Orientador: Luciano Barbanti / Coorientador: Geraldo Nunes Silva / Banca: German Jesus Lozada Cruz / Banca: Márcia Cristina Anderson Braz Federson / Resumo: Neste trabalho estabelecemos condições para a existência e unicidade de solução para equações integrais do tipo Volterra-Stieltjes não linear x(t)+ Z [a,t]T DsK(t,s) f (s,x(s)) = u(t), t E [a,b]T em escalas temporais T, usando a integral de Cauchy-Stieltjes à direita sobre funções regradas a valores em espaços de Banach / Abstract: In this work we establish conditions for the existence and uniqueness of solution a Volterra- Stieltjes integral nonlinear equations x(t)+ Z [a,t]T DsK(t,s) f (s,x(s)) = u(t), t E [a,b]Tin time scales T, using the right Cauchy-Stieltjes integral on regulated functions with values in Banach spaces / Mestre

Les surfaces sélectives en fréquence (FSS) pour la furtivité radar ou l’optique ont été largement étudiées. Depuis plus de vingt ans, des matériaux artificiels ont été conçus, permettant d’obtenir des propriétés particulières, notamment l’existence de bandes permises ou interdites, réfraction négative, ultra-réfraction. Par ailleurs, des antennes basées sur la mise en réseau d’un élément rayonnant sont plus compactes et plus facilement intégrables. Le problème de la diffraction d’une onde plane par des réseaux tridimensionnels bipériodiques peut être résolu par éléments finis ou par équations intégrales bipériodiques ; il l’est souvent par une méthode hybride combinant la méthode des éléments finis et la méthode aux équations intégrales. Nous avons choisi de développer une méthode hybride utilisant deux variantes de la méthode aux équations intégrales. Les domaines semi-infinis (l’extérieur du réseau) sont traités par des équations intégrales bipériodiques (EI3D2D), et les domaines bornés (l’intérieur du réseau) sont traités par des équations intégrales tridimensionnelles (EI3D), auxquelles on impose des conditions aux limites de pseudopériodicité. Ce code numérique est développé dans le cadre du code SPECTRE de Dassault-Aviation, qui est un code généraliste 3D, afin de bénéficier de la richesse des modèles qui y ont déjà été développés (modèle composé d’un nombre quelconque de sous-domaines de formes et de matériaux quelconques, traitement des différents cas de jonctions entre sous-domaines, matériaux de faible épaisseur). L’efficacité en termes de précision et en temps de calcul de la méthode numérique est validée par comparaison des résultats avec d’autres simulations numériques et également avec des résultats de mesures. Les cas testés sont représentatifs de plusieurs des principaux phénomènes liés aux métamatériaux : surfaces sélectives en fréquence, transmission « extraordinaire », surfaces à haute impédance. Enfin, nous étudions un radôme passe-bande indépendant à l’angle d’incidence, à l’aide de la méthode numérique que nous proposons. La structure retenue se base sur un réseau de cavités coaxiales dans une couche métallique. Nous expliquons l’origine physique des résonances qui apparaissent et nous suggérons une évolution géométrique du profil des cavités, afin d’augmenter la largeur de bande passante. / Frequency selective surfaces (FSS) for radar stealth or in optics have been widely studied. For more than two decades, articial materials have been designed to highlight specific behaviour, like the existence of allowed or forbidden bands, negative refraction, ultra-refraction... Moreover, antennas based upon an array of radiating elements improve the compactness and integration of these features. The problem of the diffraction of a plane wave by 3D biperiodic scatterers can be solved by finite-elements methods (FEM) or biperiodic boundary integral equations (BIE). It is often done by hybrid methods, that combine FEM and BIE. We choose to develop a hybrid method that uses two variants of the BIE method. Semiinfinite outer domains are treated by biperiodic integral equations (3D2D IE) and inner bounded domains are treated by 3D free-space integral equations (3D IE). Pseudoperiodic boundary conditions are enforced in the scattering biperiodic structure. The numerical code is developed in the framework of Dassault Aviation’s SPECTRE code, which is a general 3D code, in order to take advantage of the various models that have already been developed : arbitrary number of sub-domains of various shapes or materials, treatment of the different types of junctions between sub-domains, thin slabs. The efficiency in terms of accuracy and computation time of the numerical code is validated by comparison of the results from other numerical simulations or measurements. All the test cases are representative of several of the main phenomena that can be observed in metamaterials : FSS, “extraordinary” transmission, high-impedance surfaces. Finally, a bandpass radome which is independent to the angle of incidence is studied. The proposed numerical method is used. The chosen structure is based upon an array of coaxial cavities in a metallic slab. We explain the physical origin of resonances that appear and we suggest a geometrical evolution of the profile of the cavities, to favor a wideband behavior.

Neste trabalho, é estudada a descrição do fluxo de um gás rarefeito em um duto cilíndrico de comprimento infinito. A formulação matemática do problema está baseada na forma integral de equações cinéticas derivadas da Equação de Boltzmann. Particularmente são estudados os modelos cinéticos conhecidos como BGK e S. Métodos espectrais são propostos para obtenção de soluções, em forma fechada, para quantidades de interesse como o perfil de velocidade do gás, bem como taxas de fluxo. As formulações espectrais são baseadas em duas abordagens: expansão clássica em termos de Polinômios de Legendre e expansão em termos de splines cúbicas de Hermite, neste caso, associada a um esquema de colocação. A implementação das propostas produz resultados computacionais satisfatórios do ponto de vista prático. Para obtenção de resultados com maior precisão, técnicas de tratamento da singularidade do núcleo da equação integral foram introduzidas, resultando em ganho computacional significativo. Finalmente, a proposta de solução espectral para problemas em geometria cilíndrica se mostrou adequada para problemas em que se admite reflexão especular na superfície do cilindro, situação onde outras abordagens clássicas disponíveis na literatura não podem ser utilizadas. / In this work, rarefied gas flows in cylindrical ducts are studied. The mathematical formulation of the problems are based on the integral form of kinetic equations derived from the Boltzmann equation. Particularly, the BGK and S models are studied. Spectral methods are proposed to obtain closed form solutions for quantities of interest as velocity profile of the gas as well as flow rates. The spectral formulations are based on two approaches: classical expansions in terms of Legendre Polynomials and Hermite cubic splines expansions. In this case, associated with a collocation scheme. The approaches provide good computational results, from the practical point of view. On the other hand, for obtaining higher accuracy, some techniques were introduced to deal with the inherent singularity of the integral kernel. In this context, a significant computational gain is achieved. Finally, this spectral approach has shown to be adequate to solve problems where specular reflection is assumed at the surface, in which cases, classical approaches available in the literature can not be used.

La reconstruction des sources de l'activité cérébrale à partir des mesures de potentiel fournies par un électroencéphalographie (EEG) nécessite de résoudre le problème connu sous le nom de « problème inverse de l'EEG ». La solution de ce problème dépend de la solution du « problème direct de l'EEG », qui fournit à partir de sources de courant connues, le potentiel mesuré au niveau des électrodes. Pour des modèles de tête réels, ce problème ne peut être résolut que de manière numérique. En particulier, les équations intégrales de surfaces requièrent uniquement la discrétisation des interfaces entre les différents compartiments constituant le milieu cérébral. Cependant, les formulations intégrales existant actuellement ne prennent pas en comptent l'anisotropie du milieu. Le travail présenté dans cette thèse introduit deux nouvelles formulations intégrales permettant de palier à cette faiblesse. Une formulation indirecte capable de prendre en compte l'anisotropie du cerveau est proposée. Elle est discrétisée à l'aide de fonctions conformes aux propriétés spectrales des opérateurs impliqués. L'effet de cette discrétisation de type mixe lors de la reconstruction des sources cérébrales est aussi étudié. La seconde formulation se concentre sur l'anisotropie due à la matière blanche. Calculer rapidement la solution du système numérique obtenu est aussi très désirable. Le travail est ainsi complémenté d'une preuve de l'applicabilité des stratégies de préconditionnement de type Calderon pour les milieux multicouches. Le théorème proposé est appliqué dans le contexte de la résolution du problème direct de l'EEG. Un préconditionneur de type Calderon est aussi introduit pour l'équation intégrale du champ électrique (EFIE) dans le cas de structures unidimensionnelles. Finalement, des résultats préliminaires sur l'impact d'un solveur rapide direct lors de la résolution rapide du problème direct de l'EEG sont présentés. / Electroencephalography (EEG) is a very useful tool for characterizing epileptic sources. Brain source imaging with EEG necessitates to solve the so-called EEG inverse problem. Its solution depends on the solution of the EEG forward problem that provides from known current sources the potential measured at the electrodes positions. For realistic head shapes, this problem can be solved with different numerical techniques. In particular surface integral equations necessitates to discretize only the interfaces between the brain compartments. However, the existing formulations do not take into account the anisotropy of the media. The work presented in this thesis introduces two new integral formulations to tackle this weakness. An indirect formulation that can handle brain anisotropies is proposed. It is discretized with basis functions conform to the mapping properties of the involved operators. The effect of this mixed discretization on brain source reconstruction is also studied. The second formulation focuses on the white matter fiber anisotropy. Obtaining the solution to the obtained numerical system rapidly is also highly desirable. The work is hence complemented with a proof of the preconditioning effect of Calderon strategies for multilayered media. The proposed theorem is applied in the context of solving the EEG forward problem. A Calderon preconditioner is also introduced for the wire electric field integral equation. Finally, preliminary results on the impact of a fast direct solver in solving the EEG forward problem are presented.

This dissertation develops and implements an effective numerical scheme, the inclusion-based boundary element method (iBEM), to investigate the mechanical and multi-physical properties of the composites containing arbitrarily shaped particles. Besides the linear elasticity and transient heat conduction problems shown in the dissertation, it can be extended to other problems, such as potential flows and Stokes flows. Through the combination of conventional boundary element method (BEM) and the Eshelby's equivalent inclusion method (EIM), the local field is obtained through superposition of the domain integral of eigen-fields and boundary integral equations. Firstly, the boundary value problems of a composite containing various fully bonding phases of subdomains is introduced. Due to the continuity of displacement (potential) and traction (flux) at the interfaces between different material phases, the interfacial continuity equations are established, which can be solved with the multi-region BEM conventionally. Thanks to Eshelby's celebrated contribution, the material difference in inhomogeneity problems is simulated by an eigenstrain on the inclusion domain but with the same material properties as the matrix. Therefore, the boundary value problems with inhomogeneities can be transformed as domain integral of Green's function with the eigenstrain over the inclusion, where can be determined by the equivalent stress conditions in EIM. Hence, the algorithm of iBEM is formulated and established on the basis of boundary conditions and equivalent stress equations instead of various continuity constraint equations, which saves efforts in computational resources and pre/post-process. The domain integral of Green's function is the key to the algorithm of iBEM, as it bridges the inhomogeneities and the boundary. The closed-form expression of domain integrals for ellipsoidal / elliptical inclusions with polynomial eigenstrain, polygonal and polyhedral inclusions with constant eigenstrain have already existed in the literature. However, it is not applicable to arbitrary particles with varying eigenstrain. This dissertation derives the closed-form domain integrals for polygon and polyhedral inclusions with polynomial eigenstrain source terms, which creates feasibility to solve the local field and effective material properties for composites with arbitrary particles. Although the EIM with polynomial-form eigenstrain has been applied to simulate the material mismatch for ellipsoidal / elliptical inhomogeneities by using the Taylor's of eigenstrain field at the particle center, when it is extended to angular particles, the inaccuracy is significantly reduced due to the rapid and complicated eigenstrain variation in the neighborhood of vertices with the strong singular effects. Therefore, the domain discretization of an angular particle is proposed to tackle the complicated distribution of elastic fields, which keeps the features of exactness (no approximation of interior field) and 𝐂⁰ continuity of eigenstrain. Hereby, the iBEM is proposed to serve as an effective and powerful tool, which takes the advantages of both BEM and EIM. The interaction of inhomogeneities is considered in the process of constructing EIM equations, and boundary effects are taken into account as the contribution to displacement of the eigen-field over inhomogeneities, hence, a complete linear equation system can be established. For the inclusion problems with a prescribed eigenstrain, no domain discretization is required because the exact elastic solution is obtained given the specific dimension of the geometry. Regarding to inhomogeneity problems, 1) the ellipsoidal / elliptical shape is versatile, which could be switched to various of shapes by adjusting the aspect ratio and orientations; 2) though the angular subdomain requires discretization, this method is rapidly convergent and no mesh is needed for the matrix. Therefore, this method enables the simulation of thousands 3𝐷 and 2𝐷 arbitrary shaped particles in a desk-top computer and the effective moduli can be obtained through virtual experiments (i.e, uni-axial loading) or periodic boundary conditions. This method can be easily extended to multi-physical problems, such as transient hear transfer, steady state heat, through changing the fundamental solutions accordingly. Three major packages have been added to the iBEM software, as transient heat transfer, closed-form 2D/3D domain integrals, and domain discretization method. Some case studies demonstrate the capability and applications of this method and software. This main contributions of the PhD studies are as follows: 1) The closed-form domain integrals for polygonal and polyhedral inhomogeneities have been derived based on the gravitational potential theory and transformed coordinates. The solutions are verified with the classic solution of circular and spherical potentials with polynomial source terms (i.e, linear and quadratic) by using many triangular and tetrahedral elements. It enables to solve the inhomogeneity problems with arbitrary particles. 2) Due to the discontinuity on the surfaces and edges of the subdomains and strong singular effects on the vertices, the variation of eigenstrain field is complicated in the neighborhood of edges and vertices. The domain discretization approach is proposed to provide a rapid convergent and effective solution in the infinite space. Different from the Taylor's expansion, the eigenstrain is assigned exactly at the nodes with shape functions instead of at the centroid of the elements, therefore, a 𝐂⁰ continuity is enforced. Here 3-node, 6-node triangular elements and 4-node, 10-node tetrahedral elements are implemented in the code of iBEM, which agree well with FEM but with much fewer of elements. Other types of element are also implementable in the same fashion. 3) The discretization method is applied to investigate the stress singularities of a vertex on an isosceles triangle embedded in an unbounded matrix. Two types of stress singularities are investigated: when the load is applied to the triangular inclusion with the same stiffness as the matrix, the singularity is caused by the irregular load distribution, namely load singularity, and can be exactly evaluated by integral of the potentials on the source with Eshelby's tensor. The second singularity, namely material singularity, is caused by the stiffness mismatch between the triangular inhomogeneity and the matrix under a uniform far field stress, in which the material mismatch is simulated by an eigenstrain. The relationship between the load singularity and material singularity is investigated, and the linkages of these singularities with line distributed force, cracking, and point force are discussed. 4) A parametric study of accuracy on stress field for uniform, linear and quadratic eigenstrain fields was performed and case studies have been presented to demonstrate the capability of iBEM for virtual experiments of ellipsoidal / elliptical inhomogeneities. Subsequently, combining the domain discretization method, iBEM is also applied to study the local elastic fields of the angular inhomogeneities. The effective material behavior is obtained with either large number of particles or periodic boundary condition (PBC) and some interesting discoveries of microstructure-dependent material behavior are reported with the aid of virtual experiments. 5) The iBEM is extended to multiphysical problems. The temperature and hear flux fields of composite materials containing phase change materials (PCM) for energy efficient buildings is demonstrated. Different from the static EIM, the thermal property mismatch between PCM particle and matrix phase is simulated with a uniformly distributed eigen-temperature gradient field and a fictitious heat source on the particle. With the equivalent heat flux conditions and the specific heat-temperature relationship, the eigen-temperature gradient and fictitious heat source can be solved and temperature field of the bounded domain can be calculated. Verified with FEM and laboratory measurements of the transient heat transfer within a building block containing a PCM capsule. Parametric studies have also been conducted to study the influences of the PCM location and volume fraction on the temperature fields of composites with multiple particles. The virtual experiments demonstrate the energy saving and phase delay by using the PCM-concrete wall panel. In summary, the proposed iBEM algorithm bridges the gap between conventional EIM and BEM for virtual experiments of composites samples. The combination of shape functions and domain integrals of polygonal / polyhedral subdomain enables its application to arbitrary shaped particles. It serves as a powerful tool to conduct virtual experiments for composite materials with various geometry and investigate the effective moduli under uni-axial load of samples with large number of particles or under the periodic boundary condition. In the future, the iBEM will be implemented for time independent and dependent nonlinear behavior of composites, such as elastoplastic, viscoelastic, and dynamic elastic problems. In addition to the current parallel computing scheme, GPU can be employed to speed up particle - particle interactions.

Cette thèse développe un cadre rigoureux qui permet de démontrer des représentations exactes associées à divers observables de la chaîne XXZ de Heisenberg de spin 1/2 à température finie. Il a était argumenté dans la littérature que l’énergie libre par site ou les longueurs de corrélations admettent des représentations intégrales où les intégrandes sont exprimées en termes de solutions d’équations intégrales non-linéaires. Les dérivations de ces représentations reposaient sur divers conjectures telles que l’existence d’une valeur propre de la matrice de transfert quantique, real, non-dégénérée, de module maximale, de l’échangeabilitée de la limite du volume infinie et du nombre de Trotter à l’infinie, de l’existence et de l’unicité des solutions des equation intégrales non-linéaires auxiliaires et finalement de l’identification des valeurs propers de la matrice de transfert quantiques avec les solutions de l’équations intégrales non-linéaires. Nous démontrons toutes ces conjectures dans le regime de haute température. Nôtre analyse nous permet aussi de démontrer que pour ces température suffisamment élevées, il est possible d’avoir une description d’un certain sous-ensemble de valeurs propres sous-dominante de la matrice de transfert quantique décrite en terme de solutions d’une chaîne de spin-1 de taille finie. / This thesis develops a rigorous framework allowing one to prove the exact representations for various observables in the XXZ Heisenberg spin-1/2 chain at finite temperature. Previously it has been argued in the literature that the per-site free energy or the correlation lengths admit integral representations whose integrands are expressed in terms of solutions of non-linear integral equations. The derivations of such representations relied on various conjectures such as the existence of a real, non-degenerate, maximal in modulus Eigenvalue of the quantum transfer matrix, the exchangeability of the infinite volume limit and the Trotter number limits, the existence and uniqueness of the solutions to the auxiliary non-linear integral equations and finally the identification of the quantum transfer matrix’s Eigenvalues with solutions to the non-linear integral equation. We rigorously prove all these conjectures in the high temperature regime. Our analysis also allows us to prove that for temperatures high enough, one may describe a certain subset of sub-dominant Eigenvalues of the quantum transfer matrix described in terms of solutions to a spin-1 chain of finite length.

We consider a collocation method for Cauchy singular integral equations on the interval based on weighted Chebyshev polynomials , where the coefficients of the operator are piecewise continuous. Stability conditions are derived using Banach algebra methods, and numerical results are given.

Les propriétés de solvatation jouent un rôle important dans les problèmes chimiques et biochimiques. La théorie fonctionnelle de la densité moléculaire (MDFT) est l'une des méthodes frontières pour évaluer ces propriétés, dans laquelle une fonction d'énergie libre de solvatation est minimisée pour un soluté arbitraire dans une boîte de solvant cubique périodique. Dans cette thèse, nous travaillons sur l'évaluation du terme d'excès de la fonctionnelle d’énergie libre sous l’approximation du fluide de référence homogène (HRF), équivalent à l'approximation de la chaîne hypernettée (HNC) dans la théorie des équations intégrales. Deux algorithmes sont proposés: le premier est une extension d'un algorithme précédent, qui permet de traiter le cas d'un solvant moléculaire à trois dimensions (en fonction de trois angles d'Euler) au lieu d'un solvant linéaire (selon deux angles); L'autre est un nouvel algorithme qui intègre le traitement de la convolution angulaire de l'équation Ornstein-Zernike (OZ) moléculaire dans MDFT, et en fait développe la densité du solvant et le gradient fonctionnel en harmoniques sphériques généralisées (GSHs). On montre que le nouvel algorithme est beaucoup plus rapide que le précédent. Les deux algorithmes sont appropriés pour des solutés arbitraires tridimensionnel dans l'eau liquide, et pour prédire l'énergie libre et la structure de solvatation d'ions et de molécules. / Solvation properties play an important role in chemical and bio-chemical issues. The molecular density functional theory (MDFT) is one of the frontier numerical methods to evaluate these properties, in which the solvation free energy functional is minimized for an arbitrary solute in a periodic cubic solvent box. In this thesis, we work on the evaluation of the excess term of the free energy functional under the homogeneous reference fluid (HRF) approximation, which is equivalent to hypernetted-chain (HNC) approximation in integral equation theory. Two algorithms are proposed: the first one is an extension of a previously implemented algorithm, which makes it possible to handle full 3D molecular solvent (depending on three Euler angles) instead of linear solvent (depending on two angles); the other one is a new algorithm that integrates the molecular Ornstein-Zernike (OZ) equation treatment of angular convolution into MDFT, which in fact expands the solvent density and the functional gradient on generalized spherical harmonics (GSHs). It is shown that the new algorithm is much more rapid than the previous one. Both algorithms are suitable for arbitrary three-dimensional solute in liquid water, and are able to predict the solvation free energy and structure of ions and molecules.

<div>In this work, we introduce a framework for constructing light-weight neural network approximations to the solution operators for partial differential equations (PDEs). Using a data-driven offline training procedure, the resulting operator network models are able to effectively reduce the computational demands of traditional numerical methods into a single forward-pass of a neural network. Importantly, the network models can be calibrated to specific distributions of input data in order to reflect properties of real-world data encountered in practice. The networks thus provide specialized solvers tailored to specific use-cases, and while being more restrictive in scope when compared to more generally-applicable numerical methods (e.g. procedures valid for entire function spaces), the operator networks are capable of producing approximations significantly faster as a result of their specialization.</div><div><br></div><div>In addition, the network architectures are designed to place pointwise posterior distributions over the observed solutions; this setup facilitates simultaneous training and uncertainty quantification for the network solutions, allowing the models to provide pointwise uncertainties along with their predictions. An analysis of the predictive uncertainties is presented with experimental evidence establishing the validity of the uncertainty quantification schema for a collection of linear and nonlinear PDE systems. The reliability of the uncertainty estimates is also validated in the context of both in-distribution and out-of-distribution test data.</div><div><br></div><div>The proposed neural network training procedure is assessed using a novel convolutional encoder-decoder model, ConvPDE-UQ, in addition to an existing fully-connected approach, DeepONet. The convolutional framework is shown to provide accurate approximations to PDE solutions on varying domains, but is restricted by assumptions of uniform observation data and homogeneous boundary conditions. The fully-connected DeepONet framework provides a method for handling unstructured observation data and is also shown to provide accurate approximations for PDE systems with inhomogeneous boundary conditions; however, the resulting networks are constrained to a fixed domain due to the unstructured nature of the observation data which they accommodate. These two approaches thus provide complementary frameworks for constructing PDE-based operator networks which facilitate the real-time approximation of solutions to PDE systems for a broad range of target applications.</div>

<p>In this work, we mainly focus on the topic related to dimension reduction, operator learning and uncertainty quantification for problems of differential equations. The supervised machine learning methods introduced here belong to a newly booming field compared to traditional numerical methods. The building blocks for our works are mainly Gaussian process and neural network. </p> <p><br></p> <p>The first work focuses on supervised dimension reduction problems. A new framework based on rotated multi-fidelity Gaussian process regression is introduced. It can effectively solve high-dimensional problems while the data are insufficient for traditional methods. Moreover, an accurate surrogate Gaussian process model of original problem can be formulated. The second one we would like to introduce is a physics-assisted Gaussian process framework with active learning for forward and inverse problems of partial differential equations(PDEs). In this work, Gaussian process regression model is incorporated with given physical information to find solutions or discover unknown coefficients of given PDEs. Three different models are introduce and their performance are compared and discussed. Lastly, we propose attention based MultiAuto-DeepONet for operator learning of stochastic problems. The target of this work is to solve operator learning problems related to time-dependent stochastic differential equations(SDEs). The work is built on MultiAuto-DeepONet and attention mechanism is applied to improve the model performance in specific type of problems. Three different types of attention mechanism are presented and compared. Numerical experiments are provided to illustrate the effectiveness of our proposed models.</p>