Les structures composites sont souvent exposées à des ambiances dynamiques plus oumoins sévères. Ces vibrations peuvent développer différentes formes d’endommagement(fracture des fibres, délamination, fissuration de la matrice. . . ). Les défauts locaux sepropagent et affectent les propriétés mécaniques de la structure modifiant ainsi soncomportement dynamique global. Ces changements peuvent induire une dégradationrapide de la structure et une réduction de sa durée de vie. La thèse a pour objectif lamise en oeuvre de modèles de comportement pour le dimensionnement de structurescomplexes intégrant des sous-structures composites susceptibles d’être endommagées.La méthode des éléments finis est utilisée pour modéliser le comportement vibratoirelinéaire et non-linéaire de ces structures et l’endommagement est introduit via un modèlebilatéral, dans un premier temps. Durant le processus de résolution, une des difficultésrencontrées est le coût de calcul très élevé. Ainsi, un méta-modèle a été développé basésur les réseaux de neurones artificiels couplé avec la méthode de condensation par sousstructurationde Craig-Bampton. Les réseaux de neurones artificiels permettent d’estimer,à moindre cout numérique, le niveau d’endommagement sans avoir recours au calculexact. Le modèle d’endommagement bilatéral n’est pas adapté au cas de chargementsalternés ou périodiques. Par conséquent, la deuxième partie de la thèse est orientée versle développement d’un modèle d’endommagement unilatéral qui donne une meilleuredescription du comportement mécanique lorsque les micro-fissures sont fermées. De plus,dans plusieurs applications industrielles, les structures composites utilisées sont de faibleépaisseur. Par conséquent, elles peuvent avoir naturellement un comportement vibratoirenon-linéaire de type grands déplacements. Le modèle de comportement dynamique engrands déplacements et en présence de la non-linéarité matérielle d’endommagement estdéveloppé et validé. A l’issue de ces travaux de thèse, un outil numérique implémentésur MATLAB® a été développé intégrant deux modèles d’endommagement, bilatéralet unilatéral et une méta-modélisation permettant la localisation et l’estimation del’endommagement ainsi que la prédiction de la réponse dynamique des structures composites, totalement ou localement, endommagées. Le méta-modèle proposé permet deréduire significativement le coût de calcul tout en assurant une bonne précision en termesde localisation et d’estimation du niveau d’endommagement. Cet outil peut s’avérer utilepour diverses applications dans le domaine de surveillance de l’état de santé des structurescomposites. / Composite structures are often exposed to more or less severe dynamic perturbations.These vibrations can develop different forms of damage (fiber fracture, delamination,cracking of the matrix, etc.). Local defects propagate and affect the mechanical propertiesof the structure resulting to modify its global dynamic behavior. These changes can leadto the degradation of the structure and the reduction in its lifetime. This thesis focuseson the implementation of behavior models for the dimensioning of complex structuresintegrating damaged composite sub-structures. The finite element method is used tomodel the linear and nonlinear vibration behavior of these structures where the damageis introduced, initially, via a bilateral model. Since the high computational costs duringthe solving process, a meta-model was developed based on artificial neural networkscoupled with the condensation method of Craig-Bampton. Artificial neural networkspermit to estimate the damage severity at a lower numerical cost without resorting toexact calculation. The bilateral damage model is not adapted to the case of periodic loads.Consequently, the second part of the thesis is oriented towards the development of aunilateral damage model which gives a better description of the mechanical behaviorwhen the micro-cracks are closed. Moreover, in several industrial applications, the usedcomposite structures have small thickness. Therefore, they can naturally have a geometricnon-linear dynamic behavior. The model of dynamic behavior in large displacements andin the presence of material non-linearity of damage is developed and validated. At theend of this thesis, a numerical tool implemented on MATLAB® software was developedintegrating two models of damage, bilateral and unilateral, and a meta-modeling allowingthe localization and the estimation of the damage as well as the prediction of the linear andnon-linear dynamic responses of composite structures, totally or locally, damaged. Theproposed meta-model reduces significantly the computational cost and ensuring a goodaccuracy in terms of localization and estimation of the damage severity. Thereby, thistool can be useful in life-time estimation and monitoring strategies of composite structures.Thèse de

Uncertainty Quantification (UQ) has been an active area of research in recent years with a wide range of applications in data and imaging sciences. In many problems, the source of uncertainty stems from an unknown parameter in the model. In physical and engineering systems for example, the parameters of the partial differential equation (PDE) that model the observed data may be unknown or incompletely specified. In such cases, one may use a probabilistic description based on prior information and formulate a forward UQ problem of characterising the uncertainty in the PDE solution and observations in response to that in the parameters. Conversely, inverse UQ encompasses the statistical estimation of the unknown parameters from the available observations, which can be cast as a Bayesian inverse problem. The contributions of the thesis focus on examining the aforementioned forward and inverse UQ problems for the low-frequency, time-harmonic Maxwell equations, where the model uncertainty emanates from the lack of knowledge of the material conductivity parameter. The motivation comes from the Controlled-Source Electromagnetic Method (CSEM) that aims to detect and image hydrocarbon reservoirs by using electromagnetic field (EM) measurements to obtain information about the conductivity profile of the sub-seabed. Traditionally, algorithms for deterministic models have been employed to solve the inverse problem in CSEM by optimisation and regularisation methods, which aside from the image reconstruction provide no quantitative information on the credibility of its features. This work employs instead stochastic models where the conductivity is represented as a lognormal random field, with the objective of providing a more informative characterisation of the model observables and the unknown parameters. The variational formulation of these stochastic models is analysed and proved to be well-posed under suitable assumptions. For computational purposes the stochastic formulation is recast as a deterministic, parametric problem with distributed uncertainty, which leads to an infinite-dimensional integration problem with respect to the prior and posterior measure. One of the main challenges is thus the approximation of these integrals, with the standard choice being some variant of the Monte-Carlo (MC) method. However, such methods typically fail to take advantage of the intrinsic properties of the model and suffer from unsatisfactory convergence rates. Based on recently developed theory on high-dimensional approximation, this thesis advocates the use of Sparse Quadrature (SQ) to tackle the integration problem. For the models considered here and under certain assumptions, we prove that for forward UQ, Sparse Quadrature can attain dimension-independent convergence rates that out-perform MC. Typical CSEM models are large-scale and thus additional effort is made in this work to reduce the cost of obtaining forward solutions for each sampling parameter by utilising the weighted Reduced Basis method (RB) and the Empirical Interpolation Method (EIM). The proposed variant of a combined SQ-EIM-RB algorithm is based on an adaptive selection of training sets and a primal-dual, goal-oriented formulation for the EIM-RB approximation. Numerical examples show that the suggested computational framework can alleviate the computational costs associated with forward UQ for the pertinent large-scale models, thus providing a viable methodology for practical applications.

La réduction du temps de calcul lors de la résolution de problèmes d’évolution dans le cadre du calcul de structure constitue un enjeu majeur pour, par exemple, la mise en place de critères de rupture des pièces dans le secteur de l’aéronautique et de l’automobile. En particulier, la prédiction du cycle stabilisé des polymères sollicités sous chargement cyclique nécessite de résoudre un problème thermo-viscoélastique à grand nombre de cycles. La présence de différentes échelles de temps telles que le temps de relaxation (viscosité), le temps caractéristique associé au problème thermique et le temps du cycle de chargement conduit à un temps de calcul significatif lorsqu’un schéma incrémental est utilisé comme c’est le cas avec la méthode des éléments finis (MEF). De plus, un nombre important de données doit être stocké (au moins à chaque cycle). L’objectif de cette thèse est de proposer de nouvelles méthodes ainsi que d’étendre des méthodes existantes. Il est choisi de résoudre un problème thermique transitoire cyclique impliquant différentes échelles de temps avec l’objectif de réduire le temps de calcul réduit. Les méthodes proposées font partie des méthodes de réduction de modèles. Tout d’abord, la méthode de décomposition propre généralisée(PGD) a été étendue à un problème transitoire cyclique 3D non linéaire, la non-linéarité a été traitée en combinant la méthode PGD à la Méthode d’interpolation empirique discrète (DEIM), stratégie numérique déjà proposée dans la littérature. Les résultats ont montré l’efficacité de la PGD pour générer des résultats précis par rapport à la solution FEM avec une erreur relative inférieure à (1%). Ensuite, afin de réduire le temps de calcul, une autre approche alternative a été développée. Cette approche est basée sur l’utilisation d’une collection de modes, les modes les plus significatifs, issus de solutions PGD pour différentes échelles de temps et différentes valeurs de paramètres. Un dictionnaire regroupant ces modes est alors utilisé pour construire des solutions pour différents temps caractéristiques et différentes conditions aux limites, uniquement par projection de la solution sur les modes du dictionnaire. Cette approche a été adaptée pour traiter un problème faiblement couplé diffuso-thermique. La nouveauté de cette approche est de considérer un dictionnaire composé de bases spatio-temporelles et non pas uniquement de bases spatiales comme dans la fameuse méthode POD. Les résultats obtenus avec cette approche sont précis et permettent une réduction notable du temps de calcul on line. Néanmoins, lorsque différents temps de cycles sont considérés, le nombre de modes dans le dictionnaire augmente, ce qui en limite son utilisation. Afin de pallier cette limitation,une troisième stratégie numérique est proposée dans cette thèse. Elle consiste à considérer comme a priori connues des bases temporelles, elle est appelée stratégie mixte. L’originalité dans cette approche réside dans la construction de la base temporelle a prior basée sur l’analyse de Fourier de différentes simulations pour différents temps et différentes valeurs des paramètres. Une fois cette étude réalisée, une expression analytique des bases temporelles fonction des paramètres tels que le temps caractéristique et le temps du cycle est proposée. Les bases spatiales associées sont calculées à l’aide d’un algorithme type PGD. Cette méthode est ensuite testée pour la résolution de problèmes thermiques 3D sous chargement cyclique linéaires et non linéaires et un problème faiblement couplé thermo-diffusion. / In the framework of structural calculation, the reduction of computation time plays an important rolein the proposition of failure criteria in the aeronautic and automobile domains. Particularly, the prediction of the stabilized cycle of polymer under cyclic loading requires solving of a thermo-viscoelastic problem with a high number of cycles. The presence of different time scales, such as relaxation time (viscosity), thermal characteristic time (thermal), and the cycle time (loading) lead to a huge computation time when an incremental scheme is used such as with the Finite Element Method (FEM).In addition, an allocation of memory will be used for data storage. The objective of this thesis isto propose new techniques and to extend existent ones. A transient thermal problem with different time scales is considered in the aim of computation time reduction. The proposed methods are called model reduction methods. First, the Proper Generalized Decomposition method (PGD) was extended to a nonlinear transient cyclic 3D problems. The non-linearity was considered by combining the PGD method with the Discrete Empirical Interpolation Method (DEIM), a numerical strategy used in the literature. Results showed the efficiency of the PGD in generating accurate results compared to the FEM solution with a relative error less than 1%. Then, a second approach was developed in order to reduce the computation time. It was based on the collection of the significant modes calculated from the PGD method for different time scales. A dictionary assembling these modes is then used to calculate the solution for different characteristic times and different boundary conditions. This approach was adapted in the case of a weak coupled diffusion thermal problem. The novelty of this method is to consider a dictionary composed of spatio-temporal bases and not spatial only as usedin the POD. The results showed again an exact reproduction of the solution in addition to a huge time reduction. However, when different cycle times are considered, the number of modes increases which limits the usage of the approach. To overcome this limitation, a third numerical strategy is proposed in this thesis. It consists in considering a priori known time bases and is called the mixed strategy. The originality in this approach lies in the construction of a priori time basis based on the Fourier analysis of different simulations for different time scales and different values of parameters.Once this study is done, an analytical expression of time bases based on parameters such as the characteristic time and the cycle time is proposed. The related spatial bases are calculated using the PGD algorithm. This method is then tested for the resolution of 3D thermal problems under cyclic loading linear and nonlinear and a weak coupled diffusion thermal problem.

Les méthodes de modification structurale sont l'ensemble des techniques qui permettent d'estimer l'influence d'une modification quelconque sur le comportement dynamique d'une structure. Dans le cadre général, le modèle de comportement de la structure cible, comme celui de la modification, peut être aussi bien numérique qu'expérimental. Dans le cadre de ce travail, seul le cas particulier du couplage d'un modèle expérimental de la structure et d'un modèle numérique de la modification sera traité.<br />Les concepts fondamentaux utilisés dans cette thèse sont ensuite présentés. Les relations de la dynamique des structures pour les problèmes discrets sont rappelées, ainsi que les principes de la synthèse modale, de la sous-structuration dynamique et de la réduction de modèle, tout comme la notion de modes d'interface. Les formulations classiques des méthodes de modification structurale sont ensuite détaillées pour en illustrer les limitations et les restrictions.<br />Une formulation originale permettant de prendre en compte les incompatibilités entre les mesures et les DDL de l'interface structure/modification et de régulariser la construction d'un modèle de comportement couplé est alors proposée. Cette première contribution de la thèse repose sur l'utilisation des techniques d'expansion de données et de réduction de modèle. Des indicateurs sont également construits pour estimer la cohérence de la prédiction réalisée. Les évolutions sont appliquées au cas d'un démonstrateur numériques et les résultats sont comparés avec les prédictions réalisées par les méthodes classiques. La méthodologie associée à cette nouvelle formulation est alors largement exposée.<br />L'influence des différents facteurs intervenant dans la construction du modèle couplé et la qualité de la prédiction est ensuite analysée en détail. Cette analyse permet de dresser une liste non exhaustive des précautions à prendre lors de la mise en œuvre de la méthode proposée, depuis la réalisation pratique de l'analyse modale expérimentale jusqu'à l'interprétation des premiers résultats.<br />Enfin, plusieurs applications sont présentées. Une première structure académique démontre la faisabilité de la méthode. Une deuxième étude, réalisée sur un cas industriel, illustre les gains de temps potentiels en comparant la prédiction avec les résultats d'une étude basée sur un modèle EF recalé de la structure. La troisième étude illustre l'application de la méthode dans un cas type. L'analyse modale de la structure cible permet de comprendre le problème, une modification est conçue, réalisée et mise en place. La prédiction est ensuite comparée aux résultats de l'analyse modale de la structure modifiée. Enfin, la dernière application montre les limites de la méthodologie. L'étude multi-objectifs sur une large bande de fréquences d'une structure industrielle permet de faire une ouverture vers la suite des travaux et montre la nature des difficultés à surmonter.

This thesis consists of six papers. The main topic of all these papers is modeling a class of linear time-invariant systems. The system class is parameterized in the context of interpolation theory with a degree constraint. In the papers included in the thesis, this parameterization is the key tool for the design of dynamical system models in fields such as spectral estimation and model reduction. A problem in spectral estimation amounts to estimating a spectral density function that captures characteristics of the stochastic process, such as covariance, cepstrum, Markov parameters and the frequency response of the process. A  model reduction problem consists in finding a small order system which replaces the original one so that the behavior of both systems is similar in an appropriately defined sense.  In Paper A a new spectral estimation technique based on the rational covariance extension theory is proposed. The novelty of this approach is in the design of a spectral density that optimally matches covariances and approximates the frequency response of a given process simultaneously.In Paper B  a model reduction problem is considered. In the literature there are several methods to perform model reduction. Our attention is focused on methods which preserve, in the model reduction phase, the stability and the positive real properties of the original system. A reduced-order model is computed employing the analytic interpolation theory with a degree constraint. We observe that in this theory there is a freedom in the placement of the spectral zeros and interpolation points. This freedom can be utilized for the computation of a rational positive real function of low degree which approximates the best a given system. A problem left open in Paper B is how to select spectral zeros and interpolation points in a systematic way in order to obtain the best approximation of a given system. This problem is the main topic in Paper C. Here, the problem is investigated in the analytic interpolation context and spectral zeros and interpolation points are obtained as solution of a optimization problem.In Paper D, the problem of modeling a floating body by a positive real function is investigated. The main focus is  on modeling the radiation forces and moment. The radiation forces are described as the forces that make a floating body oscillate in calm water. These forces are passive and usually they are modeled with system of high degree. Thus, for efficient computer simulation it is necessary to obtain a low order system which approximates the original one. In this paper, the procedure developed in Paper C is employed. Thus, this paper demonstrates the usefulness of the methodology described in Paper C for a real world application.In Paper E, an algorithm to compute the steady-state solution of a discrete-type Riccati equation, the Covariance Extension Equation, is considered. The algorithm is based on a homotopy continuation method with predictor-corrector steps. Although this approach does not seem to offer particular advantage to previous solvers, it provides insights into issues such as positive degree and model reduction, since the rank of the solution of the covariance extension problem coincides with the degree of the shaping filter. In Paper F a new algorithm for the computation of the analytic interpolant of a bounded degree is proposed. It applies to the class of non-strictly positive real interpolants and it is capable of treating the case with boundary spectral zeros. Thus, in Paper~F, we deal with a class of interpolation problems which could not be treated by the optimization-based algorithm proposed by Byrnes, Georgiou and Lindquist. The new procedure computes interpolants by solving a system of nonlinear equations. The solution of the system of nonlinear equations is obtained by a homotopy continuation method. / QC 20100721

This thesis is divided into two parts. The first part deals with theNevanlinna-Pick interpolation problem, a problem which occursnaturally in several applications such as robust control, signalprocessing and circuit theory. We consider the problem of shaping andapproximating solutions to the Nevanlinna-Pick problem in a systematicway. In the second part, we study distance measures between powerspectra for spectral estimation. We postulate a situation where wewant to quantify robustness based on a finite set of covariances, andthis leads naturally to considering the weak*-topology. Severalweak*-continuous metrics are proposed and studied in this context.In the first paper we consider the correspondence between weighted entropyfunctionals and minimizing interpolants in order to find appropriateinterpolants for, e.g., control synthesis. There are two basic issues that weaddress: we first characterize admissible shapes of minimizers bystudying the corresponding inverse problem, and then we developeffective ways of shaping minimizers via suitable choices of weights.These results are used in order to systematize feedback controlsynthesis to obtain frequency dependent robustness bounds with aconstraint on the controller degree.The second paper studies contractive interpolants obtained as minimizersof a weighted entropy functional and analyzes the role of weights andinterpolation conditions as design parameters for shaping theinterpolants. We first show that, if, for a sequence of interpolants,the values of the corresponding entropy gains converge to theoptimum, then the interpolants converge in H_2, but not necessarily inH-infinity. This result is then used to describe the asymptoticbehaviour of the interpolant as an interpolation point approaches theboundary of the domain of analyticity.A quite comprehensive theory of analytic interpolation with degreeconstraint, dealing with rational analytic interpolants with an apriori bound, has been developed in recent years. In the third paper,we consider the limit case when this bound is removed, and only stableinterpolants with a prescribed maximum degree are sought. This leadsto weighted H_2 minimization, where the interpolants areparameterized by the weights. The inverse problem of determining theweight given a desired interpolant profile is considered, and arational approximation procedure based on the theory is proposed. Thisprovides a tool for tuning the solution for attaining designspecifications. The purpose of the fourth paper is to study the topology and develop metricsthat allow for localization of power spectra, based on second-orderstatistics. We show that the appropriate topology is theweak*-topology and give several examples on how to construct suchmetrics. This allows us to quantify uncertainty of spectra in anatural way and to calculate a priori bounds on spectral uncertainty,based on second-order statistics. Finally, we study identification ofspectral densities and relate this to the trade-off between resolutionand variance of spectral estimates.In the fifth paper, we present an axiomatic framework for seekingdistances between power spectra. The axioms requirethat the sought metric respects the effects of additive andmultiplicative noise in reducing our ability to discriminate spectra.They also require continuity of statistical quantities withrespect to perturbations measured in the metric. We then present aparticular metric which abides by these requirements. The metric isbased on the Monge-Kantorovich transportation problem and iscontrasted to an earlier Riemannian metric based on theminimum-variance prediction geometry of the underlying time-series. Itis also being compared with the more traditional Itakura-Saitodistance measure, as well as the aforementioned prediction metric, ontwo representative examples. / QC 20100817

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit. As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated.

The objective of this thesis is to propose modeling techniques that enable the design and optimization of material systems which require descriptions via molecular simulations. These kinds of systems are quite common in materials and engineering research. The first step in performing design and optimization tasks on such systems is the development of accurate simulation models from experimental data. In the first part of this thesis, we present a novel simulation model for the hyperbranched polymerization process of difunctional A2 oligomers, and B3 monomers. Unlike the previous models developed by other groups, our model is able to simulate the evolution of the polymer structure development under a wide range of synthesis routes, and in the presence of cyclization and endcapping reactions. Furthermore, our results are in agreement with the experimental data, and add insight into the underlying kinetic mechanisms of this polymerization process. The second major step in our work is the development of reduced order process models that are suitable for design and optimization tasks, using simulation data. We illustrate our approach on a stochastic simulation model of epitaxial thin film deposition process. Compared to the widely used approach called equation-free modeling, our method requires fewer assumptions about the dynamic system. The assumptions required in equation-free modeling include a wide separation between the time scales of low and high order moments describing the system state, and the accuracy of the time derivatives of system properties computed from molecular simulation data, despite the potentially large amount of fluctuations in stochastic simulations. Unlike the recent similar studies, our study also includes the analysis of prediction error which is important to evaluate the predictions of the reduced order model, compared to the high dimensional molecular simulations. Hence, we address two major issues in this thesis: development of simulation models from molecular experimental data, and derivation of reduced order models from molecular simulation data. These two aspects of modeling are both necessary to design and optimize processing conditions of materials for which continuum level descriptions are not available or accurate enough.

Uncertainties in both loading and structural characteristics can adversely affect the response and reliability of a structure. Parameter uncertainties in structural dynamics can arise due to several sources. These include variations due to intrinsic material property variability, measurement errors, manufacturing and assembly errors, differences in modeling and solution procedures. Problems of structural dynamics with randomly distributed spatial inhomogeneities in elastic, mass, and damping properties, have been receiving wide attention. Several mathematical and computational issues include discretization of random fields, characterization of random eigensolutions, inversion of random matrices, solutions of stochastic boundary-value problems, and description of random matrix products. Difficulties are encountered when one has to include interaction between nonlinear and stochastic system characteristics, or if one is interested in controlling the system response. The study of structural systems including the effects of system nonlinearity in the presence of parameter uncertainties presents serious challenges and difficulties to designers and reliability engineers. In the analysis of large structures, the situation for substructuring frequently arises due to the repetition of identical assemblages (substructures), within a structure, and the general need to reduce the size of the problem, particularly in the case of non-linear inelastic dynamic analysis. A small reduction in the model size can have a large effect on the storage and time requirement. A primary structural dynamic system may be coupled to subsystems such as piping systems in a nuclear reactor or in a chemical plant. Usually subsystem in itself is quite complex and its modeling with finite elements may result in a large number of degrees of freedom. The reduced subsystem model should be of low-order yet capturing the essential dynamics of the subsystem for useful integration with the primary structure. There are two major issues to be studied: one, techniques for analyzing a complex structure into component subsystems, analyzing the individual sub-system dynamics, and from thereon determining the dynamics of the structure after assembling the subsystems. The nonlinearity due to support gap effects such as supports for piping system in nuclear reactors further complicates the problem. The second is the issue of reviewing the methods for reducing the model-order of the component subsystems such that the order of the global dynamics, after assembly, is within some predefined limits. In the reliability analysis of complex engineering structures, a very large number of the system parameters have to be considered as random variables. The parameter uncertainties are modeled as random variables and are assumed to be time independent. Here the problem would be to reduce the number of random variables without sacrificing the accuracy of the reliability analysis. The procedure involves the reduction of the size of the vector of random variables before the calculation of failure probability. The objectives of this thesis are: 1.To use the available model reduction techniques in order to effectively reduce the size of the finite element model, and hence, compare the dynamic responses from such models. 2.Study of propagation of uncertainties in the reduced order/coupled stochastic finite element dynamic models. 3.Addressing the localized nonlinearities due to support gap effects in the built up structures, and also in cases of sudden change in soil behaviour under the footings. The irregularity in soil behaviour due to lateral escape of soil due to failure of quay walls/retaining walls/excavation in neighbouring site, etc. 4.To evolve a procedure for the reduction of size of the vector containing the random variables before the calculation of failure probability. In the reliability analysis of complex engineering structures, a very large number of the system parameters are considered to be random variables. Here the problem would be to reduce the number of random variables without sacrificing the accuracy of the reliability analysis. 5.To analyze the reduced nonlinear stochastic dynamic system (with phase space reduction), and effectively using the network pruning technique for the solution, and also to use filter theory (wavelet theory) for reducing the input earthquake record to save computational time and cost. It is believed that the techniques described provide highly useful insights into the manner structural uncertainties propagate. The cross-sectional area, length, modulus of elasticity and mass density of the structural components are assumed as random variables. Since both the random and design variables are expressed in a discretized parameter space, the stochastic sensitivity function can be modeled in a parallel way. The response of the structures in frequency domain is considered. This thesis is organized into seven chapters. This thesis deals with the reduced order models of the stochastic structural systems under deterministic/random loads. The Chapter 1 consists of a brief introduction to the field of study. In Chapter 2, an extensive literature survey based on the previous works on model order reduction and the response variability of the structural dynamic systems is presented. The discussion on parameter uncertainties, stochastic finite element method, and reliability analysis of structures is covered. The importance of reducing mechanical models for dynamic response variability, the systems with high-dimensional variables and reduction in random variables space, nonlinearity issues are discussed. The next few chapters from Chapter 3 to Chapter 6 are the main contributions in this thesis, on model reduction under various situations for both linear and nonlinear systems. After forming a framework for model reduction, local nonlinearities like support gaps in structural elements are considered. Next, the effect of reduction in number of random variables is tackled. Finally influence of network pruning and decomposition of input signals into low and high frequency parts are investigated. The details are as under. In Chapter 3, the issue of finite element model reduction is looked into. The generalized finite element analysis of the full model of a randomly parametered structure is carried out under a harmonic input. Different well accepted finite element model reduction techniques are used for FE model reduction in the stochastic dynamic system. The structural parameters like, mass density and modulus of elasticity of the structural elements are considered to be non-Gaussian random variables. Since the variables considered here are strictly positive, the probabilistic distribution of the random variables is assumed to be lognormal. The sensitivities in the eigen solutions are compared. The response statistics based on response of models in frequency domain are compared. The dynamic responses of the full FE model, separated into real and imaginary parts, are statistically compared with those from reduced FE models. Monte Carlo simulation is done to validate the analysis results from SFEM. In Chapter 4, the problem of coupling of substructures in a large and complex structure, and FE model reduction, e.g., component mode synthesis (CMS) is studied in the stochastic environment. Here again, the statistics of the response from full model and reduced models are compared. The issues of non-proportional damping, support gap effects and/local nonlinearity are considered in the stochastic sense. Monte Carlo simulation is done to validate the analysis results from SFEM. In Chapter 5, the reduction in size of the vector of random variables in the reliability analysis is attempted. Here, the relative entropy/ K-L divergence/mutual information, between the random variables is considered as a measure for ranking of random variables to study the influence of each random variable on the response/reliability of the structure. The probabilistic distribution of the random variables is considered to be lognormal. The reliability analysis is carried out with the well known Bucher and Bourgund algorithm (1990), along with the probabilistic model reduction of the stochastic structural dynamic systems, within the framework of response surface method. The reduction in number of random variables reduces the computational effort required to construct an approximate closed form expression in response surface approach. In Chapter 6, issues regarding the nonlinearity effects in the reduced stochastic structural dynamic systems (with phase space reduction), along with network pruning are attempted. The network pruning is also adopted for reduction in computational effort. The earthquake accelerogram is decomposed using Fast Mallat Algorithm (Wavelet theory) into smaller number of points and the dynamic analysis of structures is carried out against these reduced points, effectively reducing the computational time and cost. Chapter 7 outlines the contributions made in this thesis, together with a few suggestions made for further research. All the finite element codes were developed using MATLAB5.3. Final pages of the thesis contain the references made in the preparation of this thesis.

A practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.