Sensitivity analysis of parametric uncertainties and modeling errors. in generalized probabilistic modeling

Transcription

1 ECCOMAS European Congress on Computational Methods in Applied Sciences and Engineering Sensitivity analysis of parametric uncertainties and modeling errors in generalized probabilistic modeling Maarten Arnst June 9, 2016 ECCOMAS, Crete Island, Greece 1 / 39

2 Motivation blanc Simulation-based design. Virtual testing. Parametric uncertainties. Modeling errors. system subsystem subsystem subsystem blanc Multidisciplinary design. Multiple components. blanc Sensitivity analysis of parametric uncertainties, modeling errors, and multiple components in the context of generalized probabilistic modeling. ECCOMAS, Crete Island, Greece 2 / 39

3 Outline Motivation. Outline. Sensitivity analysis. Generalized probabilistic modeling. First illustration: parametric uncertainties and modeling errors. Second illustration: multiple components. ECCOMAS, Crete Island, Greece 3 / 39

4 Sensitivity analysis ECCOMAS, Crete Island, Greece 4 / 39

5 Overview Sensitivity analysis. Design optimization. Model validation. blanc blanc Analysis of uncertainty 2 Problem definition UQ Propagation of uncertainty Characterization of uncertainty Optimization methods. Monte Carlo sampling. Stochastic expansion (polynomial chaos). Mechanical modeling. Statistics. Intervals. Gaussian. Γdistribution.... The computational cost of stochastic methods can be lowered via the use of a surrogate model as a substitute for a numerical model or real tests. ECCOMAS, Crete Island, Greece 5 / 39

6 Overview There exist many types of sensitivity analysis. Local sensitivity analysis: elementary effect analysis. differentiation-based sensitivity analysis.... Global sensitivity analysis: regression analysis. variance-based sensitivity analysis, correlation analysis, methods involving scatter plots,... Here, we focus here on global sensitivity analysis methods, which can help ascertain which sources of uncertainty are most significant in inducing uncertainty in predictions. References: [A. Saltelli et al. Wiley, 2008]. [J. Oakley and A. O Hagan. J. R. Statist. Soc. B, 2004]. ECCOMAS, Crete Island, Greece 6 / 39

7 Problem setting Characterization of uncertainty: Two statistically independent sources of uncertainty modeled as two statistically independent random variablesx andy with probability distributionsp X andp Y : (X,Y) P X P Y. ECCOMAS, Crete Island, Greece 7 / 39

8 Problem setting Characterization of uncertainty: Two statistically independent sources of uncertainty modeled as two statistically independent random variablesx andy with probability distributionsp X andp Y : (X,Y) P X P Y. Propagation of uncertainty: We assume that the relationship between the sources of uncertainty and the predictions is represented by a nonlinear functiong: Sources of uncertainty (X,Y) Problem Z = g(x,y) Prediction Z ECCOMAS, Crete Island, Greece 7 / 39

9 Problem setting Characterization of uncertainty: Two statistically independent sources of uncertainty modeled as two statistically independent random variablesx andy with probability distributionsp X andp Y : (X,Y) P X P Y. Propagation of uncertainty: We assume that the relationship between the sources of uncertainty and the predictions is represented by a nonlinear functiong: Sources of uncertainty (X,Y) Problem Z = g(x,y) Prediction Z The probability distributionp Z of the prediction is obtained as the image of the probability distributionp X P Y of the sources of uncertainty under the functiong: Z P Z = (P X P Y ) g 1. ECCOMAS, Crete Island, Greece 7 / 39

10 Problem setting Characterization of uncertainty: Two statistically independent sources of uncertainty modeled as two statistically independent random variablesx andy with probability distributionsp X andp Y : (X,Y) P X P Y. Propagation of uncertainty: We assume that the relationship between the sources of uncertainty and the predictions is represented by a nonlinear functiong: Sources of uncertainty (X,Y) Problem Z = g(x,y) Prediction Z The probability distributionp Z of the prediction is obtained as the image of the probability distributionp X P Y of the sources of uncertainty under the functiong: Z P Z = (P X P Y ) g 1. Sensitivity analysis: Is eitherx ory most significant in inducing uncertainty inz? ECCOMAS, Crete Island, Greece 7 / 39

11 Geometrical point of view Least-squares-best approximation of functiong with function of only one input: Assessment of the significance of the source of uncertaintyx: g X = argmin f X g(x,y) f X(x) 2 P X (dx)p Y (dy). By means of the calculus of variations, it can be readily shown that the solution is given by gx = g(,y)p Y (dy). In the geometry of the space ofp X P Y -square-integrable functions,gx is the orthogonal projection of functiong ofxandy onto the subspace of functions of onlyx: Z E{Z X} L 2 X X ECCOMAS, Crete Island, Greece 8 / 39

12 Geometrical point of view Expansion of functiong in terms of main effects and interaction effects: Extension to assessment of significance of both sources of uncertaintyx andy: where g(x,y) = g 0 + g X (x) }{{} + g Y (y) }{{} main effect ofx main effect ofy + g (X,Y) (x,y) }{{}, interaction effect ofx andy g 0 = g(x,y)p X (dx)p Y (dy), g X (x) = gx(x) g 0 = g(x,y)p Y (dy) g 0, g Y (y) = gy(y) g 0 = g(x,y)p X (dx) g 0. Because they are obtained via orthogonal projection, the functionsg 0,g X,g Y, andg (X,Y) are orthogonal functions. The property thatg 0,g X,g Y, andg (X,Y) are orthogonal provides a link with other expansions, such as the polynomial chaos expansion. ECCOMAS, Crete Island, Greece 9 / 39

13 Statistical point of view Sensitivity indices = mean-square values of main effects and interaction effects: Quantitative insight into the significance ofx andy in inducing uncertainty inz: g(x,y) g0 2 P X (dx)p Y (dy) }{{} = =σ 2 Z g X (x) 2 P X (dx) } {{ } =s X + g Y (y) 2 P Y (dy) + g (X,Y) (x,y) 2 P X (dx)p Y (dy). } {{ } } {{ } =s Y =s (X,Y ) Becauseg X,g Y, andg (X,Y) are orthogonal, there are no double product terms. Thus, the expansion ofg (geometry) reflects a partitioning of the variance ofz into terms that are the variances of the main and interaction effects ofx andy (statistics), where: s X = portion of the variance ofz that is explained as stemming fromx, s Y = portion of the variance ofz that is explained as stemming fromy. ECCOMAS, Crete Island, Greece 10 / 39

14 Statistical point of view By the conditional variance identity, we have s X = V{E{Z X}} = V{Z} E{V{Z X}}, s Y = V{E{Z Y}} = V{Z} E{V{Z Y}}, so thats X ands Y may also be interpreted as expected reductions of amount of uncertainty: s X = expected reduction of variance ofz if there were no longer uncertainty inx, s Y = expected reduction of variance ofz if there were no longer uncertainty iny. In contrast to the expansion ofg and the variance partitioning ofz, these expressions and these interpretations ofs X ands Y remain valid even ifx andy are statistically dependent. ECCOMAS, Crete Island, Greece 11 / 39

15 Example Let us consider a simple problem whereinx andy are uniform r.v. with values in[ 1,1], X U([ 1,1]), and the functiong is given by Y U([ 1,1]), z = g(x,y) = x+y 2 +xy. ECCOMAS, Crete Island, Greece 12 / 39

16 Let us consider a simple problem whereinx andy are uniform r.v. with values in[ 1,1], X U([ 1,1]), Y U([ 1,1]), Example and the functiong is given by z = g(x,y) = x+y 2 +xy. This problem has the expansion g(x,y) = g 0 +g X (x)+g Y (y)+g (X,Y) (x,y), = g(x, y) g 0 g X (x) g Y (y) g (X,Y ) (x,y) ECCOMAS, Crete Island, Greece 12 / 39

17 Let us consider a simple problem whereinx andy are uniform r.v. with values in[ 1,1], X U([ 1,1]), Y U([ 1,1]), Example and the functiong is given by z = g(x,y) = x+y 2 +xy. This problem has the expansion g(x,y) = g 0 +g X (x)+g Y (y)+g (X,Y) (x,y), = g(x, y) g 0 g X (x) g Y (y) g (X,Y ) (x,y) To this expansion corresponds the variance partitioning σ 2 Z = s X +s Y +s (X,Y), σ 2 Z = 28 45, s X = 1 3 = 53.57%σ2 Z, s Y = 8 45 = 28.57%σ2 Z, s (X,Y) = 1 9 = 17.86%σ2 Z. ECCOMAS, Crete Island, Greece 12 / 39

18 Computation Computation by means of a stochastic expansion method: s X α 0c 2 (α,0), s Y with c 2 (0,β), β 0 g(x,y) = c (α,β) ϕ α (x)ψ β (y). (α,β) Computation by means of deterministic numerical integration: s X Q X ( QY g Q X Q Y g 2), s Y Q Y ( QX g Q X Q Y g 2). Computation by means of Monte Carlo integration: s X 1 ν s Y 1 ν ( ν g(x l,y l ) 1 ν ( ν g(x l,y l ) 1 ν l=1 l=1 )( ν g(x k,y k ) g(x l,ỹ l ) 1 ν )( ν g(x k,y k ) g( x l,y l ) 1 ν k=1 k=1 ) ν g(x k,ỹ k ), k=1 ) ν g( x k,y k ). k=1 References: [B. Sudret. Reliab. Eng. Syst. Safe., 2008], [Crestaux et al. Reliab. Eng. Syst. Safe., 2009], [I. Sobol. Math. Comput. Simulat., 2001], and [A. Owen. ACM T. Model. Comput. S., 2013]. ECCOMAS, Crete Island, Greece 13 / 39

19 Outlook Observation: Most applications in the literature involve scalar-valued sources of uncertainty. Opportunity: The concepts and methods of global sensitivity analysis are valid and useful more broadly for stochastic process, random fields, random matrices, and other sources of uncertainty. ECCOMAS, Crete Island, Greece 14 / 39

20 Generalized probabilistic modeling ECCOMAS, Crete Island, Greece 15 / 39

21 Overview Force manufactured Manufactured system manufactured Manufacturing tolerances Response manufactured Designed system manufactured Modeling approximations f manufactured Predictive model manufactured u Response Uncertain system Response X Response Parametric uncertainties and modeling errors can present themselves. ECCOMAS, Crete Island, Greece 16 / 39

22 Overview Types of probabilistic approach: Parametric approaches capture parametric uncertainties by characterizing geometrical characteristics, boundary conditions, loadings, and physical or mechanical properties as random variables or stochastic processes. Nonparametric approaches capture modeling errors (and possibly the impact of parametric uncertainties) by directly characterizing the model as a random model without recourse to a characterization of its parameters as random variables or stochastic processes. In structural dynamics, Soize constructed a class of nonparametric models by characterizing the reduced matrices of (a sequence of) reduced-order models as random matrices. Output-prediction-error approaches capture modeling errors (and possibly the impact of parametric uncertainties) by adding random noise terms to quantities of interest. Generalized approaches are couplings of parametric and nonparametric approaches. References: [C. Soize. Probab. Eng. Mech., 2000]. [C. Soize. Int. J. Num. Methods Eng., 2010]. ECCOMAS, Crete Island, Greece 17 / 39

23 Overview Structural vibration. [Cottereau et al. Earthquake Engng. Struct. Dyn., 2008]. Acoustics. [Soize et al. Comput. Methods Appl. Mech. Engrg., 2008]. Nonlinear elasticity. [Capiez-Lernout et al. Comput. Methods Appl. Mech. Engrg., 2014]. Viscoelasticity. [Capillon et al. Comput. Methods Appl. Mech. Engrg., 2016]. ECCOMAS, Crete Island, Greece 18 / 39

24 Deterministic model FE model of linear dynamical behavior of dissipative structure: [M(x)]ü(t)+[D(x)] u(t)+[k(x)]u(t) = f(t,x), where x collects the parameters of the FE model, which may consist of material properties, loadings, geometrical characteristics, and so forth. u = (u 1,...,u m ) is the (generalized) displacement vector, f the (generalized) external forces vector, and[m(x)],[d(x)], and[k(x)] the mass, damping, and stiffness matrices. ECCOMAS, Crete Island, Greece 19 / 39

25 Parametric probabilistic model Parametric probabilistic approach = probabilistic representation of uncertain parameters: [M(X)]Ü(t)+[D(X)] U(t)+[K(X)]U(t) = f(t,x), where X is the probabilistic representation of the uncertain parameters, which may consist of random variables, stochastic processes, and so forth. In order to obtain the probabilistic representation of the uncertain parameters, a suitable probability distribution must be assigned to the random variables or stochastic processes. In stochastic mechanics, methods are available for deducing a suitable probability distribution from available information, such as methods dedicated to tensor-valued fields of material properties, methods dedicated to tensors in rigid-body mechanics, and methods dedicated to random loadings. ECCOMAS, Crete Island, Greece 20 / 39

26 Generalized probabilistic model Generalized probabilistic approach = enhancement of parametric probabilistic model by introducing in it a probabilistic representation of modeling errors. Step 1: Associate with the parametric probabilistic model a reduced-order probabilistic model: where [M n (X)] Q(t)+[D n (X)] Q(t)+[K n (X)]Q(t) = f n (t,x), U n (t) = [Φ(X)]Q(t), [M n (X)],[D n (X)], and[k n (X)] are the reduced mass, damping, and stiffness matrices, and[φ(x)] the matrix collecting in its columns the reduction basisϕ 1 (X),ϕ 2 (X),...,ϕ n (X). Such a reduced-order probabilistic model can be obtained, for instance, by solving the eigenvalue problem associated with the mass and stiffness matrices of the parametric probabilistic model, [K(X)]ϕ j (X) = λ j (X)[M(X)]ϕ j (X); in which case the reduced matrices of the reduced-order probabilistic are given by [M n (X)] ij = δ ij, [D n (X)] ij = ϕ i (X) [D(X)]ϕ j (X), [K n (X)] ij = λ i δ ij. ECCOMAS, Crete Island, Greece 21 / 39

27 Generalized probabilistic model Step 2: represent the reduced matrices by using random matrices: [M n (X)] Q(t)+[D n (X)] Q(t)+[K n (X)]Q(t) = f n (t,x), U n (t) = [Φ(X)]Q(t), To accommodate in the reduced matrices a probabilistic representation of the modeling errors, the generalized probabilistic approach entails representing these reduced matrices as follows: [M n (X)] = [L M (X)][Y M ][L M (X)] T, [D n (X)] = [L D (X)][Y D ][L D (X)] T, [K n (X)] = [L K (X)][Y K ][L K (X)] T, with[l M (X)],[L D (X)], and[l K (X)] the Cholesky factors of[m n (X)],[D n (X)], and[k n (X)]. ECCOMAS, Crete Island, Greece 22 / 39

28 Generalized probabilistic model To assign a suitable probability distribution to the random matrices[y M ],[Y D ], and[y K ], the generalized probabilistic approach uses the maximum entropy principle. The probability distribution thus obtained is such that the mean values of[y M ],[Y D ], and[y K ] are all equal to the identity matrix, that is, E{[Y M ]} = [I n ], E{[Y D ]} = [I n ], E{[Y K ]} = [I n ], and the amount of uncertainty expressed in[y M ],[Y D ], and[y K ] is tunable by free dispersion parametersδ M,δ D, andδ K, respectively, defined by δ M = E{ [Y M ] [I n ] 2 F}/ [I n ] 2 F, δ D = E{ [Y D ] [I n ] 2 F}/ [I n ] 2 F, δ K = E{ [Y K ] [I n ] 2 F}/ [I n ] 2 F. The dispersion parameters must be calibrated such that the amount of uncertainty expressed in[y M ],[Y D ], and[y K ] reflects the significance of the modeling errors. ECCOMAS, Crete Island, Greece 23 / 39

29 First illustration: parametric uncertainties and modeling errors ECCOMAS, Crete Island, Greece 24 / 39

30 Overview Manufacturing tolerances 4 manufactured Designed system Clamped-clamped beam manufactured Modeling approximations Manufactured system Predictive model manufactured Uncertain material properties Random field of material properties Parametric uncertainties and modeling errors present themselves. ECCOMAS, Crete Island, Greece 25 / 39

31 Deterministic model [H(ω;x)] = [ ω 2 M(x)+iωD(ω;x)+K(x)] FRF [m/n/hz] ,000 1,200 Frequency [Hz] δ M = δ D = δ K = 0.05 δ M = δ D = δ K = 0.05 ECCOMAS, Crete Island, Greece 26 / 39

32 Parametric probabilistic model [H(ω;X)] = [ ω 2 M(X)+iωD(ω;X)+K(X)] 1, wherex = random field representation of shear and bulk moduli FRF [m/n/hz] ,000 1,200 Frequency [Hz] δ M = δ D = δ K = 0.05 δ M = δ D = δ K = 0.05 ECCOMAS, Crete Island, Greece 27 / 39

33 Parametric probabilistic model [H(ω;X)] = [ ω 2 M(X)+iωD(ω;X)+K(X)] 1, wherex = random field representation of shear and bulk moduli FRF [m/n/hz] ,000 1,200 Frequency [Hz] δ M = δ D = δ K = 0.05 δ M = δ D = δ K = 0.05 ECCOMAS, Crete Island, Greece 27 / 39

34 Nonparametric probabilistic model [H(ω;X,Y )] = [Φ(X)][ ω 2 M n (X)+iωD n (ω;x)+k n (X)] 1 [Φ(X)] T, [M n (X)] = [L M (X)][Y M ][L M (X)] T, wherey = ([Y M ],[Y D ],[Y K ]) with [D n (ω;x)] = [L D (ω;x)][y D ][L D (ω;x)] T, [K n (X)] = [L K (X)][Y K ][L K (X)] T FRF [m/n/hz] ,000 1,200 Frequency [Hz] δ M = δ D = δ K = δ M = δ D = δ K = 0.05 ECCOMAS, Crete Island, Greece 28 / 39

35 Nonparametric probabilistic model [H(ω;X,Y )] = [Φ(X)][ ω 2 M n (X)+iωD n (ω;x)+k n (X)] 1 [Φ(X)] T, [M n (X)] = [L M (X)][Y M ][L M (X)] T, wherey = ([Y M ],[Y D ],[Y K ]) with [D n (ω;x)] = [L D (ω;x)][y D ][L D (ω;x)] T, [K n (X)] = [L K (X)][Y K ][L K (X)] T FRF [m/n/hz] ,000 1,200 Frequency [Hz] δ M = δ D = δ K = δ M = δ D = δ K = 0.05 ECCOMAS, Crete Island, Greece 28 / 39

36 Nonparametric probabilistic model [H(ω;X,Y )] = [Φ(X)][ ω 2 M n (X)+iωD n (ω;x)+k n (X)] 1 [Φ(X)] T, [M n (X)] = [L M (X)][Y M ][L M (X)] T, wherey = ([Y M ],[Y D ],[Y K ]) with [D n (ω;x)] = [L D (ω;x)][y D ][L D (ω;x)] T, [K n (X)] = [L K (X)][Y K ][L K (X)] T FRF [m/n/hz] ,000 1,200 Frequency [Hz] δ M = δ D = δ K = δ M = δ D = δ K = 0.05 ECCOMAS, Crete Island, Greece 28 / 39

37 Identification The dispersion parameters must be calibrated such that the amount of uncertainty expressed in[y M ],[Y D ], and[y K ] reflects the significance of the modeling errors FRF [m/n/hz] ,000 1,200 Frequency [Hz] Simulated data (3D parametric probabilistic model, blue) Generalized probabilistic model with ˆδ M = ˆδ D = ˆδ K = 0.20 (gray). ECCOMAS, Crete Island, Greece 29 / 39

38 Convergence Because the problem is of high dimension, we compute the sensitivity indices by using Monte Carlo integration, whereby we assess the convergence as a function of the number of samples. s ν Y (ω)[ ] Number of samples [-] ν s ν Y (ω) = 1 ν ν l=1 (log H jj (ω;x l,y l ) ν 1 ν k=1 log H jj (ω;x k,y k ) ) (log H jj (ω; x l,y l ) ν 1 ν k=1 log H jj (ω; x k,y k ) ) atω = 1000 Hz. ECCOMAS, Crete Island, Greece 30 / 39

39 Sensitivity analysis V{log H jj (ω;x,y ) } = s X (ω)+s Y (ω)+s (X,Y ) (ω), s X (ω) = Var X { EY {log H jj (ω;x,y ) } }, s Y (ω) = Var Y { EX {log H jj (ω;x,y ) } } σ 2 Z,s X[ ] σ 2 Z,s Y[ ] ,200 Frequency [Hz] Variance (black) ands X (ω) (blue). Parametric uncertainties ,200 Frequency [Hz] Variance (black) ands Y (ω) (blue). Modeling errors. ECCOMAS, Crete Island, Greece 31 / 39

40 Second illustration: multiple components ECCOMAS, Crete Island, Greece 32 / 39

41 Overview Stiffened panel with a hole. δ M = δ D = δ K = 0.05 δ M = δ D = δ K = 0.05 ECCOMAS, Crete Island, Greece 33 / 39

42 Deterministic model First few dynamical eigenmodes. Mode 1 at Hz. δ M = δ D = δ K = 0.05 Mode 2 at Hz. ECCOMAS, Crete Island, Greece 34 / 39

43 Nonparametric probabilistic model After a component mode synthesis, we used the nonparametric probabilistic approach to introduce uncertainties in the submodels of the main panel and the stiffeners PDF [1/Hz] Frequency [Hz] PDFs of the first and second eigenfrequencies. δ M = δ D = δ K = 0.05 ECCOMAS, Crete Island, Greece 35 / 39

44 Sensitivity analysis 4 Sensitivity index [Hz 2 ] 0.25 Sensitivity index [Hz 2 ] 2 0 Main panel Stiffeners 0 Main panel Stiffeners First eigenfrequency. δ M = δ D = δ K = 0.05 Second eigenfrequency. δ M = δ D = δ K = 0.05 ECCOMAS, Crete Island, Greece 36 / 39

45 Conclusion and acknowledgement Global sensitivity analysis methods can help ascertain which sources of uncertainty are most significant in inducing uncertainty in predictions. Although most applications in the literature involve scalar-valued sources of uncertainty, the concepts and methods of global sensitivity analysis are valid and useful more broadly for stochastic process, random fields, random matrices, and other sources of uncertainty. Generalized probabilistic modeling approaches are hybrid couplings of parametric modeling approaches (to capture parametric uncertainties) and nonparametric probabilistic modeling approaches (to capture modeling errors). We discussed global sensitivity analysis of generalized probabilistic models and demonstrated its application in two illustrations from structural dynamics. ECCOMAS, Crete Island, Greece 37 / 39

46 Conclusion and acknowledgement This presentation can be downloaded from our institutional repository: Other references: M. Arnst and J.-P. Ponthot. An overview of nonintrusive characterization, propagation, and sensitivity analysis of uncertainties in computational mechanics. International Journal for Uncertainty Quantification, 4: , M. Arnst, B. Abello Alvarez, J.-P. Ponthot, and R. Boman. Itô-SDE-based MCMC method for Bayesian characterization and propagation of errors associated with data limitations. SIAM/ASA Journal on Uncertainty Quantification, Submitted, M. Arnst and K. Goyal. Sensitivity analysis of parametric uncertainties and modeling errors in generalized probabilistic modeling. Probabilistic Engineering Mechanics, Submitted, Support of the University of Liège through a starting grant is gratefully acknowledged. ECCOMAS, Crete Island, Greece 38 / 39

47 References A. Saltelli et al. Global sensitivity analysis. Wiley, J. Oakley an A. O Hagan. Probabilistic sensitivity analysis of complex models: a Bayesian approach. Journal of the Royal Statistical Society: Series B, 66: , B. Sudret. Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety, 93: , T. Crestaux, O. Le Maître, and J.-M. Martinez. Polynomial chaos expansion for sensitivity analysis. Reliability Engineering & System Safety, 94: , I. Sobol. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, 55: , A. Owen. Better estimation of small Sobol sensitivity indices, ACM Transactions on Modeling and Computer Simulation, 23, C. Soize. Nonparametric model of random uncertainties for reduced matrix models in structural dynamics, Probabilistic Engineering Mechanics, 15: , C. Soize. Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions. International Journal for Numerical Methods in Engineering, 81: , R. Cottereau, D. Clouteau, and C. Soize. Probabilistic impedance of foundation: Impact of the seismic design on uncertain soils. Earthquake Engineering and Structural Dynamics, 37: , C. Soize et al. Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation. Computer Methods in Applied Mechanics and Engineering, 198: , E. Capiez-Lernout, C. Soize, and M. Mignolet. Post-buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation. Computer Methods in Applied Mechanics and Engineering, 271: , R. Capillon, C. Desceliers, and C. Soize. Uncertainty quantification in computational linear structural dynamics for viscoelastic composite structures. Computer Methods in Applied Mechanics and Engineering, 305: , ECCOMAS, Crete Island, Greece 39 / 39