Polynomial chaos expansions for structural reliability analysis

Transcription

1 DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Polynomial chaos expansions for structural reliability analysis B. Sudret & S. Marelli Incl. inspiring discussions with P. Kersaudy & J. Wiart (Orange Labs) Journée de la conception robuste et fiable April 10th, 2015

2 Introduction Structural reliability as a research field has emerged in the civil engineering community in the 70 s. It is nowadays a mature science with numerous domains of applications. In parallel computer power has blown up and computer simulation (e.g. finite element models) is used in all domains of engineering (e.g. civil, mechanical, electrical, chemical, etc.). Efficient computational schemes are necessary for taking the best of computational mechanics and structural reliability B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

3 Structural reliability Problem statement Random variables X Computational model M Mechanical response (load effects) M(X) geometry material prop. loading finite element models displacements strains & stresses damage B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

4 Structural reliability Probability of failure Mechanical response M(x) x 2 Failure domain D f = {x: g(x) 0} Limit state function g(m(x), x, d) Safe domain Ds x 1 Resistance X Deterministic parameters d e.g. g(m(x), t adm ) = t adm M(x) P f = IP [g(x) 0] = f X (x) dx D f ={x:g(x) 0} B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

5 Monte Carlo simulation Probability of failure P f = R M 1 Df (x) f X (x) dx E [ 1 Df (X) ] ˆPf = 1 n n k=1 1 Df (x k ) = N f n Accuracy: measured by the coefficient of variation CV Pf (N P f ) 1/2 Computational cost: N 10 k+2 for estimating P f 10 k with CV Pf 10% How to combine efficiently advanced structural models and structural reliability? B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

6 Strategies for reducing the computational cost Estimation of P f through geometrical approximations of the limit state surface (FORM/SORM) N = Advanced simulation methods Directional simulation / line sampling Importance sampling Subset simulation N = Use of surrogate models P f = IP [ g(x) 0] = f X (x) dx D 0 f ={x : g(x) 0} where g(x) g(x) in some sense B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

7 Surrogate models for uncertainty quantification A surrogate model M is an approximation of the original computational model: It is built from a limited set of runs of the original model M called the experimental design X = { x (i), i = 1,..., n } It assumes some regularity of the model M and some general functional shape Name Shape Parameters Polynomial chaos expansions M(x) = y α Ψ α(x) α A Kriging M(x) = βt f (x) + Z(x, ω) β, σz 2, θ m Support vector machines M(x) = y i K(x i, x) + b y, b i=1 y α B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

8 Polynomial chaos basis Computation of the coefficients Consider the input random vector X (dim X = M ) with given probability density function (PDF) f X (x) = M i=1 f X i (x i) Assuming that the random output Y = M(X) has finite variance, it can be cast as the following polynomial chaos expansion: Y = y α Ψ α(x) α N M where : y α : coefficients to be computed (coordinates) Ψ α(x) : basis of multivariate polynomials in X B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

9 Classical orthogonal polynomials Polynomial chaos basis Computation of the coefficients Type of variable Weight function Orthogonal polynomials Uniform 1 ] 1,1[ (x)/2 Legendre P k (x) Gaussian 1 e x2 /2 2π Hermite H ek (x) Gamma x a e x 1 R +(x) Laguerre Lk a(x) Beta 1 ] 1,1[ (x) (1 x)a (1+x) b Jacobi J a,b (x) B(a) B(b) k The basis { Ψ α(x), α N M} is made of multivariate polynomials. Given a multi-index α = {α 1,..., α M }: The basis is orthonormal Ψ α(x) def = M Ψ i α i (x i) i=1 E [Ψ α(x)ψ β (X)] = δ αβ (Kronecker symbol) B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

10 Practical implementation Polynomial chaos basis Computation of the coefficients The input random variables are first transformed into reduced variables (e.g. standard normal variables N (0, 1), uniform variables on [-1,1], etc.): X = T (ξ) dim ξ = M (isoprobabilistic transform) The model response is cast as a function of the reduced variables and expanded: Y = M(X) = M T (ξ) = y α Ψ α(ξ) α N M A truncation scheme is selected and the associated finite set of multi-indices is generated, e.g. : A M,p = {α N M : α p} ( ) M + p where card A M,p (M + p)! P = = p M!p! B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

11 Application example (1) Polynomial chaos basis Computation of the coefficients Computational model Y = M(X 1, X 2) Probabilistic model X i N (µ i, σ i) Isoprobabilistic transform X i = µ i + σ i ξ i ξ i N (0, 1) Hermite polynomials Recurrence: H 1(x) = H 0(x) = 1 H n+1(x) = x H n(x) n H n 1(x) where H n 2 = n! First (normalized) polynomials: Hen (x) x P 0(x) = 1 P 1(x) = x P 2(x) = (x 2 1)/ 2 P 3(x) = (x 3 3x)/ 6 He1 He2 He3 He4 He5 B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

12 Application example (2) Polynomial chaos basis Computation of the coefficients Third order truncature p = 3 All the polynomials of ξ 1, ξ 2 that are product of univariate Hermite polynomials and whose total degree is less than 3 are considered j α Ψ α Ψ j 0 [0, 0] Ψ 0 = 1 1 [1, 0] Ψ 1 = ξ 1 2 [0, 1] Ψ 2 = ξ 2 3 [2, 0] Ψ 3 = (ξ1 2 1)/ 2 4 [1, 1] Ψ 4 = ξ 1 ξ 2 5 [0, 2] Ψ 5 = (ξ2 2 1)/ 2 6 [3, 0] Ψ 6 = (ξ1 3 3ξ 1)/ 6 7 [2, 1] Ψ 7 = (ξ1 2 1)ξ 2/ 2 8 [1, 2] Ψ 8 = (ξ2 2 1)ξ 1/ 2 9 [0, 3] Ψ 9 = (ξ2 3 3ξ 2)/ 6 Ỹ M PC (ξ 1, ξ 2 ) = a 0 + a 1 ξ 1 + a 2 ξ 2 + a 3 (ξ1 2 1)/ 2 + a 4 ξ 1 ξ 2 + a 5 (ξ2 2 1)/ 2 + a 6 (ξ1 3 3ξ 1)/ 6 + a 7 (ξ1 2 1)ξ 2/ 2 + a 8 (ξ2 2 1)ξ 1/ 2 + a 9 (ξ2 3 3ξ 2)/ 6 B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

13 Least-square minimization Polynomial chaos basis Computation of the coefficients The exact (infinite) series expansion is considered as the sum of a truncated series and a residual: M(X) = y α Ψ α(x) + ε P Y T Ψ(X) + ε P α A where : Y = {y α, α A} and Ψ(x) = {Ψ α(x), α A} Solution where: Ŷ L.S = arg min 1 n n ( Y T Ψ(x (i) ) M(x (i) ) ) 2 i=1 = (A T A) 1 A T M M = { M(x (1) ),..., M(x (n) ) } T A ij = Ψ j ( x (i) ) i = 1,..., n ; j = 0,..., A B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

14 Polynomial chaos basis Computation of the coefficients Sparse PC expansions Blatman & S., PEM (2010) Blatman & S., J. Comp. Phys (2011) Premise Standard least-square analysis requires to choose the truncation scheme A a priori, and to have an experimental of size n > A, i.e. typically n = 3 A Sparse expansions Finding the relevant sparse basis among a large candidate basis A is a variable selection problem: l 1 penalization The Least Angle Regression (LAR) has proven efficient: A sufficient large candidate truncated basis A is selected LAR provides a sequence of less and less sparse models The best one is finally selected from leave-one-out cross-validation ɛ LOO = 1 n n { M(x (i) ) M PC\i (x (i) ) } 2 i=1 B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

15 Error estimation: bootstrap method Adaptive design Principle A PC expansion is built up in the standard normal space, with sufficient global accuracy controlled by the LOO error Monte Carlo simulation is used to evaluate ˆP f by substituting the PCE for the limit state function: P f = P ( g PC (X) 0 ) = f X (x) dx N f D f ={x : g PC (x) 0} n where e.g. n = 10 7 B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

16 Example: simply supported beam Error estimation: bootstrap method Adaptive design V = 5 pl EI The maximum deflection V at midspan shall be smaller than a threshold t V = 5 pl 4 32 E bh 3 I = bh3 12 Basic random variables (independent lognormal) Variable Mean Std. deviation CV b 0.15 m 7.5 mm 5% h 0.3 m 15 mm 5% L 5 m 50 mm 1% E MPa MPa 15% p 10 kn/m 2 kn/m 20% B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

17 Exact solution Introduction Error estimation: bootstrap method Adaptive design The product of lognormals is a lognormal: ( ) 5 pl 4 ( 5 ln = ln + ln(p) + 4 ln(l) ln(e) ln(b) 3 ln(h) 32 Ebh 32) 3 For lognormal variables, the above sum is a Gaussian with parameters: ( 5 λ V = ln + λ p + 4λ L λ E λ b 3λ h 32) ζ V = ζ 2 p + (4ζ L ) 2 + ζ 2 E + ζ2 b + (3ζ h) 2 The deflection has a lognormal distribution V LN (λ V, ζ V ) For a given threshold t, the probability of failure reads: ( ) λv log t P (V t) = Φ ζ V B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

18 One-shot polynomial chaos results Error estimation: bootstrap method Adaptive design Sparse PC expansion based on a LHS experimental design of size 200 Probability of failure computed by crude MCS (n = 10 7 ) Threshold (mm) P f,exact β exact P f,pc β PC e e e e e e e e e e B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

19 Approximation of the tail Error estimation: bootstrap method Adaptive design PDF of the maximal deflection Reference PCE Reference PCE logfv(v) logfv(v) v (mm) v (mm) ED of size 100 ED of size 500 B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

20 Local error estimation Error estimation: bootstrap method Adaptive design Premise For a given experimental design (ED), least-square analysis provides a single set of coefficients {y α, α A} and the associated PCE By repeating the analysis (i.e. replicating it using another LHS) one gets a different set of coefficients, and a different estimation of ˆP f How to derive confidence bounds on P f,pc when using a single experimental design? B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

21 Bootstrap Introduction Error estimation: bootstrap method Adaptive design Principle Suppose a quantity θ is estimated form a random sample set X = {x 1,..., x n}, say θ = f (X ) Bootstrap consists in drawing B new samples from the original data by resampling with replacement: X = {x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 } X (1) = {x 9 x 3 x 5 x 2 x 5 x 6 x 7 x 10 x 1 x 4 } X (2) = {x 9 x 3 x 5 x 2 x 5 x 6 x 2 x 10 x 1 x 4 }... From the B bootstrap samples X (1),..., X (B) one gets a sample set of θ s: θ = {θ 1,..., θ B } θ i = f (X (i) ) B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

22 Bootstrap PCE Introduction Error estimation: bootstrap method Adaptive design From a single experimental design X, LAR is applied and a PCE is computed, say Y = Ψ α(x) α A y (0) α Notin et al. (2010) Using the same basis and B bootstrap ED { X (b), i = 1,..., B } one gets B different PCE: Y (b) = y α (b) Ψ α(x) α A From each PCE, an estimate of P f is obtained: P (1) f,..., P (B) f Confidence intervals are obtained from the empirical quantiles B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

23 Example (simply supported beam) Error estimation: bootstrap method Adaptive design Sparse PC expansion based on a LHS experimental design of size 200 Probability of failure computed by crude MCS (n = 10 7 ) Threshold (mm) β exact Bootstrap bounds on β [ ; ] [ ; ] [ ; ] [ ; ] [ ; ] The obtained bounds do not contain the exact solution for large thresholds! B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

24 Adaptive design Introduction Error estimation: bootstrap method Adaptive design Boostrap PCE (bpce) provides B different PC expansions at the cost of a single experimental design In an adaptive ED setting, it is of interest to add new points iteratively. Candidate points are selected where the sign of the prediction varies the most along b = 1,..., B Green points: all bpce give g PC (b) x > 0 b Black points: all bpce give g PC (b) x < 0 b Interesting points are those for which sign changes B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

25 Illustration: 2D example Error estimation: bootstrap method Adaptive design B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

26 Application: Simply supported beam 3 x 10 2 Error estimation: bootstrap method Adaptive design 1.5 P f Limit state function: g(x) = 20 mm V (x) Model evaluations Initial design of size 20 Convergence after 50 iterations (total cost 70 model evaluations) B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

27 Conclusions Introduction Error estimation: bootstrap method Adaptive design Polynomial chaos expansion is a versatile tool that allows one to build surrogate models for moment- and sensitivity analysis It can be used in one-shot for not too small probabilities of failure (e.g. P f 10 4 ) A local error criterion may be constructed using bootstrap, which leads to confidence bounds on P f An enrichment criterion based on the sign variations of the bpce predictions can be used For small probabilities of failure, importance sampling can be used... Work in progress B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31

28 Introduction Error estimation: bootstrap method Adaptive design Questions? Thank you very much for your attention! Chair of Risk, Safety & Uncertainty Quantification UQLab The Uncertainty Quantification Laboratory Now available! B. Sudret (Chair of Risk, Safety & UQ) PC expansions in structural reliability April 10th, / 31