Christian Soize

Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure

The basic random variables on which random uncertainties can in a given model depend can be viewed as defining a measure space with respect to which the solution to the mathematical problem can be defined. This measure space is defined on a product measure associated with the collection of basic random variables. This paper clarifies the mathematical structure of this space and its relationship to the underlying spaces associated with each of the random variables. Cases of both dependent and independent basic random variables are addressed. Bases on the product space are developed that can be viewed as generalizations of the standard polynomial chaos approximation. Moreover, two numerical constructions of approximations in this space are presented along with the associated convergence analysis.

A nonparametric model of random uncertainties for reduced matrix models in structural dynamics

Random uncertainties in finite element models in linear structural dynamics are usually modeled by using parametric models. This means that: (1) the uncertain local parameters occurring in the global mass, damping and stiffness matrices of the finite element model have to be identified; (2) appropriate probabilistic models of these uncertain parameters have to be constructed; and (3) functions mapping the domains of uncertain parameters into the global mass, damping and stiffness matrices have to be constructed. In the low-frequency range, a reduced matrix model can then be constructed using the generalized coordinates associated with the structural modes corresponding to the lowest eigenfrequencies. In this paper we propose an approach for constructing a random uncertainties model of the generalized mass, damping and stiffness matrices. This nonparametric model does not require identifying the uncertain local parameters and consequently, obviates construction of functions that map the domains of uncertain local parameters into the generalized mass, damping and stiffness matrices. This nonparametric model of random uncertainties is based on direct construction of a probabilistic model of the generalized mass, damping and stiffness matrices, which uses only the available information constituted of the mean value of the generalized mass, damping and stiffness matrices. This paper describes the explicit construction of the theory of such a nonparametric model.

Maximum entropy approach for modeling random uncertainties in transient elastodynamics.

A new approach is presented for analyzing random uncertainties in dynamical systems. This approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics. The information used does not require the description of the local parameters of the mechanical model. The probability model is deduced from the use of the entropy optimization principle, whose available information is constituted of the algebraic properties related to the generalized mass, damping, and stiffness matrices which have to be positive-definite symmetric matrices, and the knowledge of these matrices for the mean reduced matrix model. An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads. The fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed. Finally, an example is presented.

Structural Acoustics and Vibration

This book is devoted to mechanical models, variational formulations and discretization for calculating linear vibrations in the frequency domain of complex structures with arbitrary shape, coupled or not with external and internal acoustic fluids at rest. Such coupled systems are encountered in the area of internal and external noise prediction, reduction and control problems. The excitations can arise from different mechanisms such as mechanical forces applied to the structure, internal acoustic sources, external acoustic sources and external incident acoustic plane waves. These excitations can be deterministic or random. We are interested not only in the low-frequency domain for which modal analysis is suitable, but also in the medium-frequency domain for which additional mechanical modeling and appropriate solving methods are necessary. The main objective of the book is to present appropriate theoretical formulations, constructed so as to be directly applicable for developing computer codes for the numerical simulation of complex systems.

The Fokker-Planck equation for stochastic dynamical systems and its explicit steady state solutions

Stochastic Canonical Equation of Multidimensional Nonlinear Dissipative Hamiltonian Dynamical Systems Fundamental Examples of Nonlinear Dynamical Systems and Associated Second Order Equation Brief Review of Probability and Random Variables Probabilistic Tools I. Classical Stochastic Processes Probabilistic Tools II. Mean-Square theory of linear integral transformations and of linear Differential Equations Probabilistic Tools III. Diffusion Processes and Fokker-Planck Equation Probabilistic Tools IV. Stochastic Integrals and Stochastic Differential Equations Stochastic Modelling with Stochastic Differential Equations FKP Equation for the Dissipative Hamiltonian Dynamical Systems Stationary Response of Dissipative Dynamical Systems, Existence and Uniqueness, Explicit Solution of an Invariant Measure Complements for the Normalization Condition, Characteristic Function and Moments of the Invariant Measure Application II. Multidimensional Linear Oscillators Subjected to External and Parametric Random Excitations Application III. Multidimensional Nonlinear Oscillators with Inertial Nonlinearity Subjected to External Random Excitations Application Ill. Multidimensional Nonlinear Oscillators Subjected to External and Parametric Random Excitations Symplectic Change of Variables in the Multidimensional Unsteady FKP Equation.

A model and numerical method in the medium frequency range for vibroacoustic predictions using the theory of structural fuzzy

In linear dynamical analysis of complex mechanical systems, the structural fuzzy is defined as the set of minor subssytems that are connected to the master structure but are not accessible by classical modeling. A global probabilistic modeling of the structural fuzzy is proposed to improve the calculated estimates of the MF vibration in the master structure and of the far field radiated by itself in or out the context of the acoustic scattering. Two probabilistic constitutive equation laws of the structural fuzzy are constructed. A numerical simulation on standard structures and on submerged complex industrail structure are shown.

Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms

Mathematical justifications are given for a Monte Carlo simulation technique based on memoryless transformations of Gaussian processes. Different types of convergences are given for the approaching sequence. Moreover an original numerical method is proposed in order to solve the functional equation yielding the underlying Gaussian process autocorrelation function.

Random matrix theory and non-parametric model of random uncertainties in vibration analysis

Recently, a new approach, called a non-parametric model of random uncertainties, has been introduced for modelling random uncertainties in linear and non-linear elastodynamics in the low-frequency range. This non-parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive-definite real random matrices. This ensemble differs from the Gaussian orthogonal ensemble (GOE) and from the other known ensembles of the random matrix theory. The present paper has three main objectives. The first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the GOE. The second one is to compare this new ensemble of random matrices with the GOE in the context of the non-parametric approach of random uncertainties in structural dynamics for the low-frequency range. The last objective is to give a new validation for the non-parametric model of random uncertainties in structural dynamics in comparing, in the low-frequency range, the dynamical response of a simple system having random uncertainties modelled by the parametric and the non-parametric methods. These three objectives will allow us to conclude about the validity of the different theories.

Structural-acoustic modeling of automotive vehicles in presence of uncertainties and experimental identification and validation

The design of cars is mainly based on the use of computational models to analyze structural vibrations and internal acoustic levels. Considering the very high complexity of such structural-acoustic systems, and in order to improve the robustness of such computational structural-acoustic models, both model uncertainties and data uncertainties must be taken into account. In this context, a probabilistic approach of uncertainties is implemented in an adapted computational structural-acoustic model. The two main problems are the experimental identification of the parameters controlling the uncertainty levels and the experimental validation. Relevant experiments have especially been developed for this research in order to constitute an experimental database devoted to structural vibrations and internal acoustic pressures. This database is used to perform the experimental identification of the probability model parameters and to validate the stochastic computational model.

Reduced models in the medium frequency range for general dissipative structural-dynamics systems

Abstract This paper presents a theoretical approach for constructing a reduced model, in the medium frequency range, in the area of structural dynamics for a general three-dimensional anisotropic and inhomogeneous viscoelastic bounded medium. All the results presented can be used for beams, plates and shells. The boundary value problem in the frequency domain and its variational formulation are presented. For a given medium frequency band, an energy operator which is intrinsic to the dynamic system is introduced and mathematically studied. This energy operator depends on the dissipative part of the dynamical system. It is proved that this operator is a positive-definite symmetric trace operator in a Hilbert space and that its dominant eigensubspace allows a reduced model to be constructed using the Ritz-Galerkin method. A finite dimension approximation of the continuous case is presented (for instance using the finite element method). An effective construction of the dominant subspace using the subspace iteration method is developed. Finally, an example is given to validate the concepts and the algorithms.