ^the^reduced^basis^space^(resp.^theêrror^due^to^the^finite elementapproximation).Withintheframeworkandreducedbasisapproximationsbasedonproperorthogonaldecomposition orgreedysamplingmethods,manyworkshavebeendevotedtothecomputationofaposteriorierrorestimatorstomeasurethe errorduetotheprojectionofthefiniteelementsolutioninthereducedbasisspace.Anerrorestimatorisproposedforelliptic partialdifferentialequationsin[1,6],forparabolicproblemsin[7,8],forcomputationalhomogenizationin[9],forstochastic computations[10].However,theproposederrorestimatorsarefocusedontheestimationoftheerrorduetothereducedbasis approximatione_rbandassumethattheerrorduetothefiniteelementapproximatione_hisnegligible.Withintheframeworkof thepropergeneralizeddecomposition,aglobalerrorestimatorbasedontheconceptoferrorintheconstitutiverelation[11], hasbeenrecentlyproposedfortransientthermalproblemsin[12],andforlinearelasticproblemsin[13].Thiserrorestimator requirestodevelopadoublereducedbasisapproximationduringtheofflinestep,akinematicapproachandastaticapproach, forsolvingtheparametrizedproblem.

In thispaper, we focus on parametrized linear elastic models where the parametric bilinear form a is dependingon parameters-dependentfunctionsinanaffinemanner.Theobjectiveofthispaperistoextendtheconstitutiverelationerror estimatortoreducedbasisapproximationsbasedongreedysampling.Theuseoftheconstitutiverelationerror(CRE)requires thecomputationofanadmissiblepair(û^ˆ,σˆ)^forâny^parameterμ^during^theônlinecomputations.Unliketheapproachproposed in[12,13],weusedirectlytheinitialreducedbasis(i.e.thereducedbasisusedtocomputethesolutionu_rb)tobuildtheadmissible pair(û^ˆ,σˆ)ând^we^do^not^need^to^computeâ^second^reduced^basis^byâ^greedy^samplingâlgorithm.Additionally,twoerrorindi- catorsaredevelopedtoseparateintheglobalerrorestimatorthepartoftheerrorduetothefiniteelementapproximationfrom thepartduetothereducedbasisapproximation.Thisfamilyoferrorestimatorsallowstoconstructerrorboundsoftheenergy normandhasbeenappliedtoseparatethecontributionofthedifferentsourcesoferrorinfiniteelementcomputationsfornon- linearproblems[14–16],fordomaindecompositionproblems[17,18].Weshowthattheglobalerrorestimatorandthereduced basiserrorestimatorareupperboundsofthecorrespondingexacterrors.Tocomputeefficientlytheerrorestimateswithinthe frameworkofanoffline/onlinereducedbasismethodweneedafurtherassumption,andweassumethatthecomplementary energyis,aswellasthebilinearforma,dependingonparameters-dependentfunctionsinanaffinemanner.

Thepaperisorganizedasfollows:InSection2,weintroduceparametricproblemtobesolved,webrieflyrecallthereduced basismethodology,andwedefinetheapproximationerrorsintroduced.Theformulationoftheglobalerrorestimator,thefi- niteelementerrorindicatorandthereducedbasiserrorindicator,aswellastheoffline/onlinestrategytocomputethem,are describedinSection3.Finally,inSection4thedifferenterrorsareanalyzedthroughanumericalexample.

2. Problemtobesolved 2.1. Linearelasticmodel

Letusconsideranelasticstructuredefinedinadomain^bounded^by^.^Theêxternalâctionsôn^the^structureâre^repre- sentedbyasurfaceforcedensityTdefinedoverasubsetNoftheboundaryandabodyforcedensitybdefinedin^.^Weâssume thataprescribeddisplacementu=u_disimposedonD=

∂

−N.Thematerialisassumedtobelinearelastic,beingCthe Hooketensor.Weconsiderthattheproblemisdependentofasetofparametersμ∈D⊂R^p.Theproblemcanbeformulated as:findadisplacementfieldu(μ)∈Uandastressfieldσ⁽μ⁾^definedⁱⁿ^which^verify:

• thekinematicconstraints:

u

( μ )

=udon

D (3)

• theequilibriumequations:

div

σ ( μ )

+b

( μ )

=0in

^and

σ ( μ )

ⁿ=T

( μ )

ⁱⁿ

N (4)

• theconstitutiveequation:

σ ( μ )

⁼^C

( μ ) ε (

^u

( μ ))

ⁱⁿ

⁽⁵⁾

ndenotestheoutgoingnormalto^.U isthespaceinwhichthedisplacementfieldisbeingsought,U⁰thespaceofthe fieldsinU whicharezeroonD,andε⁽û⁾^denotes^the^linearizeddeformationassociatedwiththedisplacement:[ε(û)^]i j= 1/2(ûi,j+u_j_,_i)^.

Thestrongformoftheproblem(3-5)isequivalenttotheclassicalweakformformulation:ﬁndu∈{^v∈U; v_|

D=u_d}^such

that:

a

(

^u

( μ )

,u^∗;

μ )

=f

(

^u^∗;

μ ) ∀

^u^∗∈U⁰ ⁽⁶⁾

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where

a

(

^u

( μ )

,u^∗;

μ )

=

C

( μ ) ε (

^u

( μ ))

^:

ε (

^u^∗

)

^d

^and ^f

(

^u^∗;

μ )

=

b

( μ )

.u^∗d

+

N

T

( μ )

.u^∗d

Following[1],weassumethattheparametricbilinearformaandtheloadingfareaﬃnelydependentoffunctionsoftheparam- eterμ^,^by^which^we^shall^mean:

a

(

^u,u^∗;

μ )

= Q

q=1

^q

( μ )

^a^q

(

^u,u^∗

)

^and^f

(

^u^∗;

μ )

=

Q

q=1

H^q

( μ )

^f^q

(

^u^∗

)

⁽⁷⁾

where^q⁽μ⁾⁽^q∈{1,… ,Q})andH^q(μ⁾⁽^q∈{1,… ,Q})areknownfunctionsofμ^,^a^q⁽^q∈{1,… ,Q})arebilinearformsindependent ofμ^and^f^q⁽^q∈{1,… ,Q})arelinearfunctionsindependentofμ^.

Remark. This‘aﬃne’decompositionhasbeensuccessfullyappliedbyRozzaetal.fordifferentkindofparametrization:geometry, loadintensityanddirection,materialproperties,andmultisubdomainsformodularstructures(see[2]formoredetails).

2.2.Finiteelementapproximation

Tocomputethesolutionu(μ⁾ôfÊq.⁽⁶⁾^,^weîntroduceâ^finiteêlementapproximationu_hofuinafiniteelementspaceUh. Thefinite-dimensionspaceUhisassociatedwithafiniteelementmeshofcharacteristicsizeh.LetPhbeapartitionofînto elementsE_k.Thispartitionformedbytheunionofallelements,isassumedtocoincideexactlywiththedomainândâny^two elementsareeitherdisjointorshareacommonedge.Weassumethatu_dcanberepresentedbyadisplacementfieldinUh.The discretizedproblemis:findu_h∈{^v∈Uh;v_|

D=u_d}^such^that:

a

(

^uh

( μ )

,u^∗_h;

μ )

=f

(

^u^∗h;

μ ) ∀

^u^∗h∈U_h⁰ ⁽⁸⁾ whereU_h⁰={^v∈Uh;v|D=0}^.

Thecorrespondingstressﬁeldiscalculatedusingtheconstitutiveequation:

σ

h

( μ )

=C

( μ ) ε (

^uh

( μ ))

⁽⁹⁾

2.3.Reducedbasisapproximation

Letu_dir∈Uhbeadisplacementﬁeldsuchthatu_dir_|

D=u_d.LetusintroduceasetofsamplesintheparameterspaceS^N^s=

{μ¹,...,μ^N^s},whereμⁱ∈D,andforeachμⁿ^computeâ^finiteêlement^solutionû⁰_h(μⁿ)ⁱⁿU_h⁰describedbythecorresponding vectorofnodalvaluesqⁿ.

a

(

^u⁰h

( μ

ⁿ

)

,u^∗_h;

μ

ⁿ

)

= f

(

^u^∗h;

μ

ⁿ

)

−a

(

^udir,u^∗_h;

μ

ⁿ

) ∀

^u^∗h∈U_h⁰ (10) Thereducedbasisspaceisthendeﬁnedby

U_rb⁰ =span

{

^u⁰h

( μ

¹

)

,. . .,u⁰_h

( μ

^N^s

) }

^⊂^Uh⁰ (11)

ThechoiceofthesamplesintheparameterspaceS^N^sandoftheassociatedreducedbasisUrbdependsonthesamplingstrategy (see[1]formoredetails).Inthispaper,asdetailedinSection2.5,weusetheGreedyapproachproposedin[1,2].Thereduced basisapproximationconsistsinsolvingEq.(6)inU_rb⁰+{ûdir}^.Â^key^point^to^justify^theûseôf^the^reduced^basisapproximationis thatNsisassumedtobemuchsmallerthanN_fe(i.e.Ns<<N_fe).

a

(

^u⁰rb

( μ )

,u^∗_rb;

μ )

= f

(

^u^∗rb;

μ )

−a

(

^udir,u^∗_rb;

μ ) ∀

û^∗rb∈U_rb⁰ ⁽¹²⁾ Remark. Thecomputationofu_dirisperformedoffline.Thesimplestchoiceis,foragivenμ¯ ∈D,tofindu_dir∈{^v∈Uh;v_|

D=u_d}

suchthat

a

(

^udir,u^∗_h;

μ

¯

)

⁼⁰

∀

^u^∗h∈U_h⁰

Thecorrespondingstressﬁeldiscalculatedusingtheconstitutiveequation:

σ

dir=C

( μ

^¯

) ε (

^udir

)

⁽¹³⁾

σdirisequilibratedtozerointheFEsense:

σ

dir:

ε (

^u^∗h

)

^d

=0,

∀

^u^∗h∈U_h⁰

Letusdenotebyⁱrbthedisplacementﬁeldsu⁰_h(μⁱ)^forⁱ∈{¹,...,Ns}^.^The^reduced^basis^solution^writes u_rb

( μ )

=u_dir+

Ns

i=1

α

i

ⁱrb (14)

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Thecorrespondingstressisdeﬁnedusingtheconstitutiveequation

σ

rb

( μ )

⁼^C

( μ ) ε (

^udir

)

⁺

Ns

i=1

α

iC

( μ ) ε (

ⁱrb

)

⁽¹⁵⁾

byintroducingEq.(14)inEq.(12)webuildthealgebraicsystem

[K

( μ )

^]^[

α

^]⁼^[^F

( μ )

^] ⁽¹⁶⁾

Theelementsof[K(μ^)]ândôf^[^F⁽μ^)]âre^defined^by

Ki j

( μ )

=a

(

ⁱrb,

_rb^j;

μ )

^and^Fi

( μ )

=f

(

ⁱrb;

μ )

⁽¹⁷⁾

Thankstothe‘aﬃne’decompositionofaandf,K_ijandF_icanbewrittenasalinearcombinationofthefunctions^q⁽μ⁾⁽^q∈{1,… ,Q})andH^q(μ⁾⁽^q∈{1,… ,Q}).

K_{i j}

( μ )

⁼ Q

q=1

^q

( μ )

^a^q

(

ⁱrb,

rb^j

)

^and^Fi

( μ )

⁼

Q

q=1

H^q

( μ )

^f^q

(

ⁱrb

)

⁽¹⁸⁾

Acrucialpointinreducedbasisapproximationsistheseparationofthecomputationalprocedureintwoparts:anofflinepart devotedtothecomputationofparametersindependenttermsandperformedonlyonce,andanonlinepartdevotedtothe computationofparametersdependenttermsandperformedmanytimes.Thecomputationalcomplexityoftheofflinestageis greatanddependsonN_fe(thesizeofthefiniteelementapproximation),whilethecomputationalcomplexityoftheonlinestage issmallanddependsonNs,Q,Q.Wesummarizethetwopartsofthecomputation(amoredetaileddescriptioncanbefoundin [1])

• offline:Nsfiniteelementsolutionsⁱ_rb=u⁰_h(μⁱ)âs^wellâs^the^scalars^quantitiesâ^q(ⁱ_rb,_rb^j )ând^f^q(ⁱ_rb)âre^computedând stored.

• online:thematrix[K(μ^)]ând^the^right^hand^side^[^F⁽μ^)]âreâssembled^fromÊq.⁽¹⁷⁾^,^the^system^[^K(μ)^]^[α^]=[F(μ)^]îs^solved andthedisplacementfieldu_rb(μ⁾îs^computed^fromÊq.⁽¹⁴⁾^.

AlltheonlineoperationsareindependentofdimensionN_feanddependonlyonNs,QandQ. 2.4. Approximationerrors

Twoapproximationerrorsareintroducedinthecomputationofthereducedbasisapproximation

• anerrorduetotheprojectioninthereducedbasisspace,

• anerrorduetotheﬁniteelementapproximation.

Theglobalerroregisdeﬁnedby

eg

( μ )

=u

( μ )

−u_rb

( μ )

⁽¹⁹⁾

Thisglobalerrorcanbesplitintwoparts

eg

( μ )

⁼^eh

( μ )

⁺^erb

( μ )

⁽²⁰⁾

wheree_hande_rb arerespectivelytheerrorintroducedbytheﬁniteelementapproximationandtheerrorintroducedbythe reducedbasisapproximation.

e_h

( μ )

=u

( μ )

−u_h

( μ )

^and^erb

( μ )

=u_h

( μ )

−u_rb

( μ )

⁽²¹⁾

Theenergynormisaclassicalwaytomeasuretheseerrors

^e•

( μ )

²μ=a

(

^e•

( μ )

,e_•

( μ )

;

μ )

=

C

( μ ) ε (

^e•

( μ ))

^:

ε (

^e•

( μ ))

^d

⁽²²⁾

where• =g,horrb.Thefollowingequalityholdsfortheenergynorm:

^eg

( μ )

²_μ=

^eh

( μ )

²_μ+

^erb

( μ )

²_μ ⁽²³⁾

2.5. Choiceofthesnapshots

Thechoiceoftheelementsu_h(μⁱ⁾ôf^the^reduced^basis^(often^called^snapshots)îs^performed^following^the^Greedyâpproach

proposedin[1,2].WedenotebyDtrain⊂Dthesetofthesampleswhichwillbeusedtogenerateourreducedbasisapproximation andN_trainthenumberofelementsofDtrain.TypicallyN_trainischosensuchthatNs<<N_train.Thegenerationofthesnapshotsis performedbyaniterativegreedyprocedure.Webeginwithafirstpointμ¹ândâ^first^reduced^basis^spaceU_rb¹ =span{û⁰h(μ¹)}, then,forN=2,...,Ns,wecomputeu_rb(μ⁾ⁱⁿU_rb^N−1+{ûdir}^forâllμ∈Dtrainandwefind

μ

^N=argmax_μ_∈D

train

^erb

( μ )

μandu⁰_h

( μ

^N

)

=u_h

( μ

^N

)

−u_dir (24)

setS^N=S^N−1∪μ^N,andupdateU_rb^N =U_rb^N⁻¹+span{^u⁰h(μ^N)}^.

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Remark. Forpracticalpurposeitisnecessarytoreplace^erb(μ⁾μbyaninexpensiveaposteriorierrorboundasproposedin[1]. Inthispaper,weuseanerrorboundbasedontheconceptoferrorintheconstitutiverelationthatwillbedevelopedinSection (3.5).

3. Errorintheconstitutiverelation

Inthissection,weproposeanerrorestimationmethodbasedontheconceptoferrorintheconstitutiverelation.Weshow thatthiserrorestimatorisanupperboundoftheerrorintheenergynorm.Animportantpointtodevelopaneﬃcienterror estimatorforreducedbasisapproximationsisthatthecostofconstructionoftheerrorestimatorsisindependentonN_fe,and weneedtointroduceasafurtherassumptionthatthecomplementaryenergyisalsoaﬃnelydependentoffunctionsofthe parameterμ^.

C

( μ )

⁻¹

σ

¹^:

σ

²^d

=

Q

q=1

^q_σ

( μ )

S^q

σ

¹^:

σ

²^d

⁽²⁵⁾

where^q_σ(μ)âre^known^scalar^functionsând^S^qâre^fourthôrder^tensor^fields^definedôn^. 3.1. Definitionoftheerrorestimator

Theapproachbasedontheconstitutiverelationerror[11]reliesonapartitionoftheequationsoftheproblemtobesolved intotwogroups.Inlinearelasticity,thefirstgroupconsistsofthekinematicconstraintsandtheequilibriumequations;the constitutiveequationconstitutesthesecondgroup.Foragivenμ∈D,letusconsideranapproximatesolutionoftheproblem, denotedby(u^c,σê^),^which^verifies^the^first^groupôfêquations⁽³⁾ând⁽⁴^).^This^solution^will^be^saidâdmissible^.Îf⁽û^c^,σê⁾^verifies

theconstitutiveequation(Eq.(5))in^then(û^c,σê)=(û(μ),σ(μ))^.Îf⁽û^c^,σê⁾^does^not^verify^theconstitutiveequation,the qualityofthisadmissiblesolutionismeasuredbytheerrorintheconstitutiverelation

η

^g^which^is^deﬁned^with^respect^to^the

constitutiveequation

η

^g=ecr

(

^u^c,

σ

^e

)

=

| σ

^e−C

( μ ) ε (

^u^c

)) |

μ (26) where

| σ

^e−C

( μ ) ε (

^u^c

)) |

μ=

( σ

^e−C

( μ ) ε (

^u^c

))

^:^C⁻¹

( μ )( σ

^e−C

( μ ) ε (

^u^c

))

^d

1/2

(27)

ThePrager–Syngetheoremgivesarelationshipbetweentheexactandadmissiblesolution[19]

e²_cr

(

^u^c,

σ

^e

)

=

| σ

−

σ

^e

|

²_μ+

^u−u^c

²_μ ⁽²⁸⁾

3.2.Admissiblesolution

Akeypointtodeveloptheerrorestimatoristheconstruction,foragivenμ∈D,ofanadmissiblesolution(u^c,σ^e⁾^from^the

reducedbasissolutionu_rb(μ⁾^and^the^data.^Since^the^reduced^basis^solution^veriﬁes^the^kinematicconstraints,onetakes:

u^c=u_rb

( μ )

ⁱⁿ

⁽²⁹⁾

Amethodtorecoverequilibratedstressfieldsσê^from^the^finiteêlement^solutionând^the^data^have^beenûnderdevelopment forseveralyears[19–21].Inclassicallinearelasticityfiniteelementcomputation,theadmissiblestressfieldcanbeconstructed directlyfromthefiniteelementstressfield[19]becausethefiniteelementstressfieldverifiestheequilibriumequationsofthe finiteelementmodel.Inthecaseofareducedbasistheconstructionofanadmissiblestressfieldσê^requires^two^distinct^steps.

• Inaﬁrstoﬄinestep,thereducedbasisUrbisusedtobuildanadmissiblereducedbasisforthestresses.Thisreducedbasisis builtwitharecoverytechniqueforconstructingequilibratedstressesinstarpatchesproposedin[20].

• Inasecondonlinestep,theadmissiblestressiscomputedbyaminimizationofanenergynorm.

3.3.Constructionofanadmissiblestressﬁeld

Theﬁrststep(oﬄine)consistsinbuildinganadmissiblereducedbasisforthestresses.

Letμ⁰^denoteâ^particular^setôf^parameters^such^thatμ⁰∈/{μ¹,...μ^N^s},andletusintroduceasetofdisplacementfields u^q_h(μ⁰)∈U_h⁰,forq∈{¹,...,Q},suchthat

a

(

^u^q_h

( μ

⁰

)

,u^∗_h;

μ

⁰

)

= f^q

(

^u^∗h

) ∀

û^∗h∈U_h⁰ ⁽³⁰⁾ wheref^qistheq^thelementoftheaffinedecompositionoftheloading(Eq.(7)).Thestressfieldsσ^qf=C(μ⁰)ε(û^q_h(μ⁰))^verifyân equilibriumequationintheFEsense(Eq.(31))

σ

^qf :

ε (

^u^∗h

)

^d

= f^q

(

^u^∗h

) ∀

^u^∗h∈U_h⁰ (31)

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Foreachq∈{¹,...,Q},arecoverytechnique[20]isusedtobuildastressfieldσˆ^qfthatverifiestheequilibriumequationdefined bythefollowingequation:

σ

ˆ^qf:

ε (

^u^∗

)

^d

=f^q

(

^u^∗

) ∀

û^∗∈U⁰ ⁽³²⁾ Similarly,astressfieldσˆdirequilibratedtozeroisbuiltfromthestressfieldσdir(Eq.(13))suchthat

σ

ˆdir:

ε (

^u^∗

)

^d

=0

∀

û^∗∈U⁰ ⁽³³⁾ Thesetofstressesσˆ^qf isusedtobuildanadmissiblestressfieldσˆf(μ)^forâny^loading^f⁽û^∗^;μ^).^Letûs^define^the^stress^field σˆf(μ),whichdependsexplicitlyonμ^by

σ

ˆf

( μ )

=

Q

q=1

H^q

( μ ) σ

^ˆ^qf+

σ

ˆdir (34)

whereH^q(μ⁾âre^known^functionsôfμ^definedⁱⁿÊq.⁽⁷⁾^.Îf^weîntroduceÊq.⁽³²⁾ⁱⁿÊq.⁽⁷⁾^,^thanks^to^the^‘affine’decomposition, weobtain

∀

^u^∗∈U⁰ f

(

^u^∗;

μ )

=

Q

q=1

H^q

( μ )

σ

ˆ^qf:

ε (

^u^∗

)

^d

=

_Q

q=1

H^q

( μ ) σ

^ˆ^qf :

ε (

^u^∗

)

^d

⁽³⁵⁾

Hence,thestressfieldσˆf(μ)îsâdmissible^for^the^loading^f⁽û^∗^;μ^).^Similarlyσf(μ⁾^defined^byÊq.⁽³⁶⁾îsânequilibratedstress fieldforthefiniteelementmodel.

σ

f

( μ )

⁼

Q

q=1

H^q

( μ ) σ

^qf+

σ

dir (36)

Thestressfieldσˆf(μ)^could^beûsed^directly^to^computeânûpper^boundôf^theêrror,^howeverîtîsêasy^toîmprove^thisûpper boundadaptingthetechniqueproposedin[12]forapropergeneralizeddecompositionmethod,toareducedbasisapproach.Let usconsiderthesetofstressfieldscomputedfromthesnapshotsolutionsUrb

σ

ⁱrb=C

( μ

ⁱ

) ε (

^uh

( μ

ⁱ

))

^forⁱ^∈

{

¹^,^.^.^.^,^N^s

}

⁽³⁷⁾

Thestressfieldσⁱ_rbîsequilibratedintheFEsensefortheloadingf(û^∗_h;μⁱ)^.^The^recovery^technique^[20]îsâgainûsed^to^compute admissiblestressfieldsσˆⁱrb(i∈{¹,...,Ns}⁾^such^that

σ

ˆⁱrb:

ε (

^u^∗

)

^d

⁼^f

(

^u^∗^;

μ

ⁱ

) ∀

^u^∗^∈^U⁰ ⁽³⁸⁾

Foraﬁxedμⁱ^Eq.⁽³⁴⁾^leads^to

σ

ˆf

( μ

ⁱ

)

^:

ε (

^u^∗

)

^d

⁼ ^f

(

^u^∗^;

μ

ⁱ

) ∀

^u^∗^∈^U⁰ ⁽³⁹⁾

BysubtractingEq.(39)fromEq.(38)weobtain

σ

^ˆⁱrb:

ε (

^u^∗

)

^d

⁼⁰

∀

^u^∗^∈^U⁰ ⁽⁴⁰⁾

whereσ^ˆⁱrb=σˆⁱrb−σˆf(μⁱ)⁽ⁱ∈{¹,...,Ns}⁾îsâ^setôf^stress^fieldsequilibratedtozero.Similarly,wecanbuildasetofstressfields equilibratedtozerointheFEsensebysettingσⁱ_rb=σⁱ_rb−σf(μⁱ)^.

Thesecondstepoftheconstructionisperformedduringtheonlineprocedure.Foreachparameterμ^an^admissible^stress

fieldσêîs^soughtⁱⁿ^the^reduced^basisôfequilibratedstressfields

σ

^e⁼

σ

^ˆf

( μ )

+

Ns

i=1

β

i

σ

^ˆⁱrb (41)

FromEqs.(40)and(34)itfollowsthatσêîsequilibratedwiththeloadingf(u^∗;μ⁾^forâll

β

i∈R.

Thecoeﬃcients

β

iarecomputedinordertominimizethe“distance” betweenthestressﬁeldcomputedinthereducedbasis

σrb(μ)=C(μ)ε(ûrb)ândâ^stress^fieldequilibratedintheFEsenseσ˜(

β

1,...,

β

N_s)

J

( β

¹^,^.^.^.^,

β

^Ns

)

=

|

^C

( μ ) ε (

^urb

)

−

σ

˜

( β

¹^,^.^.^.^,

β

^Ns

) |

μ (42) where

σ

˜

( β

¹^,^.^.^.^,

β

^Ns

)

=

σ

f

( μ )

+

Ns

i=1

β

i

σ

ⁱrb (43)

(8)

TheminimizationofEq.(42)leadstothealgebraicsystem [A

( μ )

]

β

=[G

( μ )

] (44)

where A_{i j}=

C⁻¹

( μ ) σ

rb^j :

σ

ⁱrbd

⁽⁴⁵⁾

and Gj=

σ

_rb^j ^:

( ε (

^urb

)

−C⁻¹

( μ ) σ

f

( μ ))

^d

⁽⁴⁶⁾

Remark. Theequilibratedstressfieldσê^couldâlso^beûsedⁱⁿÊq.⁽⁴²⁾însteadôf^the^weaklyequilibratedstressfieldσ˜,toobtain abettereffectivityindexfortheglobalerrorestimator.However,theuseoftheweaklyequilibratedstressfieldleadstominimize theerrorestimatoronthereducedbasisapproximation(asshowninEq.(61))whichisusedtobuildthesnapshots.

3.4.Computationoftheerrorestimatorandupperboundproperty

AdirectapplicationofthePrager–Syngetheorem(Eq.(28))withu^c=u_rbandσê^defined^byÊq.⁽⁴¹⁾^leads^to^the^following

upperboundpropertyfortheenergynorm

^e^g

( μ )

μ=

^u⁻^urb

μ≤

η

^g⁼^e^cr

(

^urb,

σ

^e

)

⁽⁴⁷⁾

FromEq.(26)theexpressionoftheglobalerrorestimatorisgivenby e²_cr

(

^urb,

σ

^e

)

⁼

C⁻¹

( μ ) σ

^e^:

σ

^e^d

⁺

C

( μ ) ε (

^urb

)

^:

ε (

^urb

)

^d

⁻²

σ

^e^:

ε (

^urb

)

^d

⁽⁴⁸⁾

Thecomputationoftheglobalerrorestimatorisdonebyanoﬄine–onlineprocedure

• Duringtheofflineprocedurethestressfieldsσˆ^qf(q∈{¹,...,Q}⁾ândσ^ˆⁱrb(i∈{¹,...,Ns})âre^computedând^stored,âs^well asthefollowingscalarquantitiesfor(ⁱ,j)∈{¹,...,Ns},p∈{¹,...,Q}ând^r∈{¹,...,Q}

ari j=

S^r

σ

ⁱrb:

σ

_rb^j ^d

a⁰_{r p j}=

S^r

σ

0^p:

σ

rb^j d

ci j=

σ

ⁱrb:

ε (

_rb^j

)

^d

(49)

and ˆ a_{ri j}=

S^r

σ

^ˆⁱrb:

σ

^ˆrb^j d

ˆ

a⁰_{r pi}=

S^r

σ

ˆ^pf:

σ

^ˆⁱrbd

ˆ

a⁰⁰_{r pq}=

S^r

σ

ˆ^pf:

σ

ˆ^qfd

bˆ_{si j}=

C^s

ε (

ⁱrb

)

^:

ε (

_rb^j

)

^d

cˆi j =

σ

^ˆⁱrb:

ε (

_rb^j

)

^d

ˆ

c⁰_pi=

σ

ˆ^pf :

ε (

ⁱrb

)

^d

⁽⁵⁰⁾

• Duringtheonlineprocedureforeachparameterμ^,^thanks^to^the^‘affine’decomposition(Eq.(25)),wecomputethecoefficients ofmatrix[A(μ^)]ând^[^G⁽μ^)]^byintroducingthecoefficientsdefinedbyEq.(49)inEqs.(45)and(46):

Ai j=

Q

r=1

^qσ

( μ )

^ari jandGj=

Ns

i=1

α

i

( μ )

^ci j−

Q

p=1 Q

r=1

H^p

( μ )

^r_σ

( μ )

^a⁰r p j (51)

then,thesystem[A(μ)^] β

=[G(μ)^]^is^solved.

Theerrorestimatorecriscomputedduringtheonlineprocedurefromthecoeﬃcients

β

i,andthecoeﬃcientsinEq.(50)are computedduringtheoﬄineprocedure.

e²_cr

(

^urb,

σ

^e

)

=

Q

r=1 Q

p=1 Q

q=1

^r_σ^H^p^H^q^a^ˆ⁰⁰r pq+

Q

r=1 Q

p=1 Ns

i=1

^r_σ^H^p

β

iaˆ⁰_{r pi}