This thesis centers on the development and analysis of numerical multiscale methods for multiscale problems arising in steady heat conduction, heat transfer and wave propagation in heterogeneous media. In a multiscale problem several scales interact with each other to form a system which has variations over a wide range of scales. A direct numerical simulation of such problems requires resolving the small scales over a computational domain, typically much larger than the microscopic scales. This demands a tremendous computational cost. We develop and analyse multiscale methods based on the heterogeneous multiscale methods (HMM) framework, which captures the macroscopic variations in the solution at a cost much lower than traditional numerical recipes. HMM assumes that there is a macro and a micro model which describes the problem. The micro model is accurate but computationally expensive to solve. The macro model is inexpensive but incomplete as it lacks certain parameter values. These are upscaled by solving the micro model locally in small parts of the domain. The accuracy of the method is then linked to how accurately this upscaling procedure captures the right macroscopic effects. In this thesis we analyse the upscaling error of existing multiscale methods and also propose a micro model which significantly reduces the upscaling error invarious settings. In papers I and IV we give an analysis of a finite difference HMM (FD-HMM) for approximating the effective solutions of multiscale wave equations over long time scales. In particular, we consider time scales T^ε = O(ε−k ), k =1, 2, where ε represents the size of the microstructures in the medium. In this setting, waves exhibit non-trivial behaviour which do not appear over short time scales. We use new analytical tools to prove that the FD-HMM accurately captures the long time effects. We first, in Paper I, consider T^ε =O(ε−2 ) and analyze the accuracy of FD-HMM in a one-dimensional periodicsetting. The core analytical ideas are quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic wave equations.The analysis naturally reveals the role of consistency in HMM for high order approximation of effective quantities over long time scales. Next, in paperIV, we consider T^ε = O(ε−1 ) and use the tools in a multi-dimensional settingto analyze the accuracy of the FD-HMM in locally-periodic media where fast and slow variations are allowed at the same time. Moreover, in papers II and III we propose new multiscale methods which substantially improve the upscaling error in multiscale elliptic, parabolic and hyperbolic partial differential equations. In paper II we first propose a FD-HMM for solving elliptic homogenization problems. The strategy is to use the wave equation as the micro model even if the macro problem is of elliptic type. Next in paper III, we use this idea in a finite element HMM setting and generalize the approach to parabolic and hyperbolic problems. In a spatially fully discrete a priori error analysis we prove that the upscaling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media. / <p>QC 20150216</p> / Multiscale methods for wave propagation

Ce travail porte sur la modélisation micromécanique des élastomères chargés. On cherche principalement à d'une part identifier l'influence des propriétés des différentes phases (morphologie et comportement) sur la réponse macroscopique, et d'autre part explorer les mécanismes d'interactions qui peuvent avoir lieu au sein de la micro-structure. Pour ce faire, on a mené une étude à deux échelles d'observations et ce à l'aide de simulations numériques basées sur l'homogénéisation. Le premier niveau correspond à une échelle mésoscopique pour laquelle on considère un Volume Élémentaire Représentatif (VER) bi-phasique, constitué d'un agglomérat de charge dissipatif, noyé dans une matrice hyperélastique. Le second niveau consiste, à une plus petite échelle, à explorer le comportement d'un agglomérat idéalisé, constitué de particules de charges infiniment rigides liées entre elles par une mince couche de gomme. Cette micro-structure est générée de manière aléatoire par un tirage de polygones de Voronoï. Des calculs éléments finis sont réalisés en élasticité linéaire et non-linéaire dans un contexte d'homogénéisation numérique en utilisant diverses techniques de localisation. Les différentes analyses menées montrent notamment que l'hypothèse d'affinité n'est pas adaptée à ce type de micro-structures et que le caractère incompressible de la gomme ainsi que son confinement jouent un rôle prépondérant sur le comportement mécanique de l'agglomérat. / This work focuses on the micro mechanical modeling of filled elastomers. The major question to be identified: firstly the influence of the properties of different phases (morphology and behavior) on the macroscopic response, and also to explore the mechanisms of interactions that take place within the micro-structure. To do this, we conducted a study at two scales of observations and using the numerical simulations based on homogenization. The first level corresponds to a mesoscopic scale for which we consider a representative elementary volume (REV), biphasic, consisting of a homogeneous dissipative inclusion (agglomerate) embedded in a hyperelastic matrix. The second level is at a smaller scale, to explore the behavior of an idealized agglomerate, consisting of infinitely rigid filler particles bounded together by a thin layer of rubber. This micro-structure is randomly generated by a random Voronoï polygons. Finite element calculations are performed in linear elasticity and nonlinear in the context of numerical homogenization using various localization techniques. The results show in particular that the assumption of affinity is not suitable for this type of micro-structures and the incompressibility of the rubber and its containment play an important role on the mechanical behavior of the agglomerate.

Dans cette thèse on utilise une méthode de Galerkin discontinue (GD) pour modéliser la propagation des ondes dans un matériau endommagé. Deux modèles différents pour la description de l’endommagement ont été considérés. Dans la première partie de la thèse on utilise un modèle d’endommagent assez général, basé sur une modélisation micromécanique. Pour ce modèle on établit un critère de stabilité basé sur une densité critique de fissuration. On développe aussi une méthode numérique GD capable de capturer les instabilités au niveau microscopique. On construit une solution exacte pour analyser la précision de la méthode proposée.Plusieurs résultats numériques vont permettre d’analyser la propagation des ondes dans les configurations planes et anti-planes. Dans la deuxième parte de la thèse on étudie la propagation des ondes dans un milieux fissuré (microfissures en contact avec frottement). La méthode numérique développée utilise une technique GD et la méthode du Lagrangien augmenté. En utilisant cette méthode on a pu calculer numériquement la vitesse de propagation moyenne dans un matériau endommagé. On a pu comparer les résultats obtenus avec les formules analytiques obtenues avec des approches micromécaniques. Finalement, on a utilisé les calculs numériques pour étudier la propagation des ondes après un impact sur une plaque céramique pour les deux modèles mécaniques considérés. / A discontinuous Galerkin (DG) technique for modeling wave propagation in damaged (brittle) materials is developed in this thesis. Two different types of mechanical models for describing the damaged materials are considered. In the first part of the thesis general micro-mechanics based damage models were used. A critical crack density parameter, which distinguishes between stable and unstable behaviors, wascomputed. A new DG-numerical scheme able to capture the instabilities and a micro-scale time step were proposed. An exact solution is constructed and the accuracy of the numerical scheme was analyzed. The wave propagation in one dimensional and anti-plane configuration was analyzed through several numerical computations. In the second part of the thesis the wave propagation in cracked materials with a nonlinear micro-structure (micro-cracks in frictional contact) was investigated. The numerical scheme developed makes use of a DG-method and an augmented Lagrangian technique. The effective wave velocity in a damaged material, obtained by a numerical upscaling homogenization method, was compared with analytical formula of effective elasticity theory. The wave propagation (speed, amplitude and pulse length) in micro-cracked materials in complex configurations was studied. Finally, numerical computations of blast wave propagation,for the both models, illustrate the role played by the micro-cracks orientation and by the friction.

In this dissertation we develop and analyze numerical method to solve general elliptic boundary value problems with many scales. The numerical method presented is intended to capture the small scales effect on the large scale solution without resolving the small scale details, which is done through the construction of a multiscale map. The multiscale method is more effective when the coarse element size is larger than the small scale length. To guarantee a numerical conservation, a finite volume element method is used to construct the global problem. Analysis of the multiscale method is separately done for cases of linear and nonlinear coefficients. For linear coefficients, the multiscale finite volume element method is viewed as a perturbation of multiscale finite element method. The analysis uses substantially the existing finite element results and techniques. The multiscale method for nonlinear coefficients will be analyzed in the finite element sense. A class of correctors corresponding to the multiscale method will be discussed. In turn, the analysis will rely on approximation properties of this correctors. Several numerical experiments verifying the theoretical results will be given. Finally we will present several applications of the multiscale method in the flow in porous media. Problems that we will consider are multiphase immiscible flow, multicomponent miscible flow, and soil infiltration in saturated/unsaturated flow.

In this dissertation we develop and analyze numerical method to solve general elliptic boundary value problems with many scales. The numerical method presented is intended to capture the small scales e&#64256;ect on the large scale solution without resolving the small scale details, which is done through the construction of a multiscale map. The multiscale method is more e&#64256;ective when the coarse element size is larger than the small scale length. To guarantee a numerical conservation, a &#64257;nite volume element method is used to construct the global problem. Analysis of the multiscale method is separately done for cases of linear and nonlinear coe&#64259;cients. For linear coe&#64259;cients, the multiscale &#64257;nite volume element method is viewed as a perturbation of multiscale &#64257;nite element method. The analysis uses substantially the existing &#64257;nite element results and techniques. The multiscale method for nonlinear coe&#64259;cients will be analyzed in the &#64257;nite element sense. A class of correctors corresponding to the multiscale method will be discussed. In turn, the analysis will rely on approximation properties of this correctors. Several numerical experiments verifying the theoretical results will be given. Finally we will present several applications of the multiscale method in the &#64258;ow in porous media. Problems that we will consider are multiphase immiscible &#64258;ow, multicomponent miscible &#64258;ow, and soil in&#64257;ltration in saturated/unsaturated &#64258;ow.

Diese Arbeit liefert ein dreidimensionales numerisches Homogenisierungskonzept, welches beim Übergang von der Mikro- zur Makroskala einen Wechsel in der Kontinuumsbeschreibung beinhaltet. Während für die Beschreibung der Makroskala das verallgemeinerte Cosserat-Kontinuum verwendet wird, basiert die Mikroskala auf der klassischen Cauchy-Theorie. Um das homogene Cosserat-Ersatzmaterial im Rahmen numerischer Simulationen nutzen zu können, erfolgt die Implementierung geeigneter Finiter Elemente in das Programmsystem Abaqus und deren Verifikation. Neben der Diskussion der bei der Homogenisierung beobachteten Effekte werden anhand eines idealisierten Modells eines biaxialverstärkten Mehrlagengestrickes die Vorteile gegenüber der klassischen Herangehensweise aufgezeigt. / This contribution provides a threedimensional homogenization approach which includes the switch of the continuum theory during the scale transition. Whereas the microscopic scale is described in the framework of the classical Cauchy theory, the macroscopic scale is based on the generalized Cosserat continuum. In order to use the obtained homogeneous Cosserat material, suitable finite elements are implemented in the commercial program system Abaqus followed by an appropriate verification. Beside the discussion of the arising effects the advantages of this approach compared to the classical procedure are shown by means of an idealized model of a biaxial woven fabric.

Diese Arbeit liefert ein dreidimensionales numerisches Homogenisierungskonzept, welches beim Übergang von der Mikro- zur Makroskala einen Wechsel in der Kontinuumsbeschreibung beinhaltet. Während für die Beschreibung der Makroskala das verallgemeinerte Cosserat-Kontinuum verwendet wird, basiert die Mikroskala auf der klassischen Cauchy-Theorie. Um das homogene Cosserat-Ersatzmaterial im Rahmen numerischer Simulationen nutzen zu können, erfolgt die Implementierung geeigneter Finiter Elemente in das Programmsystem Abaqus und deren Verifikation. Neben der Diskussion der bei der Homogenisierung beobachteten Effekte werden anhand eines idealisierten Modells eines biaxialverstärkten Mehrlagengestrickes die Vorteile gegenüber der klassischen Herangehensweise aufgezeigt. / This contribution provides a threedimensional homogenization approach which includes the switch of the continuum theory during the scale transition. Whereas the microscopic scale is described in the framework of the classical Cauchy theory, the macroscopic scale is based on the generalized Cosserat continuum. In order to use the obtained homogeneous Cosserat material, suitable finite elements are implemented in the commercial program system Abaqus followed by an appropriate verification. Beside the discussion of the arising effects the advantages of this approach compared to the classical procedure are shown by means of an idealized model of a biaxial woven fabric.

In a first part, we study numerical schemes using the finite-element method to discretize the Oldroyd-B system of equations, modelling a viscoelastic fluid under no flow boundary condition in a 2- or 3- dimensional bounded domain. The goal is to get schemes which are stable in the sense that they dissipate a free-energy, mimicking that way thermodynamical properties of dissipation similar to those actually identified for smooth solutions of the continuous model. This study adds to numerous previous ones about the instabilities observed in the numerical simulations of viscoelastic fluids (in particular those known as High Weissenberg Number Problems). To our knowledge, this is the first study that rigorously considers the numerical stability in the sense of an energy dissipation for Galerkin discretizations. In a second part, we adapt and use ideas of a numerical method initially developped in the works of Y. Maday, A.T. Patera et al., the reduced-basis method, in order to efficiently simulate some multiscale models. The principle is to numerically approximate each element of a parametrized family of complicate objects in a Hilbert space through the closest linear combination within the best linear subspace spanned by a few elementswell chosen inside the same parametrized family. We apply this principle to numerical problems linked : to the numerical homogenization of second-order elliptic equations, with two-scale oscillating diffusion coefficients, then ; to the propagation of uncertainty (computations of the mean and the variance) in an elliptic problem with stochastic coefficients (a bounded stochastic field in a boundary condition of third type), last ; to the Monte-Carlo computation of the expectations of numerous parametrized random variables, in particular functionals of parametrized Itô stochastic processes close to what is encountered in micro-macro models of polymeric fluids, with a control variate to reduce its variance. In each application, the goal of the reduced-basis approach is to speed up the computations without any loss of precision

La thèse s’intéresse au comportement thermomécanique des circuits imprimés pour des applications spatiales hyperfréquences. Dans cette étude, les circuits imprimés sont des assemblages multi-matériaux faisant intervenir des substrats diélectriques (composites tissés) et des connexions en cuivre. Les circuits étudiés sont des multicouches et l’information électrique transite d'une couche à l'autre par le biais de trous traversants : des perçages réalisés à travers les différentes couches, recouverts de cuivre par électrodéposition. Tout satellite comporte de l’électronique embarquée dont le circuit imprimé constitue le support et les connexions. Dans le cadre des applications spatiales, le circuit imprimé subira des variations importantes de température. Ces chargements engendrent des déformations qui ne sont pas homogènes dans les différents matériaux, pouvant mener à des contraintes importantes qui seront source de défaillances. En effet, les coefficients d'expansion thermique des substrats diélectriques et du cuivre sont différents. À chaque cycle thermique, le cuivre est alors entrainé sous chargement alterné. Suivant les configurations, le cuivre peut se plastifier et rompre après quelques centaines ou milliers de cycles thermiques (fatigue oligo-cyclique). On remarque que les ruptures sont souvent observées dans les trous traversant. Deux volets sont identifiables dans la thèse : un premier volet de caractérisation du comportement thermomécanique des matériaux présents dans les circuits imprimés hyperfréquences (substrats composites et cuivre), et un second volet concernant les simulations de configurations stratégiques à partir des comportements identifiés / In this thesis, the thermomechanical behavior of Printed Circuit Boards with high frequency space applications is assessed. A printed circuit board is a multi-material assembly, linking dielectric substrates and copper paths. The studied PCBs are multilayers, thus drills are made through these layers with copper electrodeposited on the wall of the hole, allowing the electrical signal to go from one layer to the other. Any satellite carries embedded electronics and the PCB is the link and the support of these electronics. During the life of the PCB in space applications, important temperature changes will drive strains which are inhomogeneous in the different materials and thus will lead to important stresses, root of the observed failures. Indeed, the coefficients of thermal expansion of the dielectric substrates are different than the one of copper. For each thermal cycle, the copper undergoes thus an alternate loading. Depending on the configuration, the copper may endure plastic strain and break after hundreds or a few thousands of cycles (oligo-cyclic fatigue). These failures happen often in the copper barrels linking the different layers.Two phases are distinguishable in the thesis: a first phase in which the thermomechanical behaviors of the materials constituting high frequency printed circuit boards is assessed (composites substrates and copper), and a second phase concerning the simulations of crucial configurations thanks to the identified behaviors of the materials

We address two computational challenges for numerical simulations of Darcy flow problems: rough and uncertain data. The rapidly varying and possibly high contrast permeability coefficient for the pressure equation in Darcy flow problems generally leads to irregular solutions, which in turn make standard solution techniques perform poorly. We study methods for numerical homogenization based on localized computations. Regarding the challenge of uncertain data, we consider the problem of forward propagation of uncertainty through a numerical model. More specifically, we consider methods for estimating the failure probability, or a point estimate of the cumulative distribution function (cdf) of a scalar output from the model. The issue of rough coefficients is discussed in Papers I–III by analyzing three aspects of the localized orthogonal decomposition (LOD) method. In Paper I, we define an interpolation operator that makes the localization error independent of the contrast of the coefficient. The conditions for its applicability are studied. In Paper II, we consider time-dependent coefficients and derive computable error indicators that are used to adaptively update the multiscale space. In Paper III, we derive a priori error bounds for the LOD method based on the Raviart–Thomas finite element. The topic of uncertain data is discussed in Papers IV–VI. The main contribution is the selective refinement algorithm, proposed in Paper IV for estimating quantiles, and further developed in Paper V for point evaluation of the cdf. Selective refinement makes use of a hierarchy of numerical approximations of the model and exploits computable error bounds for the random model output to reduce the cost complexity. It is applied in combination with Monte Carlo and multilevel Monte Carlo methods to reduce the overall cost. In Paper VI we quantify the gains from applying selective refinement to a two-phase Darcy flow problem.