L'objectif de ce travail fut le développement de nouvelles méthodes pour aborder certainsproblèmes inverses en élasticité, en tirant parti de la présence d'un petit paramètre dans ces problèmespour construire des approximation asymptotiques d'ordre élevé.La première partie est consacrée à l'identification de la taille et la position d'une inhomogénéité$BTrue$ enfouie dans un domaine élastique tridimensionnel. Nous nous concentrons sur l'étude defonctions-co^uts $Jbb(Br)$ quantifiant l'écart entre $BTrue$ et une hétérogénéité ``test'' $Br$. Unetelle fonction-co^ut peut en effet être minimisée par rapport à tout ou partie des caractéristiques del'inclusion ``test'' $Br$ (position, taille, propriétés mécaniques ...) pour établir la meilleurecorrespondance possible entre $Br$ et $BTrue$. A cet effet, nous produisons un développement asymptotique de $Jbb$en la taille $incsize$ de $Br$, qui en constitue une approximation polynomiale plus aisée à minimiser. Cedéveloppement, établi jusqu'à l'ordre $O(incsize^6)$, est justifié par une estimation du résidu. Uneméthode d'identification adaptée est ensuite présentée et illustrée par des exemples numériques portant surdes obstacles de formes simples dans l'espace libre $Rbb^3$.L'objet de la seconde partie est de caractériser une inclusion microstructurée de longueur $ltot$, modéliséeen une dimension, composée de couches de deux matériaux alternés périodiquement, en supposant que les plusbasses de ses fréquences propres de transmission (TEs) sont connues. Ces fréquences sont les valeurs propres d'unproblème dit de transmission intérieur (ITP). Afin de disposer d'un modèle propiceà l'inversion, tout en prenant en compte les effets de la microstructure, nous nous reposons sur des approximationsde l'ITP exact obtenues par homogénéisation. A partir du modèle homogénéisé d'ordre 0, nous établissonstout d'abord une méthode simple pour déterminer les paramètres macroscopiques ($ltot$ et contrastes matériaux)d'une telle inclusion. Pour avoir accès à la période de la microstructure, nous nous intéressons ensuite àdes modèles homogénéisés d'ordre élevé, pour lesquels nous soulignons le besoin de conditions aux limitesadaptées. / The purpose of this work was to develop new methods to address inverse problems in elasticity,taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptoticexpansions.The first part is dedicated to the localization and size identification of a buried inhomogeneity $BTrue$ in a 3Delastic domain. In this goal, we focused on the study of functionals $Jbb(Br)$ quantifying the misfit between $BTrue$and a trial homogeneity $Br$. Such functionals are to be minimized w.r.t. some or all the characteristics of the trialinclusion $Br$ (location, size, mechanical properties ...) to find the best agreement with $BTrue$. To this end, weproduced an expansion of $Jbb$ with respect to the size $incsize$ of $Br$, providing a polynomial approximationeasier to minimize. This expansion, established up to $O(incsize^6)$ in a volume integral equations framework, isjustified by an estimate of the residual. A suited identification procedure is then given and supported by numericalillustrations for simple obstacles in full-space $Rbb^3$.The main purpose of this second part is to characterize a microstructured two-phases layered1D inclusion of length $ltot$, supposing we already know its low-frequency transmission eigenvalues (TEs). Thoseare computed as the eigenvalues of the so-called interior transmission problem (ITP). To provide a convenient invertiblemodel, while accounting for the microstructure effects, we then relied on homogenized approximations of the exact ITPfor the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method torecover the macroscopic parameters ($ltot$ and material contrast) of such inclusion. To access to the period of themicrostructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundaryconditions.

Les matériaux composites sont largement utilisés dans l'industrie automobile comme pièces de structure. Alliant légèreté et bonnes propriétés mécaniques, ils ont remplacé les métaux classiquement adoptés dans la fabrication de moyens de transport. Toutefois, l'emploi des matériaux composites doit tenir compte de leur comportement électromagnétique. En effet, les composites à fibres conductrices, généralement moins conducteurs que les métaux, engendrent une interaction avec les ondes électromagnétiques, différente de celle introduite par les alliages métalliques. Il s'avère donc important de développer des outils de modélisation permettant de mieux appréhender le comportement électromagnétique de matériaux composites, et d'éclairer les changements qu'apportent ces matériaux sur la distribution des champs, provenant d'une multitude de sources externes, au voisinage des systèmes mécatroniques. Par ailleurs, l'étude du comportement électromagnétique de matériaux composites permet de tirer parti de leurs propriétés mécaniques attractives afin d'alléger les boîtiers de blindage en gardant un niveau d'atténuation conforme aux normes de l'industrie. Cependant, la modélisation numérique de structures composites de grande taille, telles que les boîtiers de blindage, bien que classiquement adoptée pour les structures métalliques, est rendue complexe par le fait que les composites présentent des hétérogénéités à l'échelle microscopique, et que leurs mécanismes de blindage diffèrent de ceux des conducteurs homogènes. Le calcul numérique s'avère envisageable une fois le composite remplacé par un matériau homogène ayant une réponse identique face à une sollicitation électromagnétique. Ainsi, au travers de ce travail de thèse, nous proposons une technique d'homogénéisation permettant d'estimer les propriétés électriques équivalentes que nous appliquons aux composites à fibres conductrices unidirectionnelles et tissées. Les résultats obtenus sont utilisés pour la simulation numérique d'un boîtier de blindage. / Composite materials are widely used in the automotive industry as structural components. By combining lightness and robust mechanical properties, they are increasigly replacing the conventionnally used metallic alloys, for the manufacturing of vehicle parts. However, the use of composite materials is not without consequences on the electromagnetic behavior of these parts. Since carbon fiber reinforced composites are generally worse conductors of electricity than metals, they interact differently with the electromagnetic waves which surround them. It is therefore important to develop modeling tools to better understand the electromagnetic behavior of composite materials. This is to explain the changes that these materials bring to the distribution of waves, generated by a multitude of external sources, in the vicinity of mechatronic systems. On the other hand, the study of the electromagnetic behavior of composite materials makes it possible to determine the possibility of taking advantage of their attractive mechanical properties in order to further reduce the weight of electromagnetic shielding enclosures while maintaining a level of attenuation in accordance with the standards of the industry. However, numerical modeling of large composite structures, such as shielding enclosures, although conventionally adopted for metal structures, is hindered by the fact that composites exhibit heterogeneities at the microscopic scale. The numerical calculation becomes possible once the composite is replaced by a homogeneous material that exhibits an identical response to an identical electromagnetic solicitation. In this work, we present a homogenization technique, based on finite element simulation and an optimisation method, that computes an estimate of the equivalent electrical properties of unidirectional and woven fiber reinforced composites. The results are then used to simulate the shielding effectiveness of an enclosure constructed by combining composite materials and metallic alloys.

La prévision du comportement mécanique macroscopique de matériaux hétérogènes à partir des propriétés de leurs constituants est possible pour diverses classes de comportement (élastique, viscoélastique, etc) et ce, grâce à la théorie de l'homogénéisation. Néanmoins l'extension de cette théorie pour des matériaux possédant un comportement viscoélastique non linéaire (ou élasto-viscoplastique) reste une question ouverte à laquelle nous nous attaquons dans ce travail afin de prédire le comportement macroscopique des combustibles oxydes mixtes uranium-plutonium (MOX) utilisés dans les réacteurs nucléaires à eau sous pression (REP) français. Dans cette optique des solutions analytiques et purement nunériques ont été obtenues et le modèle retenu est utilisé pour simuler le comportement des combustibles / The prediction of the macroscopic mechanical behavior of heterogeneous materials from the properties of their constituents is possible for various classes of behavior (elastic, viscoelastic, etc.) thanks to the theory of homogenization. Nevertheless, the extension of this theory for materials with a non-linear (or elasto-viscoplastic) viscoelastic behavior remains an open question that we are tackling in this work in order to predict the macroscopic behavior of uranium-plutonium (MOX) mixed oxide fuels used in french pressurized water reactors (PWRs). From this perspective analytical and purely numerical solutions have been obtained and the model adopted is used to simulate the behavior of fuels.

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This study focuses on the numerical solution of a fourth-order cell problem obtained through a two- scale expansion approach applied to a higher gradient heat equation microscopic problem involving temperature distributions. The main objective is to investigate the temperature field within the macroscale domain and compute the effective conductivity using finite element methods. The research utilizes numerical techniques, specifically finite element methods, to solve the fourth-order cell problem and obtain the temperature distribution.

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The presence of prestrain can have a tremendous effect on the mechanical behavior of slender structures. Prestrained elastic plates show spontaneous bending in equilibrium—a property that makes such objects relevant for the fabrication of active and functionalmaterials. In this paperwe studymicroheterogeneous, prestrained plates that feature non-flat equilibriumshapes. Our goal is to understand the relation between the properties of the prestrained microstructure and the global shape of the plate in mechanical equilibrium. To this end, we consider a three-dimensional, nonlinear elasticity model that describes a periodic material that occupies a domain with small thickness. We consider a spatially periodic prestrain described in the form of a multiplicative decomposition of the deformation gradient.By simultaneous homogenization and dimension reduction, we rigorously derive an effective plate model as a Γ-limit for vanishing thickness and period. That limit has the form of a nonlinear bending energy with an emergent spontaneous curvature term. The homogenized properties of the bending model (bending stiffness and spontaneous curvature) are characterized by corrector problems. For a model composite—a prestrained laminate composed of isotropic materials—we investigate the dependence of the homogenized properties on the parameters of the model composite. Secondly, we investigate the relation between the parameters of the model composite and the set of shapes with minimal bending energy. Our study reveals a rather complex dependence of these shapes on the composite parameters. For instance, the curvature and principal directions of these shapes depend on the parameters in a nonlinear and discontinuous way; for certain parameter regions we observe uniqueness and non-uniqueness of the shapes. We also observe size effects: The geometries of the shapes depend on the aspect ratio between the plate thickness and the composite period. As a second application of our theory, we study a problem of shape programming: We prove that any target shape (parametrized by a bending deformation) can be obtained (up to a small tolerance) as an energy minimizer of a composite plate, which is simple in the sense that the plate consists of only finitely many grains that are filled with a parametrized composite with a single degree of freedom.

<p dir="ltr">Multiscale topology optimization is a numerical method that enables the synthesis of hierarchical structures, offering greater design flexibility than single-scale topology optimization. However, this increased flexibility also incurs higher computational costs. Recent advancements have integrated machine learning models into MSTO methods to address this issue. Unfortunately, existing machine learning-based multiscale topology optimization (ML-MSTO) approaches underutilize the potential of machine learning models to surrogate the inner optimization, analysis, and numerical homogenization of arbitrary non-periodic microstructures. This dissertation presents an ML-MSTO method featuring displacement-driven topology-optimized microstructures (TOMs). The proposed method solves an outer optimization problem to design a homogenized macroscale structure and multiple inner optimization problems to obtain spatially distributed, non-periodic TOMs. The inner problem formulation employs the macroscale element densities and nodal displacements to define constraints and boundary conditions for microscale density-based topology optimization problems. Each problem yields a free-form TOM. To reduce computational costs, artificial neural networks (ANNs) are trained to predict their homogenized constitutive tensor. The ANNs also enable sensitivity coefficients to be approximated through a variety of standard derivative methods. The effect of the neural network-based derivative methods on topology optimization results is evaluated in a comparative study. An explicit dehomogenization approach is proposed, leveraging the TOMs of the ML-MSTO method. The explicit approach also features two post-processing schemes to improve the connectivity and clean the final multiscale structure. A 2D and a 3D case study are designed with the ML-MSTO method and dehomogenized with the explicit approach. The resulting multiscale structures are non-periodic with free-form microstructures. In addition, a second implicit dehomogenization approach is developed in this dissertation that allows the projection of homogenized mechanical property fields onto a discrete lattice structure of arbitrary shape. The implicit approach is capable of dehomogenizing any homogenized design. This is done by incorporating an optimization algorithm to find the lattice thickness distribution that minimizes the difference between a local target homogenized property and a corresponding lattice homogenized stiffness tensor. The result is a well-connected, functionally graded lattice structure, that enables control over the length scale, orientation, and complexity of the final microstructured design.</p>