Towards error bounds of the failure probability of elastic structures using reduced basis models

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Towards error bounds of the failure probability of elastic structures using reduced basis models

L. Gallimard, Éric Florentin, David Ryckelynck

To cite this version:

L. Gallimard, Éric Florentin, David Ryckelynck. Towards error bounds of the failure probability

of elastic structures using reduced basis models. International Journal for Numerical Methods in

Engineering, Wiley, 2017, 112 (9), pp.1216-1234. �10.1002/nme.5554�. �hal-01633909�

Towards error bounds of the failure probability of elastic structures using reduced basis models

L. Gallimard

, E. Florentin

and D. Ryckelynck

1LaboratoireEnergétiqueMécaniqueElectromagnétismeEA4416,UniversitéParisNanterre,50ruedeSèvres, Villed’Avray,92410,France

2INSA-CVL/Universited’Orléans–PRISME,88,BoulevardLahitolle,Bourges,F-18020,France

3CentredesMatériaux,MINESParisTech,PSL,ResearchUniversity,CNRSUMR7633,Paris,France

SUMMARY

Structuralreliabilitymethodsaim atcomputingtheprobability offailureofsystemswith respecttopre- scribedlimitstatefunctions.AcommonpracticetoevaluatetheselimitstatefunctionsisusingMonteCarlo simulations.Themaindrawbackofthisapproachisthecomputationalcost,becauseitrequirescomputing alargenumberofdeterministicfiniteelementsolutions.Surrogatemodels,whicharebuiltfromalimited numberofrunsoftheoriginalmodel,havebeendeveloped,assubstituteoftheoriginalmodel,toreducethe computationalcost.However,thesesurrogatemodels,whiledecreasingdrasticallythecomputationalcost, mayfailincomputinganaccuratefailureprobability.Inthispaper,wefocusonthecontroloftheerrorintro- ducedbyareducedbasissurrogatemodelonthecomputationofthefailureprobabilityobtainedbyaMonte Carlosimulation.Weproposeatechniquetodetermineboundsofthisfailureprobability,aswellasastrat- egyofenrichmentofthereducedbasis,basedonlimitingtheboundsoftheerrorofthefailureprobability foramulti-materialelasticstructure.

KEY WORDS: structuralreliability;finiteelementanalysis;modelreduction;reducedbasis;errorbounds;

failure probability

1. INTRODUCTION

Many applications in structural analysis require taking into account stochastic properties of material, geometry, or loads. Given a probabilistic description of the stochastic properties of the structure (i.e., a random vectorΘassociated with a probability density functionf_Θ), reliability analysis aims at computing the probability of failure of systems with respect to a prescribed limit state function G. The failure is defined as an eventF = {G(Θ) ⩽ 0}, and the failure probability P_f is defined byPf = Prob({G(Θ) ⩽ 0}) = ∫_G(𝜽)⩽0fΘ(𝜽)d𝜽. The computation of this failure probability by a direct Monte Carlo simulation would necessitate between 10³and 10⁶finite element (FE) simulations (depending on the value ofPf) and is too computationally expensive in practice. In order to reduce the number of simulation runs, different alternatives have been proposed.

A first approach is the first-order reliability method and the second-order reliability method (SORM) [1, 2] that consists in building a simple analytical approximation of the limit-state function around the so-called design point followed by a direct estimation of the failure probability [3 – 9].

A second approach consists in building a surface response as a surrogate of the limit state function (quadratic response surfaces, polynomial chaos expansions, kriging surrogates, etc.) [10 – 14]. The Monte Carlo simulation can be then applied on this surrogate model.

*Correspondence to: L. Gallimard, LEME, Université Paris Nanterre, 50 rue de Sèvres, 92410 Ville d’Avray, France.

†E-mail: [email protected]

A thirdapproachconsistsinbuildingareduced-order model(ROM)ofthecomplexFEmodel [15 – 19]andusingtheMonteCarlomethodalongwiththeROM.

Threemainsourcesoferrorsareintroducedinareliabilityanalysis:anerrorduetotheFEanal- ysis,anerrorduetotheapproximationsintroducedbythemethodusedtoreducethenumberofFE simulationruns,andfinallyanerrorduetotheMonteCarloprocess(orduetotheanalyticalformula usedtocomputethefailureprobabilityforthefirst-orderreliabilitymethod/second-orderreliability method).MethodshavebeendevelopedovermanyyearstoevaluatethequalityoftheFEanalysis [20 – 22]andhavebeenextendedmorerecentlytothestochasticframework[23 – 25];thequalityof theMonteCarlosimulationcanbeassessedviathecentrallimittheorem.Errorestimatorshavebeen developedforROMappliedtoparametrizedmodel[26 – 30]andforROMappliedtotheestimation ofmeanvalues[17].However,asfarasweknow,thereisnoworkontheestimationoftheerror introducedbyareduced-ordermodelingonthecomputationofthefailureprobabilityP_f.

Inthispaper,wefocusonthecontroloftheerrorintroducedbyareducedbasisapproximation [30] onthecomputationofthefailureprobabilityPf.Weproposetousetheconceptoferrorinthe constitutiverelationtoprovideboundsofthefailureprobabilityforagivenreducedbasis.Inafirst step,theseboundsareobtainedfollowingthemethodologyproposedin[26]byextractingareduced basisthatisequilibratedintheFEsensefromaclassicalreducedbasis(whichsatisfiesthekinematic boundaryconditions).Inasecondstep,weproposeastrategytoconstructthisreducedbasisalong withtheMonteCarlosimulation,inordertoprovidecontrolledboundsofthefailureprobabilityPf. Thepaperisorganizedasfollows:Section2describestheproblemtobesolved.InSection3,the reducedbasisformulationusedtosolvetheproblemispresented.Thetechniqueusedtocompute boundsonquantitiesofinterestisremindedinSection4.Itsextensiontotheboundingofafailure probabilityas wellasthe algorithmused to build the reducedbasisare developedin Section 5.

Finally,Section6presentsanapplicationtomultilayeredstructuresin2Dplaneelasticity.

2. THEPROBLEMTOBESOLVED 2.1. Linearelasticmodel

LetusconsideranelasticstructuredefinedinadomainΩboundedbyΓ.Theexternalactionsonthe structurearerepresentedbyasurfaceforcedensityTdefinedoverasubsetΓNoftheboundaryand abodyforcedensitybdefinedinΩ.Weassumethataprescribeddisplacementu=u_d isimposed on Γ_D = 𝜕Ω − Γ_N.The material is assumedtobelinear elastic, beingC theHooke tensor.We considerthattheproblemisdependent ofavector𝜽 ∈ D ⊂ R^p ofuncertainsystemparameters.

Theseparametersarecharacterizedinaprobabilisticmannerbymeansofajointprobabilitydensity functionf(𝜽).Theproblem canbeformulatedasfollows:Findadisplacement fieldu ∈ U a nda stressfield 𝝈 definedinΩthatverify

• the kinematic constraints

u(x,𝜽) =ud(x)onΓD, (1)

• the equilibrium equations

div𝝈(x,𝜽) +b(x,𝜽) =0inΩand𝝈(x,𝜽)n=T(x,𝜽)inΓ_N, (2)

• the constitutive equation

𝝈(x,𝜽) =C(x,𝜽)𝜺(u(x,𝜽))inΩ. (3) ndenotes the outer normal toΩ._Uis the space in which the displacement field is being sought;

U⁰is the space of the fields inUthat are zero onΓD; and𝜺(u)denotes the linearized deformation associated with the displacement:[𝜺(u)]ij=1∕2(ui,j+uj,i).

We assume that the Hooke tensor C, the body forces b, and the traction forces T can be decomposedasthesumoffunctionsof 𝜽 multipliedbydeterministicfunctionsofx.

C(x,𝜽) =

Q^c

∑

q=1

Θ^c_q(𝜽)Cq(x), b(x,𝜽) =

Q^b

∑

q=1

Θ^b_q(𝜽)bq(x) T(x,𝜽) =

Q^T

∑

q=1

Θ^Tq(𝜽)Tq(x), (4) whereΘ^c_q,Θ^b_q,andΘ^T_q are known functions of𝜽; eachCq(x)is a fourth-order tensor defined inΩ;

bq(x)are vector fields defined inΩ; andTq(x)are vector fields defined onΓN. Remark

This decomposition can be obtained by the use of the Karhunen – Loeve expansion [31] as proposed in [25].

Remark

For the sake of simplicity, assume thatu_d is a deterministic value (i.e.,u_d(x,𝜽) = u_d(x)). For an interested reader, the development of a reduced basis algorithm whereu_d is also described by a stochastic approach can be found in [27].

Furthermore, we assume that the inverse of the Hooke tensor can be decomposed as the sum of functions of𝜽multiplied by deterministic functions ofx.

C⁻¹(x,𝜽) =

Q^s

∑

q=1

Θ^s_q(𝜽)S_q(x), (5)

where eachS_q(x)is a fourth-order tensor defined onΩ.

The strong form of the problem (Equations (1) – (3)) is equivalent to the classical weak form formulation: Findu∈ {v∈_U; v_|ΓD=u_d}such that

a(u(𝜽),u^∗;𝜽) =f(u^∗;𝜽) ∀u^∗∈_U⁰, (6) where

a(u(𝜽),u^∗;𝜽) =

∫_ΩC(𝜽)𝜺(u(𝜽)) ∶𝜺(u^∗)dΩandf(u^∗;𝜽) =

∫_Ωb(𝜽) ·u^∗dΩ +

∫_ΓNT(𝜽) ·u^∗dΓ. To compute the solutionu(𝜽)of Equation (6), an FE approximationuhofuis introduced such that uh ∈Uh⊂U. LetPh be a partition ofΩinto elementsEk(k∈ {1,…,NFE}). This partition formed by the union of all elements is assumed to coincide exactly with the domainΩ, and any two elements are either disjoint or share a common edge. We assume thatudcan be represented by a displacement field inUh. The discretized problem is as follows: Find a displacement fielduh(𝜽) ∈Uhand a stress field𝝈h(𝜽)defined inΩthat verify

• the kinematic constraints

uh(x,𝜽) =ud(x)onΓD, (7)

• the FE equilibrium equations

∫_Ω𝝈h(𝜽) ∶𝜺(u^∗_h)dΩ =

∫_Ωb(𝜽) ·u^∗_h dΩ +

∫_ΓNT(𝜽) ·u^∗_hdΓ ∀u^∗_h ∈U_h⁰, (8)

• the constitutive equation

𝝈h(x, 𝜽) = C(x, 𝜽)𝜺(u_h(x, 𝜽))inΩ. (9)

The classical weak form formulation is finduh ∈ {v∈Uh; v_|ΓD=ud}such that

a(uh(𝜽),u^∗_h;𝜽) =f(u^∗_h;𝜽) ∀u^∗_h ∈U_h⁰, (10) where_Uh⁰ = {v∈_Uh;v|Γ_D =0}.

2.2. Quantity of interest

The goal of the computation is often to obtain (or characterize) some specific quantity of interest (QoI). Thus, in this context, the QoI is also a random output. The QoI is denoted byQ(uh(𝜽);𝜽), and it is given as a linear output ofuh(𝜽), namely,

Q(uh(𝜽);𝜽) =

∫_ΩbΣ(𝜽) ·uh(𝜽)dΩ +

∫_ΓNTΣ(𝜽) ·uh(𝜽)dΓ, (11) wherebΣandTΣare the given extractors that define the output of interest and can be decomposed as the sum of functions of𝜽multiplied by deterministic functions ofx.

bΣ(x,𝜽) =

Q^bΣ

∑

q=1

Θ^b_q^Σ(𝜽)bΣ_q(x), TΣ(x,𝜽) =

Q^TΣ

∑

q=1

Θ^T_q^Σ(𝜽)TΣ_q(x).

2.3. Structural reliability model

In this section, we describe the problem of reliability assessment [1, 13, 32]. Let us denote by S(𝜽) =Q(uh(𝜽);𝜽)the computational output of interest andR(𝜽)the threshold associated with this computational output.

Classically, the structure failure state is defined by a limit state function

G(𝜽) =R(𝜽) −S(𝜽) (12)

such that

• G(𝜽)<0 is a failure state for the structure;

• G(𝜽) =0 is the limit state;

• G(𝜽)>0 is a safe state for the structure.

According to this definition, the system fails when G is lower or equal to zero. The failure probabilityPfis then given by

P_f =

∫_Df={𝜽∈Rⁿ_{;G(𝜽)⩽0}}f_Θ(𝜽)d𝜽, (13) wherenis the dimension of the vector𝜽andf_Θ(𝜽)is a density probability function. The evaluation of the integral defined by Equation (13) is not easy because it represents a very small quantity and because the integration domain is defined implicitly. Monte Carlo simulation is the main approach to solving the reliability problem. Recasting Equation (13) as

Pf =

∫_Rⁿ1_Df(𝜽)fΘ(𝜽)d𝜽=E[ 1_Df]

, (14)

where1_Df is the failure indicator function being equal to one ifG(𝜽) ⩽0 and zero otherwise. The probability of failure is equal to the expectation of1_Df. The Monte Carlo technique consists in gen- erating a numberN_MCof realizations𝜽^kof the random vectorΘusing the actual probability density function, then computing the Monte Carlo estimatorP̂_f:

P̂_f = 1 N_MC

N_MC

∑

k=1

1_Df(𝜽^k). (15)

According to the central limit theorem, this estimator is asymptotically unbiased and normally distributedwithvariance

Var[P̂f

]=P̂_f(1−P̂_f) NMC−1 .

When the failure probability is small, the coefficient of variation of the estimator is

𝛿=

√

Var[P̂_f]

P̂_f ≈ 1

√NMC

√P̂f

. (16)

From Equation (16), it can be seen that obtaining a coefficient of variation𝛿⩽10%for a probability of failure of 10⁻ⁿrequires about 10ⁿ⁺²FE simulations.

3. REDUCED BASIS FORMULATION

The purpose of a reduced basis method is to provide a fast evaluationu_rb(𝜽)of the value of the dis- placement fieldu_h(𝜽)and hence a fast evaluationQ(u_rb(𝜽);𝜽)of any QoIQ(u_h(𝜽);𝜽). In Section 3.1, we recall a classical displacement approach that consists in performing a Galerkin projection onto a reduced basis space that is assumed to represent accurately the solutions of the problem to be solved.

In this paper, the construction of this reduced basis space, which will be detailed in Section 5, is aimed to control the accuracy of the computed failure probability. To control this accuracy, we define and compute an error estimator that is an upper bound of the error on the QoI|Q(uh(𝜽);𝜽)−Q(urb(𝜽);𝜽)|. The computation of this upper bound, as shown in Section 4, necessitates the construction of a stress field that satisfies the FE equilibrium equations (Equation (8)). In order to obtain a fast evaluation of this stress field, we propose in Section 3.2 to build, from the displacement reduced basis, a stress reduced basis that satisfies the FE equilibrium equations.

3.1. Displacement approach

Letu_dir∈_Uhbe a displacement field such thatu_dir|ΓD =u_d. Let us introduce a set of samples in the parameter space_S^Ns = {𝜽1,… ,𝜽N_s}, where𝜽i∈_D, and for each𝜽ncompute a FE solutionu⁰_h(𝜽n) in_Uh⁰described by the corresponding vector of nodal valuesqⁿ.

a(u⁰_h(𝜽n),u^∗_h;𝜽n) =f(u^∗_h;𝜽n) −a(udir,u^∗_h;𝜽n) ∀u^∗_h ∈U_h⁰. (17) On this space, we perform a Gram – Schmidt orthonormalizing process for the inner producta(u,v;𝜽) (𝜽being a fixed value of the random vector that is selected a priori as shown in Section 5.2) to build a reduced basis𝝓ⁱ. The reduced basis space is then defined by

U_rb⁰ =span{𝝓¹,… ,𝝓^Ns}⊂U_h⁰. (18) The choice of the samples in the parameter space_S^Nsand of the associated reduced basis_Urb⁰ depends on the sampling strategy (see [30] for more details). In this paper, we propose a reduced basis strategy that permits to control the quality of the reliability analysis. This reduced basis strategy will be introduced in Section 5. The reduced basis approximation consists in solving Equation (6) inU_rb⁰ + {udir}. A key point to justify the use of the reduced basis approximation is thatNsis assumed to be much smaller than the number of degree of freedom of the FE modelNFE(i.e.,Ns ≪NFE).

a(u⁰_rb(𝜽),u^∗_rb;𝜽) =f(u^∗_rb;𝜽) −a(u_dir,u^∗_rb;𝜽) ∀u^∗_rb∈_Urb⁰. (19)

Remark

Thecomputationofudiris performedoff-line.Thesimplestchoiceis,foragiven𝜽 ∈ _D, to fi nd udir∈ {v∈Uh; v_|ΓD =ud}suchthat

a(u_dir,u^∗_h;𝜽) =0 ∀u^∗_h∈_Uh⁰. (20) The reduced basis solution writes

urb(𝜽) =udir+

N_s

∑

i=1

𝛼i𝝓ⁱ. (21)

The corresponding stress is defined using the constitutive equation 𝝈rb(𝜽) =C(𝜽)𝜺(udir) +

Ns

∑

i=1

𝛼iC(𝜽)𝜺(𝝓ⁱ). (22)

By introducing Equation (21) in Equation (19), we build the algebraic system

[K(𝜽)] [𝜶] = [F(𝜽)]. (23)

The elements of[K(𝜽)]and of[F(𝜽)]are defined by

Kij(𝜽) =a(𝝓ⁱ,𝝓^j;𝜽)andFi(𝜽) =f(𝝓ⁱ;𝜽). (24) Thanks to the decomposition (Equation (4)) ofC,b, andT,K_ij andF_i can be written as a linear combination of the functionsΘ^c_q(𝜽),Θ^b_q(𝜽), andΘ^T_q(𝜽)

Kij(𝜽) =

Q^c

∑

q=1

KijqΘ^q_c(𝜽)andFi(𝜽) =

Q^b

∑

q=1

biqΘ^b_q(𝜽) +

Q^T

∑

q=1

TiqΘ^T_q(𝜽), (25) where

Kijq=

∫Ω

Cq𝜺(𝝓ⁱ) ∶𝜺(𝝓^j)dΩ, biq=

∫Ω

bq·𝝓ⁱdΩ, andTiq=

∫Γ_N

Tq·𝝓ⁱdΓ.

3.2. Stress approach

The first step consists in building a stress𝝈neu(𝜽)that verifies the FE equilibrium∀𝜽. Let us introduce two sets of displacement fields(u^b_q,u^T_q′) ∈_Uh⁰, for(q,q^′) ∈ {1,… ,Q^b} × {1,… ,Q^T}, such that

a(u^b_q,u^∗_h;𝜽) =

∫_Ωbq·u^∗_hdΩanda(u^T_q′,u^∗_h;𝜽) =

∫_ΓNTq^′·u^∗_hdΓ ∀u^∗_h∈U_h⁰. (26) The stress field𝝈neu(𝜽)defined by

𝝈neu(𝜽) =

Q^b

∑

q=1

Θ^b_q(𝜽)C(𝜽)𝜺(u^b_q) +

Q^T

∑

q^′=1

Θ^T_q′(𝜽)C(𝜽)𝜺(u^T_q′) (27) verifies the equilibrium equation in the FE sense (Equation (28))

∫_Ω𝝈neu(𝜽) ∶𝜺(u^∗_h)dΩ =f(u^∗_h;𝜽) ∀u^∗_h ∈U_h⁰. (28) Let us consider the set of stress fields computed from the snapshot solutions (Equation (17))

𝝈ⁱ_rb=C(𝜽i)𝜺(u⁰_h(𝜽i) +udir)fori∈ {1,…,N_s} (29) and the set of stress fields defined by

Δ𝝈ⁱ_rb=𝝈ⁱ_rb−𝝈neu(𝜽i). (30)

FromEquations(17)and(28), itfollowsthat{Δ𝝈ⁱ_rb, fori ∈ {1, … , N_s}}isasetofstressfields equilibratedtozero intheFEsense.Anorthonormalbasis{𝜻¹, … , 𝜻^Ns } isbuiltfromΔ𝝈ⁱ_rbbya Gram – Schmidtprocesswithrespecttoaninternal

e

product ∫_Ω𝝈1 ∶C⁻¹(𝜽)𝝈2 dΩ.

Foreachparameter 𝜽,anadmissiblestressfield 𝝈_rb issoughtinthereducedbasisofequilibrated stressfieldsintheFEsense.

𝝈^e_rb(𝜽) =𝝈neu(𝜽) +

N_s

∑

i=1

𝛽i𝜻ⁱ. (31)

The coefficients𝛽iare computed in order to minimize a distance between the stress field computed in the reduced basis𝝈rb(𝜽) =C(𝜽)𝜺(u_rb(𝜽))and𝝈^e_rb(𝜽)

J(𝛽1,… , 𝛽N_s) =

∫_Ω(𝝈rb(𝜽) −𝝈^e_rb(𝜽)) ∶C⁻¹(𝜽)(𝝈rb(𝜽) −𝝈^e_rb(𝜽))dΩ. (32) The minimization of Equation (32) leads to the algebraic system

[S(𝜽)] [𝜷] = [G(𝜽)]. (33)

The elements of[S(𝜽)]and of[G(𝜽)]are defined by Sij=

∫_ΩC⁻¹(𝜽)𝜻^j∶𝜻ⁱdΩ (34) and

Gi=

∫_Ω𝜻ⁱ∶ (𝜺(urb) −C⁻¹(𝜽)𝝈neu(𝜽))dΩ. (35) Thanks to the decompositions (Equations (4) and (5)), Sij and Gi can be written as a linear combination of the functionsΘ^s_q(𝜽),Θ^b_q(𝜽), andΘ^T_q(𝜽)

Sij=

Q^s

∑

q=1

SijqΘ^s_q(𝜽)

Gi=ci0+

N_s

∑

j=1

cij𝛼j−

Q^s

∑

j=1

⎛⎜

⎜⎝

Q^b

∑

q=1

bijqΘ^s_j(𝜽)Θ^b_q(𝜽) +

Q^T

∑

q=1

TijqΘ^s_j(𝜽)Θ^T_q(𝜽)

⎞⎟

⎟⎠ ,

where

S_ijq=

∫Ω

S^q𝜻ⁱ∶𝜻^jdΩ, c_i0 =

∫Ω

𝜻ⁱ∶𝜺(u_dir)dΩ, c_ij=

∫Ω

𝜻ⁱ∶𝜺(𝝓^j)dΩ and

bijq=

∫_Ω𝜻ⁱ∶SjC(𝜽)𝜺(u^b_q)dΩ, Tijq=

∫_ΓN𝜻ⁱ∶SjC(𝜽)𝜺(u^T_q)dΓ.

4. BOUNDS ON THE QUANTITY OF INTEREST

The computation of error bounds on linear QoI has been developed by many authors [22, 33 – 35]

for measuring the gap between an exact QoI and a QoI computed from an FE analysis. There is less work concerning the error bounds between a QoI computed from an FE analysis and a QoI computed by a reduced modeling approach [36 – 38]. For any realization of the random vector𝜽, the error introduced by the reduced approximation is given by

e_rb(𝜽) =u_h(𝜽) −u_rb(𝜽); (36)

because of the linearity assumption, one has

Q(erb(𝜽);𝜽) =Q(uh(𝜽);𝜽) −Q(urb(𝜽);𝜽). (37) Following [33], we consider the following auxiliary problem: Findu^aux_h ∈_Uh⁰such that

a(u^∗_h,u^aux_h (𝜽);𝜽) =Q(u^∗_h(𝜽);𝜽) ∀u^∗_h∈U_h⁰. (38) Aserb(𝜽) ∈U_h⁰, we obtain a first representation of the error on the QoI

Q(erb(𝜽);𝜽) =a(erb(𝜽),u^aux_h (𝜽);𝜽). (39) Let us compute an approximation ofu^aux_h in the reduced basisU_rb⁰

a(u^∗_rb,u^aux_rb (𝜽);𝜽) =Q(u^∗_rb(𝜽);𝜽) ∀u^∗_rb∈_Urb⁰; (40) by using Equations (10) and (19) and remarking thatu^aux_rb (𝜽) ∈U_rb⁰ ⊂U_h⁰, we obtain

a(e_rb(𝜽),u^aux_rb (𝜽);𝜽) =0. (41) A second representation of the error on the QoI is obtained by subtracting Equation (41) from Equation (39):

Q(erb(𝜽);𝜽) =a(erb(𝜽),e^aux_rb (𝜽);𝜽), wheree^aux_rb (𝜽) =u^aux_h (𝜽) −u^aux_rb (𝜽). (42) Following [35], the error in the QoI can be written as

Q(e_rb(𝜽);𝜽) = 1

4||se_rb+s⁻¹e^aux_rb ||²_u,𝜽−1

4||se_rb−s⁻¹e^aux_rb ||²_u,𝜽, (43) wheresis a scaling factor and||u||u,𝜽=a(u,u;𝜽). Then, for anys, we have the following bounding properties

−1

4||se_rb−s⁻¹e^aux_rb ||²_u,𝜽⩽Q(e_rb(𝜽);𝜽)⩽ 1

4||se_rb+s⁻¹e^aux_rb ||²_u,𝜽. (44) In order to determine the bounds for the error in the QoI, Equation (44) indicates that it is sufficient to bound the global energy norms ofserb−s⁻¹e^aux_rb andserb+s⁻¹e^aux_rb . In this paper, the methodology proposed in [26] that is based on the concept of constitutive relation error estimator [39] is used to build bounds of these errors.

It can be shown that (we refer the reader to [22, 26, 40] for more details)

||se_rb+s⁻¹e^aux_rb ||²_u,𝜽⩽||(s𝝈^e_rb+s⁻¹𝝈^aux_rb ^,e) − (s𝝈rb+s⁻¹𝝈^aux_rb )||²_𝜎,𝜽,

||se_rb−s⁻¹e^aux_rb ||²_u,𝜽⩽||(s𝝈^e_rb−s⁻¹𝝈^aux,e_rb ) − (s𝝈rb−s⁻¹𝝈^aux_rb )||²_𝜎,𝜽, (45) where𝝈rband𝝈^aux_rb are the stresses computed from the reduced basis solutionsu_rbandu^aux_rb through the constitutive equation (Equation (22)), and||𝜎||𝜎,𝜽is defined by

||𝜎||²_𝜎,𝜽=<𝝈, 𝝈>𝜎,𝜽with<𝝈1, 𝝈2>𝜎,𝜽=

∫_Ω𝝈1∶C⁻¹(𝜽)𝝈2dΩ.

𝝈^e_rb(𝜽)and𝝈^aux,e_rb (𝜽)are stress fields equilibrated in the FE sense that are computed, for a given value of𝜽, as proposed in Section 3.2

∫Ω

𝝈^e_rb(𝜽) ∶𝜺(u^∗_h)dΩ =f(u^∗_h;𝜽)and

∫Ω

𝝈^aux,e_rb (𝜽) ∶𝜺(u^∗_h)dΩ =Q(u^∗_h;𝜽). The value that minimizes the bounds is

s^opt=

√||𝝈^aux_rb^,e−𝝈^aux_rb ||𝜎,𝜽

||𝝈^e_rb−𝝈rb||𝜎,𝜽 .

Finally, introducings^optin Equation (45) provides us the following computable bounds 1

4||serb+s⁻¹e^aux_rb ||²_u,𝜽⩽e⁺_rb(𝜽), 1

4||serb−s⁻¹e^aux_rb ||²_u,𝜽⩽e⁻_rb(𝜽),

(46)

where

e⁺_rb(𝜽) = 1 2

(||𝝈^e_rb−𝝈rb||𝜎,𝜽 ||𝝈^aux,e_rb −𝝈^aux_rb ||𝜎,𝜽+<𝝈^e_rb−𝝈rb,𝝈^aux,e_rb −𝝈^aux_rb >𝜎,𝜽)

e⁻_rb(𝜽) = 1 2

(||𝝈^e_rb−𝝈rb||𝜎,𝜽 ||𝝈^aux,e_rb −𝝈^aux_rb ||𝜎,𝜽−<𝝈^e_rb−𝝈rb,𝝈^aux,e_rb −𝝈^aux_rb >𝜎,𝜽) .

(47)

From Equations (37) and (44), it follows that the QoIQ(u_h(𝜽);𝜽)is bounded by

Q(urb(𝜽);𝜽) −e⁻_rb(𝜽)⩽Q(uh(𝜽);𝜽)⩽Q(urb(𝜽);𝜽) +e⁺_rb(𝜽). (48) Remark

IfSdoes not depend linearly on uh, bounding of the quantity interest is much more difficult and there are few works addressing the subject. In the case whereQis anL2norm of the stress or a mean von Mises stress over a subdomain, it is possible to derive approximate upper bounds as shown in [41].

5. ERROR BOUNDS ON THE FAILURE PROBABILITY AND REDUCED BASIS ALGORITHM

In the following section, let us assume for the sake of simplicity that the thresholdRis a constant and that the limit state function defined in Equation (12) is

G(𝜽) =R(𝜽) −Q(uh(𝜽);𝜽);

the failure domainDf is defined by

Df = {𝜽; R(𝜽) −Q(uh(𝜽);𝜽)⩽0}.

The failure probability computed by a Monte Carlo method is given by Equation (15) P̂f = 1

NMC N_MC

∑

k=1

1_Df(𝜽^k). (49)

5.1. Error bounds on the failure probability

The approximate limit state function computed from the reduced basis analysis is Grb(𝜽) =R(𝜽) −Q(urb(𝜽);𝜽).

The reduced basis failure domain is defined by

D^rb_f = {𝜽; R(𝜽) −Q(urb(𝜽);𝜽)⩽0}.

The approximate failure probability computed by a Monte Carlo method is given by P̂^rb_f = 1

NMC N_MC

∑

k=1

1_D^rb

f (𝜃^k), (50)

and from Equation (48), the following computable bounds of the limit state function G(𝜽) are obtained

G⁻_rb(𝜽)⩽G(𝜽)⩽G⁺_rb(𝜽), (51)

where

G⁻_rb(𝜽) =G_rb(𝜽) −e⁺_rb(𝜽)andG⁺_rb(𝜽) =G_rb(𝜽) +e⁻_rb(𝜽). (52) Let us introduce the failure domain associated with the lower bound and the upper bound of the limit state function and the associated failure probabilities

D⁺_f = {𝜽; Grb(𝜽) −e⁺_rb(𝜽)⩽0}andD⁻_f = {𝜽; Grb(𝜽) +e⁻_rb(𝜽)⩽0}, P̂⁺_f = 1

N_MC

N_MC

∑

k=1

1_D⁺

f(𝜽^k)andP̂⁻_f = 1 N_MC

N_MC

∑

k=1

1_D⁻_f(𝜽^k). (53) AsD⁻_f ⊂Df ⊂D⁺_f, we obtain the following bounds for the failure probability

P̂⁻_f ⩽P̂f ⩽P̂⁺_f. (54)

Remark

It must be noticed that the obtained bounds concern the failure probabilityP̂_f computed from a Monte Carlo simulation. The quality of the Monte Carlo simulation will be classically assessed in the examples presented in Section 6 via the central limit theorem and the computation of a confidence interval.

Furthermore, to control the reduced basis algorithm, an error estimator𝜀G(𝜽), which is an upper bound on the error on the limit state functioneG(𝜽) =|G(𝜽) −Grb(𝜽)|, is introduced

e_G(𝜽)⩽𝜀G(𝜽) =max(e⁻_rb(𝜽),e⁺_rb(𝜽)). (55) Properties:

1. If for all𝜽^k, 𝜀G(𝜽^k) =0 thenP̂⁻_f =P̂⁺_f =P̂f.

2. IfP̂⁻_f =P̂⁺_f, then𝜀G(𝜽^k) =0 for all𝜽^kandP̂f =P̂⁻_f =P̂⁺_f. 5.2. Reduced basis algorithm

The initial step of the algorithm consists in computing a displacement fieldudirsatisfying the bound- ary conditions (Equation (20)) and a stress field𝝈neu(𝜽)satisfying the equilibrium in the FE sense (Equations (26) and (27)). The computation ofudirand𝝈neu(𝜽)involves the choice of a particular value of𝜽denoted𝜽; a simple choice is to use the mean value of the random vector𝜽=E[𝜽]. This value will also be used in the Gram – Schmidt orthonormalizing algorithm. The Monte Carlo simu- lation process starts generating a sample of𝜽denoted𝜽1, which is used to compute the initial vector of the displacement reduced basis𝝓¹=u⁰_h(𝜽1)from Equation (17) and the initial vector of the stress reduced basis𝜻¹= Δ𝝈¹_rbfrom Equations (29) – (30).

From this initial reduced basis, the idea is to improve the reduced basis during the Monte Carlo simulation process, increasing the number of elements in the reduced basis only when it is required.

An admissibility criterion is required to control the algorithm. A prescribed tolerance𝜏is set. The reduced basis will be improved for a given Monte Carlo sample𝜽iif the state of the structure (safe or failure) cannot be certified and the error estimator on the limit state function𝜀G(𝜽i)is greater than 𝜏. The algorithm is described in Algorithm 1.

Let us denote by_Urb^0,N = span{𝝓¹,… ,𝝓^N}and _Srb^0,N = span{𝜻¹,… ,𝜻^N}the reduced basis spaces that have been generated along with thenMonte Carlo samples𝜽i(i∈ {1,… ,n}). For the next Monte Carlo sample𝜽n+1, an approximate solutionu_rb(𝜽n+1)ofu_h(𝜽n+1) is computed in the reduced basis space_Urb^0,N. The approximate limit state functionGrb(𝜽n+1)is computed as well as its upper boundG⁺_rb(𝜽n+1)and lower boundG⁻_rb(𝜽n+1)(Equation (52)). In order to estimate the failure probability, we need to count the number of samples that belongs to the failure domain. Let us denote N_f^rbthe number of samples𝜽iwhereGrb(𝜽i)⩽0,N_f⁺the number of samples whereG⁻_rb(𝜽i)⩽0, and N_f⁻the number of samples whereG⁺_rb(𝜽i)⩽0. Three cases need to be distinguished:

• IfG⁻_rb(𝜽n+1) > 0, then G(𝜽n+1) > 0 (Equation (52)), and the accuracy of the reduced basis approximation is sufficient to ensure that the structure is in a safe state;

• ifG⁺_rb(𝜽n+1) ⩽ 0, then G(𝜽n+1) ⩽ 0 (Equation (52)), and the accuracy of the reduced basis approximation is sufficient to ensure that the structure is in a failure state;

• ifG⁻_rb(𝜽n+1) ⩽0 andG⁺_rb(𝜽n+1) > 0, then the accuracy of the reduced basis approximation is not sufficient to certify the state of the structure.

In the first and second cases, there is no need to improve the reduced basis and the Monte Carlo simulations are continued by generating the next sample. In the third case, the reduced basis is improved only if the error estimator on the limit state function is greater than a fixed tolerance 𝜀G(𝜽n+1)⩾𝜏: Equation (17) is solved in the FE space, and new reduced basis spaces are constructed U_rb^0,N+1=span{𝝓¹,…,𝝓^N+1}andS_rb^0,N+1 =span{𝜻¹,… ,𝜻^N+1}.

• If a new vector of the reduced basis is generated from𝜽i, thenuh(𝜽i)is computed and no error due to the reduced basis approximation is introduced: G(𝜽i) = G⁺_rb(𝜽i) = G⁻_rb(𝜽i). Hence, if G_rb(𝜽i)⩽0,N_f⁺,N_f^rbandN_f⁻are increased by one altogether.

• If u_rb(𝜽i) is computed from the current reduced basis, G⁻_rb(𝜽n+1) ⩽ 0,G⁺_rb(𝜽n+1) > 0, and 𝜀G(𝜽n+1)< 𝜏, then the failure state of the structure is not determined accurately by the reduced basis approximation and only the upper boundN_f⁺is increased by one (the value ofN_f^rbbeing modified accordingly to the value ofGrb(𝜽i)).

The reduced basis failure probabilityP̂^rb_f and the boundsP̂⁻_f andP̂⁺_f of the failure probabilityP̂f

are then easily computed

P̂^rb_f = N_f^rb N_MC, ̂P⁺_f =

N_f⁺

N_MC, ̂P⁻_f = N_f⁻ N_MC.

6. NUMERICAL EXAMPLES 6.1. Example 1

The first example is a plate with two rectangular holes proposed in [21]. The structure is submitted, in plane strain, to a normal tractionP = 20 MPa applied along the vertical edge, and the plate is considered to be composed of three different materials. The symmetry of the problem allows to study only one-fourth of the plate as shown on Figure 1. The Poisson ratios are fixed𝜈1 =𝜈2 =𝜈3 =0.30.

The Young’s moduliE_i are random independent variables such that their cumulative distribution function is uniform on[184.5,225.5]GPa. The QoI considered here is the average displacement on the lineL_𝜔.

Q(u(𝜽),𝜽) = 1

mes(L_𝜔)∫_L𝜔u(𝜽) ·n_𝜔dl. The limit state function is given by

G(𝜽) =u−Q(𝜽),

whereuis set to 10⁻⁴. Figure 2 shows the mesh used for the FE analysis. A direct Monte Carlo simu- lation is performed, with the complete FE model, usingnMC =1000 as a reference value. The failure

Figure 1. Example 1: thick plate studied.

Figure 2. Example 1 mesh: six-node triangular elements – 2122 DoF.

Table I. Example 1: computed failure probability and bounds.

𝜏 0.1 0.05 10⁻² 10⁻³ 10⁻⁴ Reference P̂^rb_f (10⁻²) 6.7 9.1 8.5 8.5 8.5 8.5 P̂⁺_f (10⁻²) 24.1 10.9 8.8 8.6 8.5 — P̂⁻_f (10⁻²) 1.6 7.5 8.4 8.5 8.5 —

N^maxs 2 3 3 4 5 —

probability obtained isP̂f =8.5×10⁻²and the 95%confidence interval is[6.77×10⁻²,10.2×10⁻²].

The coefficient of variation𝛿obtained from Equation (16) is𝛿=10.8%and shows that the number of Monte Carlo simulations is sufficient for this example.

Five Monte Carlo simulations (n_MC =1000) are then performed with the reduced basis algorithm for different values of the tolerance𝜏. For each simulation, we use the same set of random values for the Young’s moduli. The results are shown in Table I where the fixed tolerance𝜏, the failure probability computed with the reduced basis approachP̂^rb_f , its upper and lower boundsP̂⁺_f andP̂⁻_f, and the maximum number of computed snapshotsN_s^maxhave been reported. The number of elements in the basis increases when the tolerance𝜏decreases. However, the same number of elements in the basis does not lead to the same bounds as it is shown for𝜏=0.05 and𝜏 =0.01 where the number of elements in the basis is 3 and the obtained bounds are clearly different. It can be observed that five snapshots are sufficient to obtain a gap between the bounds equal to zero and thus to ensure that the failure probability computed by reduced basis approach is equal to the failure probability that will be obtained by an FE approach. (The elements of the generated reduced basis are shown in Figure 3). That means that the whole Monte Carlo algorithm requires only five complete FE computations. If we take into account that the 95%confidence interval is[6.77×10⁻²,10.2×10⁻²], it is clear that obtaining an exact evaluation ofP̂f is an overkill and that tolerance 𝜏 = 10⁻² is sufficient to obtain reasonable bounds forP̂_f(i.e.,P̂⁺_f−P̂⁻_f is small compared with the 95%confidence interval).

The evolution of the number of snapshots as a function of the number of Monte Carlo throws for different values of𝜏is shown in Figure 4.