MODEL UNCERTAINTY ISSUES FOR PREDICTIVE MODELS

Transcription

1 MODEL UNCERTAINTY ISSUES FOR PREDICTIVE MODELS Christian SOIZE Mechanics Laboratory University of Marne-la-Vallée (Paris) - France soize@univ-mlv.fr WORSHOP ON THE ELEMENTS OF PREDICTABILITY Organized by The Johns Hopkins University Sandia National Laboratories Johns Hopkins University, November 13-14, 2003

2 I - INTRODUCTION

3 In a predictive model of a complex system, among all the sources of uncertainties, there are: data uncertainties model uncertainties Today, it is well understood that the parametric probabilistic approach is required for modeling data uncertainties in predictive models. For predictive models of complex systems, a large part of the lack of predictability is due to model uncertainties. For a complex system: If an additional smaller spatial scale is introduced in the predictive model for reducing model uncertainties, then data uncertainties increase due to the increasing of the number of parameters. If several spatial scales are introduced, model uncertainties will always exist for the smaller spatial scale. For the smaller spatial scale introduced in the predictive model, the model uncertainties cannot be taken into account with the parametric probabilistic approach of data uncertainties. Consequently, in a predictive model of a complex system, there will always have a spatial scale for which model uncertainties will have to be taken into account for increasing the predictability.

4 EXAMPLE 1 : A SYSTEM WITH MODEL UNCERTAINTIES AT THE MACRO SCALE Complex joints in structural dynamics Dynamical system: Two dural plates connected together through a complex joint constituted of 2 smaller dural plates tightened by 2 lines of 20 bolts. Excitation in bending modes: point force at node 5 in plate 2. Observations: normal displacements at nodes 1,2,... Mean mechanical model: Plates 1 and 2 modeled by homogeneous isotropic thin plates. Complex joint modeled by an equivalent homogeneous orthotropic plate. Complex joint: mass density and elastic parameters of the equivalent plate updated using experiments. Mean finite element model: Uniform 4 nodes thin plate elements: dofs. Modal density: 126 modes in [20,2000] Hz such that: 64 modes in [20,1000] Hz 13 modes in [1000,1200] Hz 49 modes in [1200,2000] Hz

5 TRANSIENT LINEAR DYNAMICS : Shock Response Spectrum (SRS) at node 2 in plate 1 due to a point force applied to note 5 in plate 2 EXPERIMENTAL IMPULSIVE LOAD defined as a point force applied to note 5 in plate 2. Frequency domain representation (top). Time domain representation (down). SHOCK RESPONSE SPECTRUM at note 2 in plate 1. Solid lines: Experimental SRS corresponding to 21 random distributions of screw-bolt prestresses. Dashed line: Numerical prediction of the SRS using the mean finite element model. In spite of a rather large number of dofs utilized for the mean finite element model, significant errors occur between experimental and numerical predictions (in spite of experimental updating). These errors are due to the simplified model of the complex joint: MODEL ERRORS.

6 LINEAR DYNAMICS IN THE FREQUENCY DOMAIN : Frequency Response Function (FRF) through the complex joint CROSS FREQUENCY RESPONSE FUNCTION: PLATE 2 - PLATE 1 THROUGH THE JOINT Thin solid lines: Upper and lower envelopes of the 21 experimental FRF associated with 21 random distributions of screw-bolt prestresses. Thick solid line: Numerical prdiction of the FRF using the mean finite element model. In spite of a rather large number of dofs utilized in the mean finite element model, significant errors occur between experimental and numerical predictions (in spite of experimental updating). These errors are due to the simplified model used for modeling the complex joint: MODEL ERRORS

7 EXAMPLE 2 : A COMPLEX SYSTEM FOR WHICH MODEL UNCERTAINTIES HAVE TO BE TAKEN INTO ACCOUNT FOR IMPROVING PREDICTIVE MODELS Vehicles acoustic performances at low frequencies and at medium frequencies The refined finite element model (developed for the crash analysis) can be utilized for the vehicle acoustic performances at low frequencies (0-200) Hz (booming noise) and at medium frequencies ( ) Hz (power-train, road excitations). 307 sw car body PSA company

8 Sources of model uncertainties in macro scale predictive models for vehicles acoustic performances (highly complex vibroacoustic system) Structure (car body) Sound proofing scheme Internal acoustic fluid Internal body Sources of model uncertainties Level Structure (car body) Medium Coupling interface S-SPS High Sound proofing scheme High Coupling interface SPS-IAF High Internal acoustic cavity Medium Coupling interface IAF-IB High Internal body Small Coupling interface: SPS-IAF Coupling interface: IAF-IB Coupling interface: S-SPS

9 A fact concerning the predictive models of vehicles acoustic performances Due to presence of a high level of model uncertainties in such a fluid-structure interaction problem, probabilistic parametric approach for data uncertainties and - is not sufficient - does not allow the design consequences to be understood - does not allow the manufacturing process dispersion to be understood - model uncertainties have to be included in predictive models (such a research is in progress jointly by LaM-UMLV and PSA)

10 WHAT COULD BE A PROBABILISTIC APPROACH WHICH WOULD ALLOW MODEL UNCERTAINTIES TO BE TAKEN INTO ACCOUNT IN PREDICTIVE MODELS? A nonparametric probabilistic approach is proposed for model uncertainty issues in predictive models. This nonparametric probabilistic approach can be used at any spatial scale is based on a direct construction of the probability measures of the random operators of the predictive model, instead of constructing these probability measures as transformations of the probability measures of the random parameters (data uncertainties) needs to develop new ensembles of random matrices in the context of the random matrix theory needs to develop adapted inverse methods and mathematical statistics for experimental identification should allow advanced representation of stochastic processes and stochastic fields, such as Chaos decomposition, to be used.

11 A FEW REFERENCES CONCERNING NONPARAMETRIC MODEL OF RANDOM UNCERTAINTIES Fundamentals concerning nonparametric approach of random uncertainties and developing the set of positive definite random matrices: C. Soize PEM 15(3) (2000). Algebraic closure, convergence analysis as dimension goes to infinity, transient linear elastodynamics of stochastic systems: C. Soize JASA 109(5) (2001), C. Soize PEM 16(4) (2001). Dynamic substructuring method for non homogeneous random uncertainties and experimental validation: H. Chebli & C. Soize REEF (2002), C. Soize & H. Chebli JEM-ASCE 129(4) (2003), H. Chebli & C. Soize JASA (submitted July 2002, revised July 2003). Uncertain dynamical system in the medium frequency range: C. Soize JEM-ASCE 129(9) (2003) Random eigenvalues and the non adaptation of the GOE for low- and medium-frequency dynamics: C. Soize JSV (2003). Nonparametric probabilistic model for nonlinear dynamical systems and transient nonlinear dynamics of stochastic systems: C. Soize e-journal UEM 1(1) 1-38 (2001), C. Desceliers & C. Soize & Cambier EESD Journal (accepted in June 2003). Model uncertainties in dynamics of systems with cyclic symmetry and application to mistuned bladed disk: E. Capier- Lernout & C. Soize JEGTP (submitted in June 2002, revised in July 2003). Random matrix theory for modeling uncertainties in computational mechanics, identification of the parameters of the nonparametric probabilistic model with experiments. C. Soize CNAME Special Issue in CSMRA (submitted in October 2003).

12 II - MAIN IDEAS AND FOUNDATIONS OF THE NONPARAMETRIC PROBABILISTIC APPROACH OF RANDOM UNCERTAINTIES

13 Mathematical-mechanical modeling of a physical mechanical system 1 PMS: Physical Mechanical System for which a predictive model has to be constructed. BVP: Boundary Value Problem resulting from the mathematicalmechanical modeling of the PMS. Weak formulation of the BVP introduces several linear operators (such as the stiffness operator for a bounded elastic medium). 0 = one of these operators. exp = operator corresponding to 0 for the PMS. - This operator is unknown. - Only an approximate model 0 of exp can be constructed.

14 Mean reduced model 2 The reduced model is deduced from the BVP by using the Ritz-Galerkin projection on a finite dimension subspace Àn of the admissible space. The projection of operator 0 is represented by the real square matrix [A 0,n ] in Ån(Ê). The mean reduced model is defined as the reduced model constructed by using the nominal values of the parameters. Matrix [A 0,n ] corresponding to the nominal mean reduced model is rewritten as [A 0,n ].

15 Updating the mean reduced model using experimental data 3 [A exp,n ] = the matrix of the projection of exp on Àn. Matrix [A exp,n ] in Ån(Ê) is assumed to be experimentally identified (indirect experimental identification). [B 0,n ] = [A 0,n ] [A exp,n ] = bias between the PMS and the mean reduced model. For complex system:. norm [B 0,n ] of bias [B 0,n ] is not small.. bias has to be reduced in updating the nominal parameters.. updating yields the updated matrix [A n ] of nominal matrix [A 0,n ].. bias [B n ] = [A n ] [A exp,n ] has a norm [B n ] < [B 0,n ] which is generally not sufficiently small. Then, a probabilistic model of uncertainties has to be introduced.

16 Parametric probabilistic approach 4 Consisting in introducing random variables and stochastic fields as parameters in the BVP in order to model data errors. Using a statistical reduction method, in which: [A n ] becomes a random matrix [A n (X)]. X is an Ê m -valued random variable with probability measure P X. support D m of P X is such that D m Ê m. x [A n (x)] is a mapping from Ê m into Ån(Ê).

17 5 Experimental S par,n S n The range of mapping x [A n (x)] is a subset Ë par,n of Ën such that Ë par,n Ën Ån(Ê) (for instance Ën = Å + n (Ê)). If it can be assumed that [A exp,n ] is surely in Ën, then due to random uncertainties, [A exp,n ] is generally not in Ë par,n.

18 [A par,n ] = random matrix with values in Ån(Ê) corresponding to the parametric probabilistic model of data errors and defined by 6 [A par,n ] = [A n (X)] Then, the error between the parametric probabilistic model of random uncertainties and the experimental data can, for instance, be measured by E{ [A par,n ] 1 [A exp,n ] 1 2 } = [a n ] 1 [A exp,n ] 1 2 P par ( da n ) Ë par,n where P par ( da n ) is the image of P X by x [ A n (x)] and is such that E{ [ A par,n ] 1 2 } = [a n ] 1 2 P par ( da n ) < + Ë par,n In general, D m and probability measure P X on D m are given. Due to the model errors, the error is generally not sufficiently small.

19 Introduction of the nonparametric probabilistic approach 7 Introducing an approach of random uncertainties induced by data and model errors allowing the mean-square error to be reduced. Consisting in. substituting [A par,n ] by a random matrix [A nonpar,n ],. for which the probability measure P nonpar on Ën is directly constructed by using the random matrix theory,. such that E{[A nonpar,n ]} = [A n ], E{ [ A nonpar,n ] 1 2 } < +. Then, the error between the nonparametric probabilistic model of random uncertainties and the experimental data is given by E{ [A nonpar,n ] 1 [A exp,n ] 1 2 } = [a n ] 1 [A exp,n ] 1 2 P nonpar ( da n ) Ë n

20 Capability of the nonparametric probabilistic approach 8 Experimental S par,n S n Since Ë par,n Ën, we can take P nonpar = P par. In this case, [A nonpar,n ] = [A par,n ] which proves that the nonparametric model has the capability to take into account data errors. In addition, since the support of P nonpar is Ën Ë with par,n Ën, the nonparametric model allows a larger class of random matrices to be constructed and consequently, has, a priori, the capability to take into account model errors.

21 9 Experimental S par,n S n For instance, assume that the model errors are sufficiently high for that [A exp,n ] Ë par,n but, we have [A exp,n ] Ën. Then the error cannot be reduced even if probability measure P par could be arbitrary chosen. On the other hand, since [A exp,n ] belongs to Ën, then there are probability measures P nonpar on Ën which allow the error to be reduced.

22 10 Experimental S par,n S n For instance, the probability measure P nonpar ( da n ) = δ 0 ([a n ] [A exp,n ]) leads the error to be zero because [A exp,n ] belongs to Ën. Of course, P nonpar cannot be arbitrary chosen on Ën but has to be constructed using the random matrix theory. Then there is P nonpar on Ën verifying the required properties, such that [a n ] 1 [A exp,n ] 1 2 P nonpar ( da n ) < [a n ] 1 [A exp,n ] 1 2 P par ( da n ) Ë n which means that Ë par,n E{ [A nonpar,n ] 1 [A exp,n ] 1 2 } < E{ [A par,n ] 1 [A exp,n ] 1 2 }

23 III - RANDOM MATRIX ENSEMBLES FOR UNCERTAINTIES MODELING IN PREDICTIVE MODELS

24 Notation for matrix sets 1 Å Ån,m(Ê) : set of all the (n m) real matrices Ån(Ê) : set of all the square (n n) real matrices S n(ê) : set of all the (n n) real symmetric matrices +0 n (Ê) : set of all the (n n) real symmetric semipositive definite matrices Å + n (Ê) : set of all the (n n) real symmetric positive-definite matrices Å Å + n (Ê) Å +0 n (Ê) Å S n(ê) Ån(Ê) Operator norm : A = sup x 1 [A ] x, x Ê m Frobenius norm: A 2 F = [A ],[A ] = tr{[a ]T [A ]} = n j=1 m k=1 [A ]2 jk

25 New ensembles of random matrices constructed for the nonparametric probabilistic approach of random uncertainties 2 Random set Set of values Second-order r.m Given mean value Invertibility SG + : [G + n n (Ê), E{ [G n] 2 F }<+, E{[G ] Å n]}=[ I n ], E [G n ] 1 2 F <+ { } SE + : [A + n n (Ê), E{ [A n] 2 F }<+, E{[A n]}=[a ] Å n ], E [A n ] 1 2 F <+ Å ] Å + n (Ê) SE +0 : [A +0 n n (Ê), E{ [A n] 2 F }<+, E{[A n]}=[a n ] +0 n (Ê) Å SE + lf : [B n ] Å + n (Ê), E{ [B n] 2 F }<+, E{[B n]}=[b n ], E { } { } [B n ] 1 2 F <+ f([b n ])=b + n n (Ê) Å } SE inv : [A m,n ], E{ [A m,n ] 2 F }<+, E{[A m,n]}=[a m,n ],E { [A m,n ] 1l 2 <+ Å m,n (Ê) Å m,n (Ê) left pseudo-inverse For each ensemble, the probability measure is explicitely constructed by using the maximum entropy principle for which the constraints are defined by the available information (algebraic properties, probabilistic properties). We will limit the presentation to ensemble SG + and SE + and for the three other ensembles, see CNAME Special Issue in CSMRA (submitted in October 2003).

26 Probability measure on ensemble SG + of random matrices 3 Probability measure of random matrix [G n ] P [Gn ] = p [Gn ]([G n ]) dg n defined by a probability density function p [Gn ]([G n ]) with respect to dg n which is the measure on Å S n(ê) defined by the Euclidean structure on Å S n(ê), such that dg n = 2 n(n 1)/4 Π 1 i j n d[g n ] ij

27 Probability density function with respect to dg n 4 p [Gn ]([G n ]) = ½ Å + n (Ê) ([G n]) C Gn ( det[g n ] ) (n+1) (1 δ 2 ) 2δ 2 exp{ (n+1) 2δ 2 tr [G n ]} Constant of normalization: C Gn = (2π) n(n 1)/4 { ( n+1 2δ 2 ) n(n+1)(2δ2 ) 1 ( Π n j=1 Γ n+1 2δ 2 +1 j 2 Parameter controlling the dispersion of random matrix [G n ]: )} δ = { E{ [Gn ] [G n ] 2 F } } 1/2 = [G n ] 2 F { E{ [Gn ] [ I n ] 2 F } n } 1/2 0 < δ < (n + 1)(n + 5) 1

28 Invariance under real orthogonal transformations 5 [Φ n ]: any real orthogonal matrix [Φ n ] T [Φ n ] = [Φ n ] [Φ n ] T = [ I n ]. Let [G n] be the random matrix with values Å in + n (Ê) defined by [G n] = [Φ n ] T [G n ] [Φ n ] We then have p [G n ]([G n]) dg n = p [Gn ]([G n]) dg n

29 Convergence property when dimension goes to infinity 6 n n 0, E{ [G n ] 1 2 } C δ < Graph of function n E{ [G n ] 1 2 } for δ = 0.1 (lower line), 0.3 (mid line) and 0.5 (upper line), in which the horizontal axis is dimension n of the reduced matrix model.

30 Order statistics of the random eigenvalues of [G n ] [G n ] = Λ for [G n ] in SG + 7 Λ 1 Λ 2... Λ n : order statistics for [G n ] For j = 1,...n, graphs of the probability density functions p Λj, calculated for δ A = 0.5, n = 30 and n s = samples.

31 Algebraic representation of random matrix belonging to SG + 8 Any random matrix [G n ] in SG + can be written as [G n ] = [L n ] T [L n ], in which [L n ] = upper triangular random matrix with values in Ån(Ê) Random variables {[L n ] jj, j j } are independent. For j < j, [L n ] jj = σ n U jj in which σ n = δ(n + 1) 1/2 and where U jj is a real-valued Gaussian random variable with zero mean and variance equal to 1. For j = j, [L n ] jj = σ n 2Vj where V j is a positive-valued gamma random variable whose probability density function with respect to dv is p Vj (v) = ½ Ê +(v) 1 Γ ( n+1 2δ + 1 j ) v n+1 2δ 2 1+j 2 e v 2 2

32 Probability measure on ensemble SE + of random matrices 9 Any random matrix [A n ] belonging to SE + is such that: and is written as [A n ] is with values in Å + n (Ê), E{ [A n ] 2 F } < +, E{[A n ]} = [A n ] Å + n (Ê), E { [A n ] 1 2 F} < + [A n ] = [L An ] T [G n ] [L An ] with [L An ] the upper triangular matrix such that [A n ] = [L An ] T [L An ] and where [G n ] is a random matrix in SG + such that [G n ] = E{[G n ]} = [ I n ]. The dispersion parameter allowing the dispersion of random matrix [A n ] to be controlled is parameter δ of random matrix [G n ].

33 IV - NONPARAMETRIC MODEL OF RANDOM UNCERTAINTIES FOR LINEAR AND NONLINEAR DYNAMICS

34 Mean finite element model The nominal finite element model of the nonlinear (linear) dynamical system is called the mean finite element model 1 [ Å]ÿ(t) + [ ]ẏ(t) + [ Ã]y(t) + f NL (y(t), ẏ(t)) = f(t) y: unknown time response vector of the m DOFs. f(t): known external load vector of the m inputs. [ Å ], [ ], [ Ã]: mass, damping and stiffness positive-definite symmetric real (m m) matrices of the linear part of the model. (y, z) f NL (y, z): nonlinear mapping.

35 Mean reduced matrix model 2 Projection on the subspace spanned by the elastic modes { 1,..., n } of the underlying linear system ([ Ã] = ω 2 [ Å] ), associated with the n smallest eigenfrequencies (with n m). Nonlinear matrix equation for the mean reduced matrix model y n (t) = n α=1 q n α (t) α [M n ] q n (t) + [D n ] q n (t) + [K n ]q n (t) + F n NL (qn (t), q n (t))=f n (t)

36 Nonparametric model of random uncertainties 3 Using the principle introduced previously: the generalized mass, damping and stiffness matrices of the mean reduced matrix models are substituted by random matrices [M n ], [D n ] and [K n ] for which the probability model is constructed using only the available information. if the local nonlinear forces of the model are uncertain, then the usual parametric model can be used and yields to a nonparametric-parametric mixed formulation.

37 Stochastic response of the nonlinear dynamical system with the nonparametric probabilistic model of random uncertainties 4 For all fixed t, stochastic response is the random variable Y n (t) with values in m, such that Ê n Y n (t) = Q n α(t) α α=1 in which random variable Q n (t) = (Q n 1(t),...,Q n n(t)) with values in Ê n, is such that [M n ] Q n (t) + [D n ] Q n (t) + [K n ]Q n (t) + F n NL (Qn (t), Q n (t))=f n (t)

38 Available information for the construction of the probability model of random generalized matrices [M n ], [D n ] and [K n ] 5 (C1) - Random generalized matrices [M n ], [D n ] and [K n ] are defined on a probability space (A, T, P), and are with values in Å + n (Ê). (C2) - Mean values are known (values of the nominal model): E{[M n ]} = [ M n ], E{[D n ]} = [ D n ], E{[K n ]} = [ K n ] (C3) - Second-order moments E { [M n ] 1 2 F} <+, E { [Dn ] 1 2 F} <+, E { [Kn ] 1 2 F} <+ have to exist in order that stochastic solution Q n (t) be a second-order stochastic process.

39 Probability model of random generalized matrices [M n ], [D n ] and [K n ] 6 (1) Random matrices [M n ], [D n ] and [K n ] are independent (2) Each random matrix has then to belong to set SE + of random matrices. (3) The parametersδ M, δ D andδ K allow the dispersion of random matrices [M n ], [D n ] and [K n ] to be controlled.

40 Construction of the stochastic solution and convergence 7 Stochastic harmonic response for the linear case: frequency-byfrequency construction with the Monte Carlo numerical simulation method. Transient response with initial conditions for the linear and nonlinear cases: Monte Carlo numerical simulation method with an implicit step-bystep integration method (newmark method) Convergence is numerically checked by calculating { 1 (n s, n) Conv(n s, n) = ns T n s k=1 0 Qn (t, θ k ) 2 dt } 1/2

41 V - APPLICATIONS COMPUTATIONAL LINEAR AND NONLINEAR DYNAMICS

42 APPLICATION 1 Nonparametric probabilistic approach of model uncertainties for complex joints in structural dynamics Jérome DUCHEREAU, Christian SOIZE (for transient linear dynamics) Hamid CHEBLI, Christian SOIZE (for linear dynamics in the frequency domain)

43 EXAMPLE 1 : Dynamical system, mean mechanical model and mean finite element model defined in Introduction Section Model uncertainties are larger in the complex joint than in plates 1 and 2. Therefore: Uncertainties are non homogeneous The nonparametric approach of random uncertainties are then implemented for each substructure by using Dynamic substructuring (Craig and Bampton method)

44 TRANSIENT LINEAR DYNAMICS : Shock Response Spectrum (SRS) at node 2 in plate 1 due to a point force applied to note 5 in plate 2 EXPERIMENTAL IMPULSIVE LOAD defined as a point force applied to note 5 in plate 2. Frequency domain representation (top). Time domain representation (down). SHOCK RESPONSE SPECTRUM at note 2 in plate 1. Solid lines: Experimental SRS corresponding to 21 random distributions of screw-bolt prestresses. Dashed line: Numerical prediction of the SRS using the mean finite element model. Grey region: 95% confidence region of the stochastic response computed with the nonparametric approach: For plates 1 and 2: δ M = 0, δ D = 0.3, δ K = 0.15 For complex joint: δ M = 0, δ D = 0.8, δ K = 0.8

45 LINEAR DYNAMICS IN THE FREQUECY DOMAIN: Cross Frequency Response Function (FRF) through the complex joint CROSS FREQUENCY RESPONSE FUNCTION: PLATE 2 - PLATE 1 THROUGH THE JOINT Solid lines: Experimental FRF corresponding to 21 random distributions of screw-bolt prestresses. Thick solid line: Numerical prediction of the FRF using the mean finite element model. Grey region: 95% confidence region of the stochastic response computed with the nonparametric approach: For plates 1 and 2: δ M = 0, δ D = 0.1, δ K = 0.15 For complex joint: δ M = 0, δ D = 0.8, δ K = 0.8

46 APPLICATION 2 Nonparametric-parametric probabilistic approach of model and data uncertainties for nonlinear dynamics of a reactor coolant system Christophe DESCELIERS, Christian SOIZE Simon CAMBIER University of Marne-La-Vallée Mechanics Laboratory EDF R&D Analysis in Mechanics and Acoustics Department

47 Model uncertainty for a predictive model in transient nonlinear dynamics of a reactor coolant system (PWR) in order to identify and to quantify the design margins Reactor coolant system Steam generator Hot leg Reactor Ground motion 3-dimensional Model of the building with soil structure interaction Displacements, Velocity and Accelerations at the supports of the structure U leg Pump Cold leg

48 Reactor coolant system model Nonlinear dynamical system: Multisupported system: 12 time-dependent Dirichlet conditions Nonlinearities : 28 nonlinear dofs with elastic stops Model Finite element model with 5022 dofs 828 Lagrange dofs

49 Why a nonparametric probabilistic approach has to be used? Many model uncertainties and data uncertainties Complex Structure Nonlinearities => A lot of modes have to be used in the analysis ( 200 modes between 1,4 Hz et 174 Hz ) Difficult to choose the parameters having important effects on the dynamic behavior Example of model uncertainties: the steam generators?

50 Nonlinear stochastic dynamical response computation Monte Carlo numerical simulation method Construction of the mean reduced matrix model Simulation of the random matrices and of the random variables from the nonlinearity Solving the corresponding transient problem ( reduced system) Monte Carlo loop Computation of the system response Computation of response normalized spectra Computation of the statistics observation nodes : A, B, C, D,E T=15s t = s time steps

51 Convergence n 1 S T Z&& n t 2 dt n S 0 j (; θ i ) i= 1 T && n { 0 j } && = ( ) n 2 2 Z j E Z t dt n=5 modes n=50 modes n=200 modes n=100 modes Impact node D (upper stop of one Steam Generator), x2 direction number of the Monte-Carlo simulation samples

52 Result: confidence region for each Shock Response Spectrum (SRS) Proba { S ( ξ, ω ) < S(ξ, ω) < S ( ξ, ω )} min max = δ [ M n ][, D n ][, K n ] = 0.2 δ kstops = 0.2 S X & ( f, ξ ) (normalized by g pseudo-acceleration) S S max min ( ξ, ω) S( ξ, ω) ( ξ, ω) node A (middle of one hot leg), x3 direction frequency (Hz) ( n=200, ns =700 )

53 Role plays by the parametric uncertainties for the 28 nonlinear elastic-stops dofs with respect to the nonparametric uncertainties for the linear model δ [ M ][, D ][, K ] δ kstops δ [ ][ ][ ] δ kstops n n n = 0.2 = 0.2 M n, D n, K n = 0.2 = 0. frequency (Hz) node B, x3 direction frequency (Hz)

54 VI - CONCLUSIONS

55 For predictive models of complex systems, a large part of the lack of predictability is due to model uncertainties. For the smaller spatial scale of a multiscale predictive model, model uncertainties have to be taken into account for increasing the predictability and cannot be described with the parametric probabilistic approach of data uncertainties. A nonparametric probabilistic approach is proposed for model uncertainty issues in predictive models.

56 Such a nonparametric probabilistic approach can be used at any spatial scale is based on a direct construction of the probability measures of the random operators of the predictive model, instead of constructing these probability measures as transformations of the probability measures of the random parameters (data uncertainties) has to be based on the use of new ensembles of random matrices in the context of the random matrix theory can easily be implemented in computational sciences and computer codes. allows advanced representation of stochastic processes and stochastic fields, such as Chaos decomposition, to be used. needs to develop adapted inverse methods and mathematical statistics for experimental identification (in progress for dynamical systems) needs many additional theoretical, numerical and experimental developments.