Computational methods for uncertainty quantification and sensitivity analysis of complex systems

Transcription

1 DEPARTMENT OF CIVIL, ENVIRONMENTAL AND GEOMATIC ENGINEERING CHAIR OF RISK, SAFETY & UNCERTAINTY QUANTIFICATION Computational methods for uncertainty quantification and sensitivity analysis of complex systems B. Sudret Labex MS2T - Université Technologique de Compiègne

2 Outline 1 to uncertainty quantification 2 Polynomial chaos basis Computing the coefficients Post-processing 3 Single model / independent variables Imbricated models: the issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices 4 Mechanical example: composite beam Tolerance analysis B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

3 Some common engineering structures Cattenom nuclear power plant (France) Airbus A380 Military satellite Cormet de Roselend dam (France) Bladed disk B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

4 Computational models Modern engineering has to address problems of increasing complexity in various fields including infrastructures (civil engineering), energy (civil/mechanical engineering), aeronautics, defense, etc. Complex systems are designed using computational models that are based on: - a mathematical description of the physics (e.g. mechanics, acoustics, heat transfer, electromagnetism, etc.) - numerical algorithms that solve the resulting set of (e.g. partial differential) equations: finite element-, finite difference-, finite volume- methods, boundary element methods) B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

5 Computational models Simulation models are calibrated and validated through comparison with lab experiments and in situ / full scale measurements. Once they are validated, these models may be run with different sets of input parameters in order to: explore the design space at low cost optimize the system w.r.t to cost criteria assess the robustness of the system w.r.t. uncertainties Sources of uncertainty Differences between the designed and the real system in terms of material/physical properties and dimensions (tolerancing) Unforecast exposures: exceptional service loads, natural hazards (earthquakes, floods), climate loads (hurricanes, snow storms, etc.) B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

6 Global framework for managing uncertainties Step B Step A Step C Quantification of Model(s) of the system Uncertainty propagation sources of uncertainty Assessment criteria Random variables Computational model Moments Probability of failure Response PDF Step C Sensitivity analysis B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

7 Global framework for managing uncertainties Step B Step A Step C Quantification of Model(s) of the system Uncertainty propagation sources of uncertainty Assessment criteria Random variables Computational model Moments Probability of failure Response PDF Step C Sensitivity analysis B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

8 Global framework for managing uncertainties Step B Step A Step C Quantification of Model(s) of the system Uncertainty propagation sources of uncertainty Assessment criteria Random variables Computational model Moments Probability of failure Response PDF Step C Sensitivity analysis B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

9 Global framework for managing uncertainties Step B Step A Step C Quantification of Model(s) of the system Uncertainty propagation sources of uncertainty Assessment criteria Random variables Computational model Moments Probability of failure Response PDF Step C Sensitivity analysis B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

10 Step A: computational models (civil & mechanical engineering) Vector of input parameters x R M Computational model M Model response y = M(x) R N geometry material properties loading analytical formula finite element model etc. displacements strains, stresses temperature, etc. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

11 Step A: computational models (civil & mechanical engineering) Vector of input parameters x R M Computational model M Model response y = M(x) R N geometry material properties loading analytical formula finite element model etc. displacements strains, stresses temperature, etc. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

12 Step B: probabilistic models of input parameters No data exist expert judgment for selecting the input PDF s of X literature, data bases (e.g. on material properties) maximum entropy principle Input data exist classical statistical inference Bayesian statistics when data is scarce but there is some prior information Data on output quantities inverse probabilistic methods and Bayesian updating techniques B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

13 Step B: probabilistic models of input parameters No data exist expert judgment for selecting the input PDF s of X literature, data bases (e.g. on material properties) maximum entropy principle Input data exist classical statistical inference Bayesian statistics when data is scarce but there is some prior information Data on output quantities inverse probabilistic methods and Bayesian updating techniques B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

14 Step B: probabilistic models of input parameters No data exist expert judgment for selecting the input PDF s of X literature, data bases (e.g. on material properties) maximum entropy principle Input data exist classical statistical inference Bayesian statistics when data is scarce but there is some prior information Data on output quantities inverse probabilistic methods and Bayesian updating techniques B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

15 Step C: principles of uncertainty propagation Input parameters x R M Computational model M Model response y = M(x) R N Random variables X Computational model M Random response Y = M(X)? B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

16 Step C: principles of uncertainty propagation Input parameters x R M Computational model M Model response y = M(x) R N Random variables X Computational model M Random response Y = M(X)? B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

17 Step C: uncertainty propagation methods Computational model Step A Mean/std. deviation µ σ Probabilistic- Rare computational model event simulation P f Probabilistic model Step B Response PDF B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

18 Step C: uncertainty propagation methods Computational model Step A Mean/std. deviation µ σ Probabilistic- Rare computational model event simulation P f Probabilistic model Step B Response PDF B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

19 Step C: uncertainty propagation methods Computational model Step A Mean/std. deviation µ σ Probabilistic- Rare computational model event simulation P f Probabilistic model Step B Response PDF B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

20 Step C: uncertainty propagation methods Computational model Step A Mean/std. deviation µ σ Probabilistic- Rare computational model event simulation P f Probabilistic model Step B Response PDF B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

21 Step C: uncertainty propagation methods Computational model Step A Mean/std. deviation µ σ Probabilistic- Rare computational model event simulation P f Probabilistic model Step B Response PDF B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

22 Monte Carlo simulation B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

23 Monte Carlo simulation B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

24 Monte Carlo simulation Principles The input random variables are sampled according to their joint PDF f X (x). For each sample x (i), the response M(x (i) ) is computed (possibly time-consuming). The response sample set M = {M(x (1) ),..., M(x (n) )} T is used to compute statistical moments, probabilities of failure or estimate the response distribution (histogram, kernel density estimation). B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

25 Monte Carlo simulation Advantages This is a universal method, i.e. which does not depend on the type of model M. It is statistically well defined: convergence, confidence intervals, etc. It is non intrusive, i.e. it is based on repeated runs of the computational model as a black box. It is suited to distributed computing (clusters of PCs). Drawbacks The scattering of Y is investigated point-by-point: if two samples x (i), x (j) are almost equal, two independent runs of the model are carried out. The convergence rate is low ( N 1/2 ). B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

26 Spectral approach Principle The random response Y = M(X) is considered as an element of a suitable vector space. A basis of this space is built up (with respect to the input joint PDF). The response random vector Y is completely determined by its coordinates in this basis: Y = y j Ψ j(x) where: j=0 y j : coefficients to be computed (coordinates) Ψ j(x) : basis Assumption: Y has a finite second moment, i.e. : E [ Y 2] = Y 2 (ω)dp(ω) = M 2 (x) f X (x) dx < B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

27 Outline Polynomial chaos basis Computing the coefficients Post-processing 1 to uncertainty quantification 2 Polynomial chaos basis Computing the coefficients Post-processing 3 4 B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

28 Univariate orthogonal polynomials Polynomial chaos basis Computing the coefficients Post-processing Definition Suppose that the input random vector has independent components. f X (x) = M f Xi (x i) i For each marginal distribution, define: φ 1, φ 2 i = D i φ 1(x) φ 2(x) f Xi (x i) dx i By classical algebra one can build a family of orthogonal polynomials {Pk, i k N} as follows: P i j, Pk i = Pj i (x) Pk(x) i f Xi (x) dx = aj i δ jk B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

29 Univariate orthogonal polynomials Polynomial chaos basis Computing the coefficients Post-processing Xiu & Karniadakis Classical families Type of variable Weight function Orthogonal polynomials Hilbertian basis ψ k (x) Uniform 1 ] 1,1[ (x)/2 Legendre P k (x) P k (x)/ 1 2k+1 Gaussian 1 e x2 /2 Hermite He (x) 2π k He (x)/ k! k Gamma x a e x Γ(k+a+1) 1 R + (x) Laguerre La k (x) La k (x)/ k! Beta 1 ] 1,1[ (x) (1 x)a (1+x) b Jacobi J a,b (x) J a,b (x)/j B(a) B(b) k k a,b,k J 2 a,b,k = 2k+a+b+1 2a+b+1 Γ(k+a+1)Γ(k+b+1) Γ(k+a+b+1)Γ(k+1) Normalization The classical orthogonal polynomials are not orthonormal: they may be normalized by dividing by aj i: Ψ i j = Pj i / aj i i = 1,..., M, j N B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

30 Multivariate polynomials Polynomial chaos basis Computing the coefficients Post-processing Tensor product of 1D polynomials M One defines the multi-indices α = {α 1,..., α M }, of degree α = i=1 The associated multivariate polynomial reads: M Ψ α(x) = Ψ i α i (x i) i=1 α i The set of multivariate polynomials {Ψ α, α N M } forms a basis of the space: Y = y α Ψ α(x) α N M B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

31 Practical implementation Polynomial chaos basis Computing the coefficients Post-processing 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 truncature scheme is selected and the associated finite set of multi-indices is generated, e.g. : ( ) A = {α N M M + p : α p} card A P = p B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

32 Outline Polynomial chaos basis Computing the coefficients Post-processing 1 to uncertainty quantification 2 Polynomial chaos basis Computing the coefficients Post-processing 3 4 B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

33 Regression approach Polynomial chaos basis Computing the coefficients Post-processing Principle The exact (infinite) series expansion is considered as the sum of a truncated series and a residual: P 1 Y = M(X) = y j Ψ j(x) + ε P Y T Ψ(X) + ε P j=0 where: Y = {y 0,..., y P 1 } Ψ(x) = {Ψ 0(x),..., Ψ P 1 (x)} Mean-square minimization The coefficients are gathered into a vector Ŷ, and computed by minimizing the mean square error: [ (Y Ŷ = arg min E T Ψ(X) M(X) ) ] 2 B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

34 Regression: discretized solution Polynomial chaos basis Computing the coefficients Post-processing The discretized mean-square minimization reads: Ŷ reg = arg min 1 n n ( Y T Ψ(x (i) ) M(x (i) ) ) 2 i=1 Select an experimental design X = { x (1),..., x (n)} T that covers at best the domain of variation of the parameters Evaluate the model response for each sample (exactly as in Monte carlo simulation) M = { M(x (1) ),..., M(x (n) ) } T Compute the experimental matrix A ij = Ψ j ( x (i) ) i = 1,..., n ; j = 0,..., P 1 Solve the least-square minimization problem Ŷ = (A T A) 1 A T M B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

35 Polynomial chaos basis Computing the coefficients Post-processing Validation of the surrogate model Blatman (2009) Blatman & Sudret ( ) The truncated series expansions are convergent in the mean square sense. However one does not know in advance where to truncate (problem-dependent). Most people truncate the series according to the total maximal degree of the polynomials, say (p=2,3,4, etc.). Several values of p are tested until some kind of convergence is empirically observed. The recent research deals with the development of error estimates: adaptive integration in the projection approach cross validation in the regression approach This has lead to the development of sparse polynomial chaos expansions. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

36 Error estimators Coefficient of determination Polynomial chaos basis Computing the coefficients Post-processing The regression technique is based on the minimization of the mean square error. The generalization error is defined as: [ (M(X) E gen = E M PC (X) ) ] 2 It may be estimated by the empirical error using the already computed response quantities: E emp = 1 n ( M(x (i) ) M PC (x (i) ) ) 2 n i=1 The coefficient of determination R 2 is often used as an error estimator: R 2 = 1 Eemp ˆV[Y] ˆV[Y] = 1 n (M(x(i) ) Ȳ)2 This error estimator may lead to overfitting B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

37 Error estimators Leave-one-out cross validation Polynomial chaos basis Computing the coefficients Post-processing Principle In statistical learning theory, cross validation consists in splitting the experimental design Y in two parts, namely a training set (which is used to build the model) and a validation set. The leave-one-out technique consists in using each point of the experimental design as a single validation point for the meta-model built from the remaining n 1 points. n different meta-models are built and the error made on the remaining point is computed, then mean-square averaged. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

38 Cross validation Implementation Polynomial chaos basis Computing the coefficients Post-processing For each x (i), a polynomial chaos expansion is built using the following experimental design: X \x (i) = {x (j), j = 1,..., n, j i}, denoted by M PC\i (.). The predicted residual is computed in point x (i) : i = M PC\i (x (i) ) M(x (i) ) The PRESS coefficient(predicted residual sum of squares) is evaluated: PRESS = The leave-one-out error and related Q 2 error estimator are computed: E LOO = 1 n n i=1 2 i n i=1 2 i Q 2 = 1 E LOO ˆV[Y] B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

39 Polynomial chaos basis Computing the coefficients Post-processing Post-processing of polynomial chaos expansions Reminder Polynomial chaos Truncated series Y = M(X) = M T (ξ) Y PC = y α Ψ α(ξ) α A The computed coefficients ( coordinates of the random variable in the Hilbertian basis) are not the quantities of interest. Depending on the situation, the PDF, the statistical moments or quantiles of Y are of interest (e.g. low quantiles in structural reliability analysis). The PC expansion must be post-processed in order to get relevant information on the model response B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

40 Statistical moments Mean value and variance Polynomial chaos basis Computing the coefficients Post-processing From the orthonormality of the polynomial chaos basis one gets: E [Ψ α] = 0 E [Ψ αψ β ] = 0 Mean value ˆµ Y = y 0 The estimated mean value is the first term of the series. Variance ˆσ Y 2 = α A\0 y 2 α The estimated variance is computed as the sum of the squares of the remaining coefficients. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

41 Probability density function Polynomial chaos basis Computing the coefficients Post-processing Principle The polynomial series expansion may be considered as a stochastic response surface, i.e. an analytical function of the input variables ξ (after some isoprobabilistic transform), that may be sampled easily using Monte Carlo simulation. A large sample set of reduced variables is drawn ξ, say of size n sim = : X sim = { ξ (j), j = 1,..., n sim } The truncated series is evaluated onto this sample: { } Y sim = y α Ψ α(ξ j ), j = 1,..., n sim α A The obtained sample set is plotted as an histogram or by kernel density smoothing. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

42 Outline Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices 1 to uncertainty quantification 2 3 Single model / independent variables Imbricated models: the issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices 4 B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

43 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Sensitivity analysis Sobol, 1993 Saltelli et al., 2000 Sobol decomposition Sensitivity analysis aims at quantifying what are the input parameters (or combinations thereof) that influence the most the response variability. Global sensitivity analysis relies on so-called variance decomposition techniques. Consider a model M : x [0, 1] M M(x) R. The Sobol decomposition reads: M M(x) = M 0 + M i(x i) + M ij(x i, x j) + + M 12...M (x) where : i=1 1 i<j M M 0 is the mean value of the function M i(x i) are univariate functions M ij(x i, x j) are bivariate functions etc. Unicity under orthogonality conditions B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

44 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Sensitivity analysis Sobol, 1993 Saltelli et al., 2000 Sobol decomposition Sensitivity analysis aims at quantifying what are the input parameters (or combinations thereof) that influence the most the response variability. Global sensitivity analysis relies on so-called variance decomposition techniques. Consider a model M : x [0, 1] M M(x) R. The Sobol decomposition reads: M M(x) = M 0 + M i(x i) + M ij(x i, x j) + + M 12...M (x) where : i=1 1 i<j M M 0 is the mean value of the function M i(x i) are univariate functions M ij(x i, x j) are bivariate functions etc. Unicity under orthogonality conditions B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

45 Sobol indices Independent input variables Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Variance decomposition Assume X i U(0, 1), i = 1,..., M (possibly after some isoprobabilistic transform) Due to the orthogonality of the decomposition: D Var [M(X)] = E [ (M(X) M 2] 0) = E = {i 1,...,i s} {1,...,M} {i 1,...,i s} {1,...,M} 2 M i1...i s (X i1,..., X is ) E [ M 2 i 1...i s (X i1,..., X is ) ] B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

46 Sobol indices Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Partial variance Consider: Then: D i1...i s = Var [M(X)] = [0,1] s M 2 i 1...i s (x i1,..., x is ) dx i1... dx is M D i + i=1 1 i<j M The Sobol indices are obtained by normalization: S i1...i s = Di 1...i s D D ij D 12...M They represent the fraction of the total variance Var [Y ] that can be attributed to each input variable i (S i) or combinations of variables {i 1... i s} B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

47 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Link with PC expansions Sudret, 2008 Sobol decomposition vs. polynomial chaos expansions The truncated polynomial series may be sorted so as to emphasize the uni-, bi-, etc.- variate functions of the Sobol decomposition, namely: M 0 = y 0 M i(x i) =... M i1...i s (x i1,..., x is ) = y α Ψ α(x) A i = {α : α i > 0, α j i = 0} α A i y α Ψ α(x) α A i1,...,is A i1...i s = {α : α k > 0 k (i 1,..., i s)} The PC expansion readily provides a functional decomposition B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

48 Link with PC expansions Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Computation of the Sobol indices The partial variances D i1...i s are obtained by summing up the square of selected PC coefficients. D i = yα 2 A i = {α : α i > 0, α j i = 0} α A i D i1...i s = A i1...i s = {α : α k > 0 k (i 1,..., i s)} α A i1,...,is y 2 α The Sobol indices come after normalization: S i1...i s = Di 1...i s D Once the PC expansion is available, the full set of Sobol indices are obtained for free! B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

49 Outline Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices 1 to uncertainty quantification 2 3 Single model / independent variables Imbricated models: the issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices 4 B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

50 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Characteristics of imbricated multiphysics models X 1 X 1.1 X 2 X 1.2 M 1 Y 1 = X X 1.n1 = X 2.1 X 2.2 X Y 2 = X 3.n3 M 3 Y 3... M 2 X 2.n2 X 2 The computational workflow consists in an imbrication of models (multiscale/multiphysics analysis). Models of different levels have common parameters (e.g. geometrical dimensions for thermal, mechanical, acoustics problems). The output of micro-scale models is the input of the macro-scale models. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

51 Impact on sensitivity analysis Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices X 1 X 1.1 X X 1.n1 = X 2.1 X X 2.n2 X 2 M 1 M2 Y 1 Y 2 = = X 2 X 3.1 X X 3.n3 M 3 Y 3 Specificity of multi-level imbricated models The PDF of the parameters of the intermediate levels (margins + dependence structure (correlations)) is implicitely defined by uncertainty propagation. The appropriate statistical tools are non-parametric representation of the marginals and copula theory. Wand & Jones (1995); Nelsen (1999) Sensitivity indices have to be defined in the case of dependent input variables. Two approaches have emerged recently: Caniou (2012) Distribution-based sensitivity indices Borgonovo (2007, 2011) Covariance decomposition Li & Rabitz (2010) B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

52 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Distribution-based sensitivity indices Borgonovo (2007) Heuristics Schemes are taken from Caniou (2012) If the PDF of the model response Y = M(X) is influenced by a given input parameter X i then the conditional distribution f Y Xi (y) shall significantly differ from f Y (y). Shift between f Y and f Y Xi : s(x i) = fy (y) f Y Xi (y) dy D Y Borgonovo s δ importance measure: [ expected shift ] δ i 1 2 E [s(xi)] = 1 2 D Xi D Y fy (y) f Y Xi (y) dy f Xi (x i) dx i B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

53 Estimating the shift from PDFs Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices (Conditional) distributions are estimated by kernel smoothing techniques based on massive Monte Carlo simulation: N 1 MC ˆfY (y) = K( y M(y(l) ) ) N MC h h l=1 e.g. K(t) = e t2 /2 / 2π h is the bandwidth parameter, e.g. h N 1/5 MC. Shift estimation (inner integral) s(x i) = fy (y) f Y Xi (y) dy D Y Computed by Gaussian quadrature Expected shift (outer integral) E [s(x i)] = D Xi s(x i) f Xi (x i) dx i N Q ω q s(x (q) i ) q=1 B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

54 Estimating the shift from PDFs Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices (Conditional) distributions are estimated by kernel smoothing techniques based on massive Monte Carlo simulation: N 1 MC ˆfY (y) = K( y M(y(l) ) ) N MC h h l=1 e.g. K(t) = e t2 /2 / 2π h is the bandwidth parameter, e.g. h N 1/5 MC. Shift estimation (inner integral) s(x i) = fy (y) f Y Xi (y) dy D Y Computed by Gaussian quadrature Expected shift (outer integral) E [s(x i)] = D Xi s(x i) f Xi (x i) dx i N Q ω q s(x (q) i ) q=1 B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

55 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Estimating the shift from CDFs Borgonovo et al. (2011) Remarks The two nested integrals of functions derived from kernel PDF estimates are extremely expensive to compute and prone to numerical noise. The inner integral may be computed analytically though. Alternative scheme Schemes are taken from Caniou (2012) s(x i) = 2 P ( f Y (y) > f Y Xi (y) ) 2 P ( f Y (y) < f Y Xi (y) ) δ i = E Xi [ FY (y 2) F Y (y 1) + F Y Xi (y 1) F Y Xi (y 2) ] where (y 1, y 2) are the intersection points of the PDF and the conditional PDF, i.e. are the roots of: f Y (y) = f Y Xi (y) B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

56 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Covariance-based sensitivity indices Li, Rabitz et al., 2010 Suppose a functional decomposition of the model exists, e.g. : M(x) = M u(x u) u {1,...,M} NB : the summands M u(x u) may not be orthogonal with respect to the probability measure associated with f X. Covariance decomposition Var [M(X)] = Var M u(x u) = u {1,...,M} Var [M u(x u)] + u {1,...,M} u {1,...,M} v u Cov [M u(x u); M v(x v)] NB: in case of independence and Sobol decomposition, the second term is zero. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

57 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Covariance-based sensitivity indices Li, Rabitz et al., 2010 Total sensitivity index Structural sensitivity index Correlative sensitivity index S (T) u = Cov [M u(x u), M(X)] /Var [Y ] S (S) u = Var [M u(x u)] /Var [Y ] S (C) u = S (T) u S (S) u S (C) u = 0 on the case of independent input parameters and classical Sobol functional decomposition. These indices are denoted respectively by S pj, S a p j and S b p j in Li, Rabitz et al. (2010). B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

58 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Covariance-based indices using PC expansions Principle Functional decomposition A truncated polynomial expansion is computed assuming that the parameters are independent: M(x) y α Ψ α(x) α A The terms are grouped according to their input parameters so as to build the functional decomposition: M u(x u) = y α Ψ α(x) A u = {α : k u α k > 0} α A u B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

59 Single model / independent variables The issue of correlation Distribution-based sensitivity indices Covariance-based sensitivity indices Covariance-based indices using PC expansions Monte Carlo estimators Caniou & Sudret (2012) A sample set X = { x (i), i = 1,..., N } of size N is drawn according to the joint PDF f X (x). Classical estimators are used: Var [Y ] = 1 N ( ) M(x (i) 2 ) µ M N 1 S u (T) = 1 N 1 S u (S) = 1 N 1 i=1 N i=1 N i=1 S u (C) = Ŝ u (T) Ŝ u (S) ( Mu(x (i) ) µ Mu ) ( M(x (i) ) µ M ) / Var [Y ] ( Mu(x (i) ) µ Mu ) 2 / Var [Y ] NB: all the functions are multivariate polynomials. N = 10 5 evaluations may be carried out in a matter of seconds. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

60 Outline Mechanical example: composite beam Tolerance analysis 1 to uncertainty quantification Mechanical example: composite beam Tolerance analysis B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

61 Simply supported composite beam Problem statement Mechanical example: composite beam Tolerance analysis A composite beam of length L and section b h is made of a fraction f of carbon fibers (E f, ρ f ) and a fraction (1 f ) of epoxyde matrix (E f, ρ f ). q E r, E m, ρ r, ρ m, f L b h The beam is loaded by its dead weight q = ρ hom gbh: Of interest is the sensitivity of the maximum midspan deflection v to its input parameters: v = 5 ql E hom I B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

62 Nested model and input variables Mechanical example: composite beam Tolerance analysis Models E hom = f E f + (1 f ) E m I = bh3 12 q = [f ρ r + (1 f ) ρ m] g h v = 5 ql E hom I Basic random variables h b ρm ρ f f E f Em L I q E hom Parameter Distribution Mean Coefficient of variation L Lognormal 2 m 1% b Lognormal 10 cm 3% h Lognormal 1 cm 3% E f Lognormal 300 GPa 15% Em Lognormal 10 GPa 15% ρ f Lognormal 1800 kg m 3 3% ρm Lognormal 1200 kg m 3 3% f Lognormal % v B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

63 Output of the intermediate variables Mechanical example: composite beam Tolerance analysis PC expansions of the intermediate variables E hom, q, I E hom = e α Ψ α(x) α A E q = I = q α Ψ α(x) α A q i α Ψ α(x) α A I Non parametric estimation of the margins (kernel smoothing with 10 4 points): f I (i) fq (q) f Ehom (e) B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

64 Output of the intermediate variables Correlation structure Mechanical example: composite beam Tolerance analysis Estimation of the dependence structure : Gaussian copula based on the Spearman s rank correlation coefficients. 1.0 ρ S = ρ S = E hom 0.4 I q q Last level uncertainty propagation ρ S (q, E) = 0.20 ρ S (q, I) = 0.71 Isoprobabilistic transform of L, I, q, E hom into standard normal variates U PC expansion of v and computation of distribution-based (resp. covariance-based) sensitivity indices B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

65 Mechanical example: composite beam Tolerance analysis Results - Distribution-based indices CDF and conditional CDF of the maximal deflection 1.0 δ 1 = δ 2 = F Y (y), F Y X1 (y) F Y (y), F Y X2 (y) y y 1.0 δ 3 = δ 4 = F Y (y), F Y X3 (y) F Y (y), F Y X4 (y) y y B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

66 Results Mechanical example: composite beam Tolerance analysis Parameter δ CDF S S U S C q L E hom I Σ v = 5 ql E hom I The homogenized Young s modulus and the moment of inertia are the most important parameters. The distribution- and covariance-based indices give consistent results. The cov.-based index of q is negative due to the strong correlation between q and I which have opposite influences in the maximal displacement. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

67 Outline Mechanical example: composite beam Tolerance analysis 1 to uncertainty quantification Mechanical example: composite beam Tolerance analysis B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

68 Tolerance analysis Mechanical example: composite beam Tolerance analysis In industrial mass production, assembled products are made of individual parts that are prone to uncertainty in their dimensions due to the manufacturing processes. Non-assembly problems may occur when some parts real dimensions differ (even slightly) from their theoretical values. After Gayton et al., Méca. Indus. (2011) Predicting the probability of non-assembly prior to the mass production is of crucial importance. Probabilistic methods derived from structural reliability analysis have been recently proposed. Gayton et al. (2011) B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

69 Connector contact clearance Mechanical example: composite beam Tolerance analysis The axial deviation of an electrical connector pin is considered. After Gayton et al., Méca. Indus. (2011) The deviation is computed as a non linear function of 14 geometrical dimensions describing the geometry of the two parts and their respective position. Each dimension is characterized by a Gaussian variable. Since different surfaces are machined during the same operation, their dimensions are highly correlated, e.g. ρ S = 0.8 between the three different diameters of the pin. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

70 Results Mechanical example: composite beam Tolerance analysis Parameter δ CDF S S U S C cm cm cm cm cm cm cm cm cm ima ima ima ima ima Σ After Caniou (2012) A set of 5 dimensions explain 94% of the variance of the pin deviation. Three of them are correlated input variables for which the correlated contribution of the cov.-based indices is larger than the uncorrelated part. The δ-indices are consistent. The results may be used for tolerance allocation, i.e. allocate smaller tolerance intervals to important dimensions. B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

71 Conclusions Mechanical example: composite beam Tolerance analysis Complex systems are nowadays modelled by computational workflows that involve various submodels / physics / scales of description. The existing uncertainty propagation methods have to be adapted to the specific characteristics of these models: large CPU demand: need for surrogate models, e.g. polynomial chaos expansions statistical dependence between the meaningful intermediate parameters of the workflow Sensitivity indices have been developed for the case of models with dependent input parameters and can be used in the context of complex workflows: distribution-based δ-indices covariance-based indices B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

72 Conclusions Mechanical example: composite beam Tolerance analysis whose efficiency in moment-, reliability- and sensitivity analysis is well established show a new feature in this context, namely the functional decomposition of the computational model. Further work is required to make these methods available to the industry in a convenient format. Thank you very much for your attention! B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

73 Conclusions Mechanical example: composite beam Tolerance analysis whose efficiency in moment-, reliability- and sensitivity analysis is well established show a new feature in this context, namely the functional decomposition of the computational model. Further work is required to make these methods available to the industry in a convenient format. Thank you very much for your attention! B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

74 References Mechanical example: composite beam Tolerance analysis Blatman, G. (2009). Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis. Ph. D. thesis, Université Blaise Pascal, Clermont-Ferrand. Blatman, G. et B. Sudret (2010a). An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Prob. Eng. Mech. 25(2), Blatman, G. et B. Sudret (2010b). Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab. Eng. Sys. Safety 95, Blatman, G. et B. Sudret (2011). Adaptive sparse polynomial chaos expansion based on Least Angle Regression. J. Comput. Phys 230, Borgonovo, E. (2007). A new uncertainty importance measure. Reliab. Eng. Sys. Safety 92, Borgonovo, E., W. Castaings, et S. Tarantola (2011). Moment-independent importance measures: new results and analytical test cases. Risk Analysis 31(3), Caniou, Y. (2012). Global sensitivity analysis for nested and multiscale models. Ph. D. thesis, Université Blaise Pascal, Clermont-Ferrand. Gayton, N., P. Beaucaire, J.-M. Bourinet, E. Duc, M. Lemaire, et L. Gauvrit (2011). APTA: advanced probability-based tolerance analysis of products. Mécanique & Industries 12(2), B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58

75 Mechanical example: composite beam Tolerance analysis Li, G., H. Rabitz, P. Yelvington, O. Oluwole, F. Bacon, C. Kolb, et J. Schoendorf (2010). Global sensitivity analysis for systems with independent and/or correlated inputs. J. Phys.Chem. 114, Saltelli, A., K. Chan, et E. Scott (Eds.) (2000). Sensitivity analysis. J. Wiley & Sons. Sobol, I. (1993). Sensitivity estimates for nonlinear mathematical models. Math. Modeling & Comp. Exp. 1, Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Sys. Safety 93, Xiu, D. et G. Karniadakis (2002). The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), B. Sudret (Chair of Risk & Safety) Labex MS2T - Seminar February 14th, / 58