Modern technologies require the materials with combinations of properties that can not be met by conventional single phase materials. This requirement leads to the development of composite materials or other materials with engineered microstructures, such as polymer composites and nanotube. Though the well-established finite element analysis (FEA) has the ability to analyze a small portion of such material, for the whole structure, the total degrees of freedom of a finite element model can easily exceed the bearable time in analysis or the capability of the best mainstream computers. To reduce the total degrees of freedom and save the computational efforts, an efficient way is to use a simpler and coarser mesh at the structure level with the micro level complexities captured by a homogenization method. Throughout the dissertation, the homogenization is carried on by variational asymptotic method which has been developed recently as the Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH). This methodology is also expandable to the structure analysis as long as a representative structural element (RSE) can be obtained from structure. In the present research, the following problems are handled: (1) Maximizing the flexibility of choosing a RSE; (2) Bounding the effective properties of a random RSE; (3) Obtaining the equivalent plate stiffnesses for a corrugated plate from a RSE; (4) Extending the shell element of relative degree of freedom to analyze thin-walled RSE. These problems covered some important topics in homogenization theory. Firstly, the rules need to be followed when choosing a unit cell from a structure that can be homogenized. Secondly, for a randomly packed structure, the efficient way to predict effective material properties is to predict their bounds. Then, the composite material homogenization and the structural homogenization can be unied from a mathematical point of view, thus the repeating structure can be always simplified by the homogenization method. Lastly, the efficiency of analyzing thin-walled structures has been enhanced by the new type of shell element. In this research, the first two topics have been solved numerically through the finite element method under the framework of VAMUCH. The third one has been solved both analytically and numerically, and in the last, a new type of element has been implemented in VAMUCH to adapt the characteristics of a thin-walled problem. Numerous examples have demonstrated VAMUCH application and accuracy as a general-purpose analysis tool.

Unlike published shell theories, the main two parts of this thesis are devoted to the asymptotic construction of a refined theory for composite laminated shells valid over a wide range of frequencies and wavelengths. The resulting theory is applicable to shells each layer of which is made of materials with monoclinic symmetry. It enables one to analyze shell dynamic responses within both long-wavelength, low- and high-frequency vibration regimes. It also leads to energy functionals that are both positive definiteness and sufficient simplicity for all wavelengths. This whole procedure was first performed analytically. From the insight gained from the procedure, a finite element version of the analysis was then developed; and a corresponding computer program, DVAPAS, was developed. DVAPAS can obtain the generalized 2-D constitutive law and recover accurately the 3-D results for stress and strain in composite shells. Some independent works will be needed to develop the corresponding 2-D surface analysis associated with the present theory and to continue towards full verification and validation of the present process by comparison with available published works.

An asymptotically-exact methodology is presented for obtaining the cross-sectional stiffness matrix of a pre-twisted, moderately-thick beam having rectangular cross sections and made of transversely isotropic material. The beam is modelled with-out assumptions from 3-D elasticity. The strain energy of the beam is computed making use of the constitutive law and the kinematical relations derived with the inclusion of geometrical nonlinearities and initial twist. Large displacements and rotations are allowed, but small strain is assumed. The Variational Asymptotic Method (VAM) is used to minimize the energy functional, thereby reducing the cross section to a point on the reference line with appropriate properties, yielding a 1-D constitutive law. In this method as applied herein, the 2-D cross-sectional analysis is performed asymptotically by taking advantage of a material small parameter and two geometric small parameters. 3-D strain components are derived using kinematics and arranged as orders of the small parameters. Warping functions are obtained by the minimisation of strain energy subject to certain set of constraints that renders the 1-D strain measures well-defined. Closed-form expressions are derived for the 3-D non-linear warping and stress fields. The model is capable of predicting interlaminar and transverse shear stresses accurately up to first order.

A Identificação de Sistemas é uma das técnicas utilizadas para se obter a representação matemática de um sistema. Diversos métodos podem ser aplicados para se obter um modelo matemático através da identificação de sistemas, entre eles o método de identificação assintótico, também chamado de ASYM (Zhu, 1998). Este trabalho propõe aplicar o método de identificação assintótico em sistemas SISO para a obtenção de modelo de sistemas ditos caixa-preta e avaliar o seu desempenho buscando também o melhor detalhamento do método. Os modelos obtidos foram avaliados de acordo com sua nota calculada através do método ASYM, através da comparação do índice de ajuste fit para autovalidação e validação cruzada e pela variância dos parâmetros dos modelos. O método ASYM é exaustivamente testado para sua avaliação. Entre os testes realizados neste trabalho destacam-se dois experimentos tipo Monte-Carlo com mais de quinhentas identificações e a aplicação do método em uma planta real. Os testes comprovaram a viabilidade da aplicação do método assintótico na identificação de sistemas SISO do tipo caixa-preta com excelente desempenho para estruturas ARMAX. / System Identification is one of the techniques used to obtain the mathematical representation of a system. Several methods can be applied to obtain a mathematical model by the system identification, including the asymptotic method, also called ASYM (Zhu, 1998). This work proposes to apply the ASYM method for SISO systems identification, then obtain models of black-box systems called \"black box\" and evaluate its performance and show details of the method. The models obtained were evaluated according to their grade calculated using the ASYM method, by comparing the fit adjustment index, self-validation and cross validation and the variance of model parameters. The asymptotic method has been extensively tested to be evaluated. Among the tests in this work, two stand out such Monte Carlo experiments with more than five hundred identifications and a real plant identification. The tests proved the feasibility of applying the asymptotic method in the \"black box\" SISO systems identification with excellent performance for ARMAX structures.

L'objet de cette thèse est la construction de schémas numériques pour les équations cinétiques dans différents régimes de diffusion anormale. Comme le modèle devient raide en s'approchant du modèle asymptotique, les méthodes numériques standard deviennent coûteuses dans ce régime. Les schémas Asymptotic Preserving ont été introduits pour pallier à cette difficulté. Ils sont en effet stables le long de la transition du régime mésoscopique au régime microscopique. Dans le premier chapitre, nous considérons le cas d'une distribution d'équilibre qui est une fonction à queue lourde et dont le moment d'ordre 2 est infini. Le poids important des grandes vitesses de l'équilibre fait tomber la limite de diffusion usuelle en défaut, et on montre que le modèle asymptotique est une équation de diffusion fractionnaire. En nous basant sur une analyse asymptotique formelle de la convergence vers le modèle limite, nous construisons trois schémas AP pour le problème. La discrétisation en vitesse est discutée afin de prendre en compte correctement les grandes vitesses, et nous montrons que le troisième schéma est en outre uniformément précis au cours de la transition vers le régime microscopique. Dans le chapitre 2, nous étendons ces résultats au cas d'une fréquence de collision dégénérée en 0 qui mène aussi à une équation de diffusion fractionnaire. Nous adaptons ensuite ces méthodes numériques au cas d'une limite de diffusion normale avec scaling en temps anormal dans l'équation cinétique dans le chapitre 3. Dans ce cadre, la lenteur de la convergence vers le modèle asymptotique rend nécessaire une adaptation de l'approche AP des chapitres précédents. Enfin, le chapitre 4 présente un schéma AP pour l'équation cinétique dans le cas heavy-tail du chapitre 1 lorsque l'opérateur de collision est non-local. / In this thesis, we construct numerical schemes for kinetic equations in some anomalous diffusion regimes. As the model becomes stiff when reaching the asymptotic model, the standard numerical methods become costly in this regime. Asymptotic Preserving (AP) schemes have been designed to overcome this difficulty. Indeed, they are uniformly stable along the transition from the mesoscopic regime to the microscopic one. In the first chapter, we study the case of a heavy-tailed equilibrium distribution, with infinite second order moment. The importance of the high velocities in the equilibrium makes the classical diffusion limit fail, and one can prove that the asymptotic model is a fractional diffusion equation. We construct three AP schemes for this problem, based on a formal asymptotic analysis of the convergence towards the limit model. The discretization of the velocities is then discussed to take into account the high velocities. Moreover, we prove that the third scheme enjoys the stronger property of being uniformly accurate along the convergence towards the microscopic regime. In chapter 2, we extend these results to the case of a degenerated collision frequency, also leading to a fractional diffusion limit. In chapter 3, these methods are then adapted to the case of a classical diffusion limit with anomalous time scale in the kinetic equation. In this case, an adaptation of the AP approach of the previous chapter is needed, because of the slow convergence rate of the kinetic equation towards the limit model. Eventually, a AP scheme for kinetic equation with heavy-tailed equilibria and non local collision operator is presented in chapter 4.

A Identificação de Sistemas é uma das técnicas utilizadas para se obter a representação matemática de um sistema. Diversos métodos podem ser aplicados para se obter um modelo matemático através da identificação de sistemas, entre eles o método de identificação assintótico, também chamado de ASYM (Zhu, 1998). Este trabalho propõe aplicar o método de identificação assintótico em sistemas SISO para a obtenção de modelo de sistemas ditos caixa-preta e avaliar o seu desempenho buscando também o melhor detalhamento do método. Os modelos obtidos foram avaliados de acordo com sua nota calculada através do método ASYM, através da comparação do índice de ajuste fit para autovalidação e validação cruzada e pela variância dos parâmetros dos modelos. O método ASYM é exaustivamente testado para sua avaliação. Entre os testes realizados neste trabalho destacam-se dois experimentos tipo Monte-Carlo com mais de quinhentas identificações e a aplicação do método em uma planta real. Os testes comprovaram a viabilidade da aplicação do método assintótico na identificação de sistemas SISO do tipo caixa-preta com excelente desempenho para estruturas ARMAX. / System Identification is one of the techniques used to obtain the mathematical representation of a system. Several methods can be applied to obtain a mathematical model by the system identification, including the asymptotic method, also called ASYM (Zhu, 1998). This work proposes to apply the ASYM method for SISO systems identification, then obtain models of black-box systems called \"black box\" and evaluate its performance and show details of the method. The models obtained were evaluated according to their grade calculated using the ASYM method, by comparing the fit adjustment index, self-validation and cross validation and the variance of model parameters. The asymptotic method has been extensively tested to be evaluated. Among the tests in this work, two stand out such Monte Carlo experiments with more than five hundred identifications and a real plant identification. The tests proved the feasibility of applying the asymptotic method in the \"black box\" SISO systems identification with excellent performance for ARMAX structures.

A series of composite beam models are constructed for efficient high-fidelity beam analysis based on the variational-asymptotic method (VAM). Without invoking any a priori kinematic assumptions, the original three-dimensional, geometrically nonlinear beam problem is rigorously split into a two-dimensional cross-sectional analysis and a one-dimensional global beam analysis, taking advantage of the geometric small parameter that is an inherent property of the structure. The thermal problem of composite beams is studied first. According to the quasisteady theory of thermoelasticity, two beam models are proposed: one for heat conduction analysis and the other for thermoelastic analysis. For heat conduction analysis, two different types of thermal loads are modeled: with and without prescribed temperatures over the crosssections. Then a thermoelastic beam model is constructed under the previously solved thermal field. This model is also extended for composite materials, which removed the restriction on temperature variations and added the dependence of material properties with respect to temperature based on Kovalenoko’s small-strain thermoelasticity theory. Next the VAM is applied to model the multiphysics behavior of beam structure. A multiphysics beam model is proposed to capture the piezoelectric, piezomagnetic, pyroelectric, pyromagnetic, and hygrothermal effects. For the zeroth-order approximation, the classical models are in the form of Euler-Bernoulli beam theory. In the refined theory, generalized Timoshenko models have been developed, including two transverse shear strain measures. In order to avoid ill-conditioned matrices, a scaling method for multiphysics modeling is also presented. Three-dimensional field quantities are recovered from the one-dimensional variables obtained from the global beam analysis. A number of numerical examples of different beams are given to demonstrate the application and accuracy of the present theory. Excellent agreements between the results obtained by the current models and those obtained by three-dimensional finite element analysis, analytical solutions, and those available in the literature can be observed for all the cross-sectional variables. The present beam theory has been implemented into the computer program VABS (Variational Asymptotic Beam Sectional Analysis).

The issue of accurately determining the effective properties of composite materials has received the attention of numerous researchers in the last few decades and continues to be in the forefront of material research. Micromechanics models have been proven to be very useful tools for design and analysis of composite materials. In the present work, a versatile micromechanics modeling framework, namely, the Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH), has been invented and various micromechancis models have been constructed in light of this novel framework. Considering the periodicity as a small parameter, we can formulate the variational statements of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. Finally, we employed the finite element method to solve the numerical solution of the constrained minimization problem. If the local fields within the unit cell are of interest, the proposed models can also accurately recover those fields based on the global behavior. In comparison to other existing models, the advantages of VAMUCH are: (1) it invokes only two essential assumptions within the concept of micromechanics for heterogeneous material with identifiable unit cells; (2) it has an inherent variational nature and its numerical implementation is shown to be straightforward; (3) it calculates the different material properties in different directions simultaneously, which is more efficient than those approaches requiring multiple runs under different loading conditions; and (4) it calculates the effective properties and the local fields directly with the same accuracy as the fluctuation functions. No postprocessing calculations such as stress averaging and strain averaging are needed. The present theory is implemented in the computer program VAMUCH, a versatile engineering code for the homogenization of heterogeneous materials. This new micromechanics modeling approach has been successfully applied to predict the effective properties of composite materials including elastic properties, coefficients of thermal expansion, and specific heat and the effective properties of piezoelectric and electro-magneto-elastic composites. This approach has also been extended to the prediction of the nonlinear response of multiphase composites. Numerous examples have been utilized to clearly demonstrate its application and accuracy as a general-purpose micromechanical analysis tool.

Rotor (helicopter/wind turbine) blades are typically slender structures that can be modeled as beams. Beam modeling, however, involves a substantial mathematical formulation that ultimately helps save computational costs. A beam theory for rotor blades must account for (i) initial twist and/or curvature, (ii) inclusion of composite materials, (iii) large displacements and rotations; and be capable of offering significant computational savings compared to a non-linear 3D FEA (Finite Element Analysis). The mathematical foundation of the current effort is the Variational Asymptotic Method (VAM), which is used to rigorously reduce the 3D problem into a 1D or beam problem, i.e., perform a cross-sectional analysis, without any ad hoc assumptions regarding the deformation. Since its inception, the VAM based cross-sectional analysis problem has been in a constant state of flux to expand its horizons and increase its potency; and this is precisely the target at which the objectives of this work are aimed. The problems addressed are the stress-strain-displacement recovery for spanwise non-uniform beams, analytical verification studies for the initial curvature effect, higher fidelity stress-strain-displacement recovery, oblique cross-sectional analysis, modeling of thin-walled beams considering the interaction of small parameters and the analysis of plates of variable thickness. The following are the chief conclusions that can be drawn from this work: 1. In accurately determining the stress, strain and displacement of a spanwise non-uniform beam, an analysis which accounts for the tilting of the normal and the subsequent modification of the stress-traction boundary conditions is required. 2. Asymptotic expansion of the metric tensor of the undeformed state and its powers are needed to capture the stiffnesses of curved beams in tune with elasticity theory. Further improvements in the stiffness matrix can be achieved by a partial transformation to the Generalized Timoshenko theory. 3. For the planar deformation of curved laminated strip-beams, closed-form analytical expressions can be generated for the stiffness matrix and recovery; further certain beam stiffnesses can be extracted not only by a direct 3D to 1D dimensional reduction, but a sequential dimensional reduction, the intermediate being a plate theory. 4. Evaluation of the second-order warping allows for a higher fidelity extraction of stress, strain and displacement with negligible additional computational costs. 5. The definition of a cross section has been expanded to include surfaces which need not be perpendicular to the reference line. 6. Analysis of thin-walled rotor blade segments using asymptotic methods should consider a small parameter associated with the wall thickness; further the analysis procedure can be initiated from a laminated shell theory instead of 3D. 7. Structural analysis of plates of variable thickness involves an 8×8 plate stiffness matrix and 3D recovery which explicitly depend on the parameters describing the thickness, in contrast to the simplistic and erroneous approach of replacing the thickness by its variation.

Le verre feuilleté est constitué de deux ou plusieurs feuilles de verre assemblées par une ou plusieurs couches de polymères viscoélastiques intermédiaires qui sont à l’origine des interactions entre les mécanismes de cisaillement, de flexion locale et de flexion globale qui régissent le comportement d’ensemble du verre feuilleté. Dans ces travaux, nous proposons un modèle de plaques dites "fortement contrastées" intégrant ces différents mécanismes et leur couplage. Dès lors qu’il existe une séparation d’échelle entre l’épaisseur de la plaque et la taille caractéristique des variations des champs cinématiques, la méthode des développements asymptotiques est applicable pour l’étude des stratifiées. Le comportement effectif de la plaque est dérivé de la loi de comportement 3D des matériaux, associée à une formulation de développements asymptotiques et à une mise à l’échelle appropriée du contraste de rigidité entre les couches raides de verre et des couches souples de polymère viscoélastique. Le procédé fournit une formulation explicite et cohérente, intégrant les efforts duaux des descripteurs macroscopiques (équations constitutives), les équations d’équilibres hors-plan et dans le plan et les différentes lois de comportements du stratifié. Le modèle est d’abord construit dans le cas d’une plaque de verre feuilleté à 2 couches raides puis différents cas de chargements statiques et dynamiques sont appliqués. Il est ensuite étendu aux plaques à 3 couches raides puis généralisé aux plaques à N couches raides. Des campagnes expérimentales sont mises en place pour valider les modèles obtenus. Enfin un cas d’étude de dimensionnement de plancher en verre feuilleté est proposé. / Widely used in recent years, the glass material makes it possible to erect light, aesthetic and resistant structures, in particular with glass lamination method, which consists in assembling two or more sheets of glass by one or more intermediate viscoelastic polymers layer. These viscoelastic interface layers are the source of interactions between the shear, the local bending and the global bending mechanisms that govern the overall behavior of laminated glass. For an optimal dimensioning and an accurate determination of the different fields, we propose a "highlycontrasted" plates model encapsulating these different mechanisms and their coupling. As long as a scale separation between the plate thickness and the caracteristic size of kinematics field variations exists, the asymptotic expansion method can be applied for the study of laminates. The effective plate behaviour is derived from the 3D constitutive law of the materials combined with an asymptotic expansion formulation and an appropriate scaling the tiffness contrast between stiff glass layers and soft viscoelastic polymer layers. The method provides a synthetic and consistent formulation, integrating the dual efforts of the macroscopic descriptors, in-plane and out-of-plane balance equations and the constitutive laws of the laminate. First the model is developped for a 2 stiff layers laminated glass plates, then different static and dynamic loading cases are applied. It is then extended to plates with 3 stiff layers and then generalized to plates with N stiff layers. Experimental campaigns are set up in order to validate the models. Finally, a case study of laminated glass floor design is proposed.