advanced methods for fe-reliability analysis

Advanced methods for FE-reliability analysisAdaptive polynomial chaos expansions

B. Sudret1,2, G. Blatman3

1Phimeca Engineering, Centre d’Affaires du Zenith, 34 rue de Sarlieve, F-63800 Cournond’Auvergne

2Clermont Universite, IFMA, EA 3867, Laboratoire de Mecanique et Ingenieries, BP 10448,F-63000 Clermont-Ferrand

3EDF R&D, Dpt for Materials and Mechanics of Components, Site des Renardieres, F-77818Moret-sur-Loing

December 2nd, 2009

Uncertainty assessmentPolynomial chaos expansions

Adaptive sparse polynomial chaos expansionsApplications examples

Conclusions

Outline

1 Uncertainty assessment

2 Polynomial chaos expansions

3 Adaptive sparse polynomial chaos expansions (Blatman & Sudret, 2008-9)

4 Applications in structural reliability

5 Conclusions

Bruno Sudret (Phimeca) JCSS workshop on semi-probabilistic FEM calculations December 2nd, 2009 2 / 32



Conclusions

Global framework

Step A

Model(s) of the system

Assessment criteria

Step B

Quantification of

sources of uncertainty

Step C

Uncertainty propagation

Random variables Mechanical model Moments

Probability of failure

Response PDF

...

Step C’

Sensitivity analysis

Step C’





Conclusions

Global framework

Step A

Model(s) of the system

Assessment criteria

Step B

Quantification of

sources of uncertainty

Step C

Uncertainty propagation

Random variables Mechanical model Moments

Probability of failure

Response PDF

...

Step C’


Step C’





Conclusions

Mechanical models (Step A)

Computationalmodel M

Vector of inputparametersx ∈ R

M

Model responsey = M(x) ∈ R

N

geometry

materialproperties

loading

analytical formula

finite elementmodel

etc.

displacements

strains, stresses

temperature,etc.




Conclusions

Mechanical models (Step A)

Computationalmodel M

Vector of inputparametersx ∈ R

M

Model responsey = M(x) ∈ R

N

geometry

materialproperties

loading

analytical formula

finite elementmodel

etc.

displacements

strains, stresses

temperature,etc.




Conclusions

Uncertainty propagation methods (Step C)

Mechanical model

Step A

Mean/std.

deviationµ

σ

Probabilistic-

mechanical

model

Probability

of failure Pf

Probabilistic model

Step B Response

PDF




Conclusions


Mechanical model

Step A

Mean/std.

deviationµ

σ

Probabilistic-

mechanical

model

Probability

of failure Pf

Probabilistic model

Step B Response

PDF




Conclusions


Mechanical model

Step A

Mean/std.

deviationµ

σ

Probabilistic-

mechanical

model

Probability

of failure Pf

Probabilistic model

Step B Response

PDF




Conclusions

From Monte Carlo simulation to the spectral approach

Monte Carlo simulation

It is based on the sampling of the input random variables according totheir joint PDF fX (x).

For each sample x(i), the response is computed M(x(i)) (possiblytime-consuming).

The response sample set M = M(x(1)), . . . ,M(x(n))T is used tocompute statistical moments, probabilities of failure or estimate theresponse distribution (histograms, kernel densities).

Spectral approach

The response random vector Y = M(X) is considered as an element of asuitable functional space.

A basis of this space is build up (with respect to the input joint PDF).

Y is completely determined by its coordinates in this basis.




Conclusions

From Monte Carlo simulation to the spectral approach

Monte Carlo simulation

It is based on the sampling of the input random variables according totheir joint PDF fX (x).

For each sample x(i), the response is computed M(x(i)) (possiblytime-consuming).

The response sample set M = M(x(1)), . . . ,M(x(n))T is used tocompute statistical moments, probabilities of failure or estimate theresponse distribution (histograms, kernel densities).

Spectral approach

The response random vector Y = M(X) is considered as an element of asuitable functional space.

A basis of this space is build up (with respect to the input joint PDF).

Y is completely determined by its coordinates in this basis.




Conclusions

Generalized polynomial chaos basisComputation of the expansion coefficientsPost-processing of the coefficients

Principles

Random model response

Y = M(X) dim X = M, dim Y = 1

X : random vector of input parameters of prescribed PDF fX (X)

Y : scalar random response

If E[Y 2

]< ∞, then Y ∈ L2(Ω,F , PX ), the Hilbert space of second order

random variables equipped with the inner product < Y, Z >= E [Y Z]

Hilbertian basisY =

∑j∈IN

yj Ψj(X)

where:

Ψj(X): polynomial chaos basis

yj : “coordinates”, i.e. coefficients to be computed




Conclusions


Principles

Random model response

Y = M(X) dim X = M, dim Y = 1

X : random vector of input parameters of prescribed PDF fX (X)

Y : scalar random response

If E[Y 2

]< ∞, then Y ∈ L2(Ω,F , PX ), the Hilbert space of second order

random variables equipped with the inner product < Y, Z >= E [Y Z]

Hilbertian basisY =

∑j∈IN

yj Ψj(X)

where:

Ψj(X): polynomial chaos basis

yj : “coordinates”, i.e. coefficients to be computed




Conclusions


Multivariate orthogonal polynomials

Orthogonal polynomials

Suppose X has independent components: fX (x) =∏M

k=1 fXi(xi) anddefine the natural inner product w.r.t to fXi :

< φ1 , φ2 >=

∫φ1(x) φ2(x)fXi(x) dx

Define the one-dimensional orthonormal polynomials ψik, k ∈ IN

satisfying < ψij , ψi

k >= δjk

Marginal distribution Orth. polynomialsUniform 1[−1,1](x) Legendre

Gaussian 1/√

2πe−x2/2 Hermite

Multivariate polynomials obtained by tensor product:

Ψα(x) =

M∏i=1

ψiαi

(xi)




Conclusions


Multivariate orthogonal polynomials

Orthogonal polynomials

Suppose X has independent components: fX (x) =∏M

k=1 fXi(xi) anddefine the natural inner product w.r.t to fXi :

< φ1 , φ2 >=

∫φ1(x) φ2(x)fXi(x) dx

Define the one-dimensional orthonormal polynomials ψik, k ∈ IN

satisfying < ψij , ψi

k >= δjk

Marginal distribution Orth. polynomialsUniform 1[−1,1](x) Legendre

Gaussian 1/√

2πe−x2/2 Hermite

Multivariate polynomials obtained by tensor product:

Ψα(x) =

M∏i=1

ψiαi

(xi)




Conclusions


Generalized polynomial chaos basis Soize & Ghanem, 2004

TheoremThe set of multivariate polynomials in the input random vectorΨα(X), α ∈ INM forms a basis of L2(Ω,F , PX )

Y =∑

α∈INM

yα Ψα(X)

In practice:

The univariate polynomials are computed

The multi-indices α ∈ A are generated, where A is a finite subset of INM

The random response is approximated by a truncated series:

Y ≈ Y =∑α∈A

yα Ψα(X)

NB: if X has correlated components, the Nataf transform may be used to recast the

problem in terms of standard normal random variables.




Conclusions




Y =∑

α∈INM

yα Ψα(X)

In practice:




Y ≈ Y =∑α∈A

yα Ψα(X)






Conclusions




Y =∑

α∈INM

yα Ψα(X)

In practice:




Y ≈ Y =∑α∈A

yα Ψα(X)






Conclusions


Application example (1)

Model: Y = M(X1, X2) where Xi ∼ N (µi, σi)

Isoprobabilistic transform: X = T (ξ) : Xi = µi + σi ξi

Hermite polynomials

Obtained by recurrence formula:

H−1(x) = H0 = 1

Hn+1(x) = x Hn(x) − n Hn−1(x)

First polynomials are:1, x, x2 − 1, x3 − 3x, . . .

Normalize: ‖ Hk ‖2= 1, 1, 2, 6, . . .

Common truncation scheme:

A = α ∈ INM : ||α||1 =

M∑i=1

αi ≤ p Card A ≡ P =(M + p

p

)




Conclusions



Model: Y = M(X1, X2) where Xi ∼ N (µi, σi)

Isoprobabilistic transform: X = T (ξ) : Xi = µi + σi ξi

Hermite polynomials

Obtained by recurrence formula:

H−1(x) = H0 = 1

Hn+1(x) = x Hn(x) − n Hn−1(x)

First polynomials are:1, x, x2 − 1, x3 − 3x, . . .

Normalize: ‖ Hk ‖2= 1, 1, 2, 6, . . .

Common truncation scheme:

A = α ∈ INM : ||α||1 =

M∑i=1

αi ≤ p Card A ≡ P =(M + p

p

)




Conclusions



Truncated PC expansion of degree p = 3

The multivariate polynomials in (ξ1, ξ2) of total degree less than 3 arecomputed.

The series contains P =(3+23

)= 10 terms

j α Ψα ≡ Ψj

0 [0, 0] Ψ0 = 11 [1, 0] Ψ1 = ξ12 [0, 1] Ψ2 = ξ23 [2, 0] Ψ3 = (ξ2

1 − 1)/√

24 [1, 1] Ψ4 = ξ1ξ25 [0, 2] Ψ5 = (ξ2

2 − 1)/√

2

6 [3, 0] Ψ6 = (ξ31 − 3ξ1)/

√6

7 [2, 1] Ψ7 = (ξ21 − 1)ξ2/

√2

8 [1, 2] Ψ8 = (ξ22 − 1)ξ1/

√2

9 [0, 3] Ψ9 = (ξ32 − 3ξ2)/

√6

Y ≡ MPC(ξ1, ξ2) = a0 + a1 ξ1 + a2 ξ2

+ a3 (ξ21 − 1)/

√2 + a4 ξ1ξ2

+ a5 (ξ22 − 1)/

√2 + a6 (ξ3

1 − 3ξ1)/√

6

+ a7 (ξ21 − 1)ξ2/

√2 + a8 (ξ2

2 − 1)ξ1/√

2

+ a9(ξ32 − 3ξ2)/

√6




Conclusions


Regression approach Berveiller, 2005

PrincipleCompute the PC coefficients of a truncated series by regression of the modelresponse onto the PC basis:

Y = M(X) =

P−1∑j=0

yj Ψj(X) + εP ≡ YTΨ(X) + εP

where Y = y0, . . . , yP−1T and Ψ(x) = Ψ0(x), . . . , ΨP−1(x)T

Least-square minimization

Y = arg minY∈RP

E

[(M(X) − YTΨ(X)

)2]

Exact solution: Y = E [M(X)Ψ(X)] (equivalence with projection)




Conclusions


Regression approach Berveiller, 2005

PrincipleCompute the PC coefficients of a truncated series by regression of the modelresponse onto the PC basis:

Y = M(X) =

P−1∑j=0

yj Ψj(X) + εP ≡ YTΨ(X) + εP

where Y = y0, . . . , yP−1T and Ψ(x) = Ψ0(x), . . . , ΨP−1(x)T

Least-square minimization

Y = arg minY∈RP

E

[(M(X) − YTΨ(X)

)2]

Exact solution: Y = E [M(X)Ψ(X)] (equivalence with projection)




Conclusions


Regression approach (cont’)

Practical computation

Select an experimental designX = x(1), . . . , x(n)T

Compute the vector of model responseM = M(x(1)), . . . ,M(x(n))T

Compute the experimental matrixAij = Ψj(x

(i)) i = 1, . . . , n ; j = 0, . . . , P − 1

Solve the least-square minimization problem

Y = (ATA)−1ATM




Conclusions


Regression approach (cont’)

Practical computation

Select an experimental designX = x(1), . . . , x(n)T

Compute the vector of model responseM = M(x(1)), . . . ,M(x(n))T

Compute the experimental matrixAij = Ψj(x

(i)) i = 1, . . . , n ; j = 0, . . . , P − 1

Solve the least-square minimization problem

Y = (ATA)−1ATM




Conclusions


Post-processing of PC expansions

Response PDF

Sample the PC expansion y(i) = YTΨ(x(i)), i = 1, . . . , nK, where

x(i)’s are drawn according to fX (x) (nK = 106−7)

Use of kernel density estimation Wand & Jones, 1995

fY (y) =1

nK hK

nK∑i=1

K

(y − y(i)

hK

)Response

PDF




Conclusions


Post-processing of PC expansions

Response PDF

Sample the PC expansion y(i) = YTΨ(x(i)), i = 1, . . . , nK, where

x(i)’s are drawn according to fX (x) (nK = 106−7)

Use of kernel density estimation Wand & Jones, 1995

fY (y) =1

nK hK

nK∑i=1

K

(y − y(i)

hK

)Response

PDF

Statistical momentsµPC

Y ≡ E [Y ] = y0

σ2,PCY ≡ Var [Y ] =

P−1∑j=1

y2j

...

Mean/std.

deviationµ

σ




Conclusions


Post-processing of PC expansions (cont’)

Reliability analysis Sudret & Der Kiureghian, 2002Berveiller et al., 2005

Define a limit state functiong(X) ≡ g (X1, M(X2)) and theassociated probability of failure

Pf =

∫DX

1g(x)≤0(x) fX (x) dx

Compute the response PC expansion

MPC(X2) =∑

0≤|α|≤p

yα Ψα(X2)

Failure domainDf = x:g(x) ≤ 0

Safe domain Ds

Apply standard reliability methods usingg

(X1, MPC(X2)

), e.g. Monte Carlo sim-

ulation, FORM + importance sampling, etc. Probability

of failurePf




Conclusions





Pf =

∫DX



MPC(X2) =∑

0≤|α|≤p

yα Ψα(X2)


Safe domain Ds


(X1, MPC(X2)



of failurePf




Conclusions





Pf =

∫DX



MPC(X2) =∑

0≤|α|≤p

yα Ψα(X2)


Safe domain Ds


(X1, MPC(X2)



of failurePf




Conclusions

Error estimators and sparse truncation schemesAdaptive algorithms

Error estimatorsCoefficient of determination

The regression technique is based on the minimization of the mean squareerror. The generalization error is defined as:

Egen = E

[(M(X) −MPC(X)

)2]

It may be estimated by the empirical error using the already computed responsequantities:

Eemp =1

n

n∑i=1

(M(x(i)) −MPC(x(i))

)2

The coefficient of determination R2 is often used as an error estimator:

R2 = 1 − Eemp

V[Y]V[Y] =

1

n(M(x(i)) − Y)2

This error estimator leads to overfitting




Conclusions


Error estimatorsLeave-one-out cross validation

Idea: in statistical learning theory, cross validation consists in splitting theexperimental design Y in two parts, namely a training set (which is used tobuild the model) and a validation set.

The leave-one-out technique consists in:

Computing the coefficients of the PC expansion from the experimentaldesign X\x(i) (the corresponding expansion is denoted by MPC\i(X))

Evaluating the predicted residual:

∆i = M(x(i)) −MPC\i(x(i))

Estimating the leave-one-out error and related Q2 error estimator

ELOO =1

n

n∑i=1

∆2i Q2 = 1 − ELOO

V[Y]




Conclusions


The curse of dimensionality

Common truncation scheme: all the multivariate polynomials up to aprescribed order p are selected. The number of unknown coefficients growthspolynomially both in M and p.

P =

(M + p

p

)

So does the size of experimental design, e.g. n = 2P when using Latinhypercube sampling.

“Full” expansions are not tractable when M ≥ 10

Solutions:

sparse truncation schemes, based on the sparsity-of-effect principle

adaptive algorithms for the construction of the PC expansion




Conclusions


Hyperbolic truncation schemes

Degree of a multi-index α : |α| ≡ ||α||1 =∑M

i=1 αi (total degree of Ψα)

Rank of a multi-index α : ||α||0 =∑M

i=1 1αi>0

Sparsity-of-effects principle: in usual problems, only low-order interactionsbetween the input variables are relevant. One shall select PC approximationusing low-rank monomials.

Hyperbolic truncation sets

AM,p,q = α ∈ INM : ||α||q ≤ p

||α||q ≡(

M∑i=1

αqi

)1/q

, 0 < q < 1




Conclusions


Adaptive algorithm: principles

Adaptivity in the PC basis:

Build up the polynomial chaos basis from scratch, i.e. starting with asingle constant term, using an iterative algorithm

Add monomial terms (that are selected using the hyperbolic truncationscheme) one-by-one and decide whether they are kept or discarded (R2

coefficient is used)

Monitor the PC error using cross-validation (Q2 estimate) and stop whenQ2

tgt is attained.

Adaptivity in the experimental design:

In each step, the PC coefficients are computed by regression. Nestedexperimental designs (e.g. quasi-random numbers, nested LHS) are usedin order to make the regression problem well-posed.




Conclusions


Basis-and-design adaptive algorithm: scheme




Conclusions

Elastic trussFrame structureFoundation with spatial variability

Elastic truss Blatman et al., 2007

Problem statement

10 independent random variables

4 section/material properties(lognormal)

6 loads (Gumbel)

Use of Hermite polynomial chaos

Response quantity: vertical displacement at midspan

Reliability problem:

g(X) = vmax − v(X) = vmax − FEM(E1, A1, E2, A2, P1, . . . , P6)




Conclusions


Structural reliability results (adaptive algorithm)

Threshold (cm) Reference Full PCE Sparse PCE

βREF β ε (%) β ε (%)

10 1.72 1.71 0.6 1.72 0.011 2.38 2.38 0.0 2.38 0.012 2.97 2.98 0.3 2.99 0.714 3.98 4.04 1.5 4.07 2.316 4.85 4.95 2.1 5.02 3.5

1 − Q2 1 · 10−6 9 · 10−5

Number of terms 286 114Number of FE runs 443 207

Using ∼ 200 finite element runs, the structural reliability problem with various

thresholds is solved accurately up to β = 5.




Conclusions


PDF of the maximal deflection

Using ∼ 200 finite element runs, the PDF of the maximal deflection is

accurately computed in the central part as well as in the tails.




Conclusions


Frame under lateral loads

Problem statement

21 correlated randomvariables:

2 Young’s moduli(truncated Gaussian)

8 cross sections and 8moments of inertia(truncated Gaussian)

3 loads (lognormal)

Use of Nataf transform and Hermite polynomial chaos




Conclusions


Structural reliability results (adaptive algorithm)

Threshold (cm) Reference Full PC Sparse PC

βREF β ε (%) β ε (%)

4 2.27 2.26 0.4 2.29 0.95 2.96 3.00 1.4 3.01 1.76 3.51 3.60 2.6 3.61 2.87 3.96 4.12 4.0 4.11 3.88 4.33 4.58 5.8 4.56 5.3

1 − Q2 1 · 10−3 1 · 10−3

Number of terms 2,024 138Number of FE runs 53,240 3,724 450

Using ∼ 450 finite element runs, the reliability problem with variousthresholds is solved accurately up to β = 4.5.

The sparse polynomial chaos approach divides the computational cost by8 (from 3,750 to 450 runs).




Conclusions


Foundation with spatial variability Blatman et al., 2009

Elastic soil mass with spatialvariability: lognormal random fieldwith µ = 50 MPa, σ = 15 MPa andGaussian autocorrelation function( = 15 m):

ρ(x, x′) = exp

[−||x − x′||2

2

]

Model response: mean settlement under the applied load

Karhunen-Loeve expansion of the random field (38 random variables)

Sparse adaptive algorithm




Conclusions


Foundation - Results (1)

Moments Reference Sparse PCE’s Full PCEQ2

tgt = 0.99 Q2tgt = 0.995 p = 2

µY (cm) 5.93 5.93 5.93 5.93σY (cm) 1.25 1.26 1.26 1.26

γY 0.7 0.5 0.6 0.6κY 3.9 3.4 3.5 3.5

Number of terms - 254 360 780PC degree - 2 3 2

Number of FE runs 50, 000 428 579 820

Good accuracy of the sparse PC expansion, not much difference with afull second order PC expansion

Reduction of the computational cost w.r.t a full expansion of order 2 by afactor 2




Conclusions


Foundation - Results (1)

Only some modes of theKarhunen-Loeve expansion areimportant in the analysis

The antisymmetric modesw.r.t. vertical do not play anyrole




Conclusions


Moments of the vertical displacement field

Correlation coefficient between max. vertical displacement and current point




Conclusions

Conclusions

Spectral methods (e.g. polynomial chaos expansions) are ofincreasing importance in stochastic computational mechanics andmay be fruitfully applied to structural reliability.

In order to bypass the curse of dimensionality, sparse PC expansionsobtained by regression through adaptive algorithms reveal promising(up to M = 50 − 100 random variables).

The Q2 leave-one-out error is a relevant estimator to master theaccuracy of the PC expansions.

Basis-and-design adaptive algorithms coupled with hyperbolictruncation sets allow one to reduce the computational cost by afactor 2 to 10 w.r.t classical PC expansions.




Conclusions

Thank you very much for your attention!

Acknowlegment: The partnership between Phimeca, IFMA/LaMI and EDF R&D

under Contract # 8610-AAP-5910049992 is gratefully acknowledged.




Conclusions

References

Berveiller, M. (2005).

Elements finis stochastiques : approches intrusive et non intrusive pour des analyses de fiabilite.Ph. D. thesis, Universite Blaise Pascal, Clermont-Ferrand.

Blatman, G. (2009).

Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis.Ph. D. thesis, Universite Blaise Pascal, Clermont-Ferrand.in preparation.

Blatman, G. and B. Sudret (2008).

Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach.Comptes Rendus Mecanique 336(6), 518–523.

Blatman, G. and B. Sudret (2009).

Use of sparse polynomial chaos expansions in adaptive stochastic finite element analysis.Prob. Eng. Mech..(In press).

Soize, C. and R. Ghanem (2004).

Physical systems with random uncertainties: chaos representations with arbitrary probability measure.SIAM J. Sci. Comput. 26(2), 395–410.

Sudret, B. (2007).

Uncertainty propagation and sensitivity analysis in mechanical models – Contributions to structural reliability and stochasticspectral methods.Habilitation a diriger des recherches, Universite Blaise Pascal, Clermont-Ferrand, France.


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