My main research topic is about developing new domain decomposition algorithms for the solution of systems of partial differential equations. This was mainly applied to fluid dynamics problems (as compressible Euler or Stokes equations) and electromagnetics (time-harmonic and time-domain first order system of Maxwell's equations). Since the solution of large linear systems is strongly related to the application of a discretization method, I was also interested in developing and analyzing the application of high order methods (such as Discontinuos Galerkin methods) to Maxwell's equations (sometimes in conjuction with time-discretization schemes in the case of time-domain problems). As an active member of NACHOS pro ject (besides my main afiliation as an assistant professor at University of Nice), I had the opportunity to develop certain directions in my research, by interacting with permanent et non-permanent members (Post-doctoral researchers) or participating to supervision of PhD Students. This is strongly refflected in a part of my scientific contributions so far. This memoir is composed of three parts: the first is about the application of Schwarz methods to fluid dynamics problems; the second about the high order methods for the Maxwell's equations and the last about the domain decomposition algorithms for wave propagation problems.

Abstract Several numerical methods and algorithms, for solving the mathematical model of a continuous casting process, are presented, and theoretically studied, in this work. The numerical algorithms can be divided in to three different groups: the Schwarz type overlapping methods, the nonoverlapping Splitting iterative methods, and the Predictor-Corrector type nonoverlapping methods. These algorithms are all so-called parallel algorithms i.e., they are highly suitable for parallel computers. Multiplicative, additive Schwarz alternating method and two asynchronous domain decomposition methods, which appear to be a two-stage Schwarz alternating algorithms, are theoretically and numerically studied. Unique solvability of the fully implicit and semi-implicit finite difference schemes as well as monotone dependence of the solution on the right-hand side are proved. Geometric rate of convergence for the iterative methods is investigated. Splitting iterative methods for the sum of maximal monotone and single-valued monotone operators in a finite-dimensional space are studied. Convergence, rate of convergence and optimal iterative parameters are derived. A two-stage iterative method with inner iterations is analyzed in the case when both operators are linear, self-adjoint and positive definite. Several new finite-difference schemes for a nonlinear convection-diffusion problem are constructed and numerically studied. These schemes are constructed on the basis of non-overlapping domain decomposition and predictor-corrector approach. Different non-overlapping decompositions of a domain, with cross-points and angles, schemes with grid refinement in time in some subdomains, are used. All proposed algorithms are extensively numerically tested and are founded stable and accurate under natural assumptions for time and space grid steps. The advantages and disadvantages of the numerical methods are clearly seen in the numerical examples. All of the algorithms presented are quite easy and straight forward, from an implementation point of view. The speedups show that splitting iterative method can be parallelized better than multiplicative or additive Schwarz alternating method. The numerical examples show that the multidecomposition method is a very effective numerical method for solving the continuous casting problem. The idea of dividing the subdomains to smaller subdomains seems to be very beneficial and profitable. The advantages of multidecomposition methods over other methods is obvious. Multidecomposition methods are extremely quick, while being just as accurate as other methods. The numerical results for one processor seem to be very promising.

Ces travaux de recherche ont pour objectif de contribuer au développement et à l'exploitation d'outils de vérification des problèmes de mécanique linéaires dans le cadre des méthodes de décomposition de domaine sans recouvrement. Les apports de cette thèse sont multiples : * Nous proposons d'améliorer la qualité des champs statiquement admissibles nécessaires à l'évaluation de l'estimateur par une nouvelle méthodologie de reconstruction des contraintes en séquentiel et par des optimisations du calcul de l'intereffort en cadre sous-structuré.* Nous démontrons des bornes inférieures et supérieures de l'erreur séparant l'erreur algébrique (due au solveur itératif) de l'erreur de discrétisation (due à la méthode des éléments finis) tant pour une mesure globale que pour une quantité d'intérêt. Cette séparation permet la définition d'un critère d'arrêt objectif pour le solveur itératif.* Nous exploitons les informations fournies par l'estimateur et les espaces de Krylov générés pour mettre en place une stratégie auto-adaptative de calcul consistant en une chaîne de résolution mettant à profit remaillage adaptatif et recyclage des directions de recherche. Nous mettons en application le pilotage du solveur par un objectif de précision sur des exemples mécaniques en deux dimensions. / This research work aims at contributing to the development of verification tools in linear mechanical problems within the framework of non-overlapping domain decomposition methods.* We propose to improve the quality of the statically admissible stress field required for the computation of the error estimator thanks to a new methodology of stress reconstruction in sequential context and thanks to optimizations of the computations of nodal reactions in substructured context.* We prove guaranteed upper and lower bounds of the error that separates the algebraic error (due to the iterative solver) from the discretization error (due to the finite element method) for both global error measure mentand goal-oriented error estimation. It enables the definition of a new stopping criterion for the iterative solver which avoids over-resolution.* We benefit the information provided by the error estimator and the Krylov subspaces built during the resolution to set an auto-adaptive strategy. This strategy consists in sequel of resolutions and takes advantage of adaptive remeshing and recycling of search directions .We apply the steering of the iterative solver by objective of precision on two-dimensional mechanical examples.

Les méthodes en simulation numérique dans le domaine de l’ingénierie pétrolière nécessitent la résolution de systèmes linéaires creux de grande taille et non structurés. La performance des méthodes itératives utilisées pour résoudre ces systèmes représente un enjeu majeur afin de permettre de tester de nombreux scénario.Dans ces travaux, nous présentons une manière d'implémenter des méthodes itératives parallèles au dessus d’un support exécutif à base de tâches. Afin de simplifier le développement des méthodes tout en gardant un contrôle fin sur la gestion du parallélisme, nous avons proposé une API permettant d’exprimer implicitement les dépendances entre tâches : la sémantique de l'API reste séquentielle et le parallélisme est implicite.Nous avons étendu le support exécutif HARTS pour enregistrer une trace d'exécution afin de mieux exploiter les architectures NUMA, tout comme de prendre en compte un placement des tâches et des données calculé au niveau de l’API. Nous avons porté et évalué l'API sur les processeurs many-coeurs KNL en considérant les différents types de mémoires de l’architecture. Cela nous a amené à optimiser le calcul du SpMV qui limite la performance de nos applications.L'ensemble de ce travail a été évalué sur des méthodes itératives et en particulier l’une de type décomposition de domaine. Nous montrons alors la pertinence de notre API, qui nous permet d’atteindre de très bon niveaux de performances aussi bien sur architecture multi-coeurs que many-coeurs. / Numerical methods in reservoir engineering simulations lead to the resolution of unstructured, large and sparse linear systems. The performances of iterative methods employed in simulator to solve these systems are crucial in order to consider many more scenarios.In this work, we present a way to implement efficient parallel iterative methods on top of a task-based runtime system. It enables to simplify the development of methods while keeping control on parallelism management. We propose a linear algebra API which aims to implicitly express task dependencies: the semantic is sequential while the parallelism is implicit.We have extended the HARTS runtime system to monitor executions to better exploit NUMA architectures. Moreover, we implement a scheduling policy which exploits data locality for task placement. We have extended the API for KNL many-core systems while considering the various memory banks available. This work has led to the optimization of the SpMV kernel, one of the most time consuming operation in iterative methods.This work has been evaluated on iterative methods, and particularly on one method coming from domain decomposition. Hence, we demonstrate that the API enables to reach good performances on both multi-core and many-core architectures.

Dans le présent travail, des nouvelles méthodes de décomposition de domaine et des nouvelles implémentations pour des méthodes existantes sont développées. Une nouvelle méthode basée sur les méthodes antérieures de décomposition du domaine est formulée. Les méthodes classiques FETI plus FETI-2LM sont utilisées pour construire le nouveau Hybrid-FETI. L'idée de base est de développer un nouvel algorithme qui peut utiliser les deux méthodes en même temps en choisissant dans chaque interface l'état le plus adapté en fonction des caractéristiques du problème. En faisant cela, nous recherchons un code plus rapide et plus robuste qui peut fonctionner avec des configurations selon lesquelles les méthodes de base ne le géreront pas de manière optimale par lui-même. La performance est testée sur un problème de contact. La partie suivante implique le développement d'une nouvelle implémentation pour la méthode S-FETI, l'idée est de réduire l'utilisation de la mémoire de cette méthode, afin de pouvoir fonctionner dans des problèmes de taille plus important. Différentes variantes pour cette méthode sont également proposées, tout en cherchant la réduction des directions stockées chaque itération de la méthode itérative. Finalement, une extension de la méthode FETI-2LM à sa version en bloc comme dans S-FETI, est développée. Les résultats numériques pour les différents algorithmes sont présentés. / In this work new domain decomposition methods and new implementations for existing methods are developed. A new method based on previous domain decomposition methods is formulated. The classic FETI plus FETI-2LM methods are used to build the new Hybrid-FETI. The basic idea is to develop a new algorithm that can use both methods at the same time by choosing in each interface the most suited condition depending on the characteristics of the problem. By doing this we search to have a faster and more robust code that can work with configurations that the base methods will not handle it optimally by himself. The performance is tested on a contact problem. The following part involves the development of a new implementation for the S-FETI method, the idea is to reduce the memory usage of this method, to make it able to work in larger problem. Different variation for this method are also proposed, all searching the reduction of directions stored each iteration of the iterative method. Finally, an extension of the FETI-2LM method to his block version as in S-FETI, is developed. Numerical results for the different algorithms are presented.

<div>The design of advanced integrated circuits (ICs) and systems calls for multiphysics and large-scale modeling and simulation methods. On the one hand, novel devices and materials are emerging in next-generation IC technology, which requires multiphysics modeling and simulation. On the other hand, the ever-increasing complexity of ICs requires more efficient numerical solvers.</div><div><br></div><div>In this work, we propose a multiphysics modeling and simulation algorithm to co-simulate Maxwell's equations, dispersion relation of materials, and Boltzmann equation to characterize emerging new devices in IC technology such as Cu-Graphene (Cu-G) hybrid nano-interconnects. We also develop an unconditionally stable time marching scheme to remove the dependence of time step on space step for an efficient simulation of the multiscaled and multiphysics system. Extensive numerical experiments and comparisons with measurements have validated the accuracy and efficiency of the proposed algorithm. Compared to simplified steady-state-models based analysis, a significant difference is observed when the frequency is high or/and the dimension of the Cu-G structure is small, which necessitates our proposed multiphysics modeling and simulation for the design of advanced Cu-G interconnects. </div><div><br></div><div>To address the large-scale simulation challenge, we develop a new split-field domain-decomposition algorithm amenable for parallelization for solving Maxwell’s equations, which minimizes the communication between subdomains, while having a fast convergence of the global solution. Meanwhile, the algorithm is unconditionally stable in time domain. In this algorithm, unlike prevailing domain decomposition methods that treat the interface unknown as a whole and let it be shared across subdomains, we partition the interface unknown into multiple components, and solve each of them from one subdomain. In this way, we transform the original coupled system to fully decoupled subsystems to solve. Only one addition (communication) of the interface unknown needs to be performed after the computation in each subdomain is finished at each time step. More importantly, the algorithm has a fast convergence and permits the use of a large time step irrespective of space step. Numerical experiments on large-scale on-chip and package layout analysis have demonstrated the capability of the new domain decomposition algorithm. </div><div><br></div><div>To tackle the challenge of efficient simulation of irregular structures, in the last part of the thesis, we develop a method for the stability analysis of unsymmetrical numerical systems in time domain. An unsymmetrical system is traditionally avoided in numerical formulation since a traditional explicit simulation is absolutely unstable, and how to control the stability is unknown. However, an unsymmetrical system is frequently encountered in modeling and simulating of unstructured meshes and nonreciprocal electromagnetic and circuit devices. In our method, we reduce stability analysis of a large system into the analysis of dissembled single element, therefore provides a feasible way to control the stability of large-scale systems regardless of whether the system is symmetrical or unsymmetrical. We then apply the proposed method to prove and control the stability of an unsymmetrical matrix-free method that solves Maxwell’s equations in general unstructured meshes while not requiring a matrix solution.<br></div><div><br></div>

No description available.

Cette thèse est dédiée à la résolution numérique tridimensionnelle des équations de Maxwell harmoniques, par des méthodes de décomposition de domaine couplant des résolutions par équations intégrales entre elles. Pour traiter les problèmes de diffraction d'ondes, la méthode des équations intégrales est un outil précieux. Elle consiste à paramétrer le champ électromagnétique solution par une source définie sur la surface de l'objet diffractant, solution d'une nouvelle équation linéaire (l'équation intégrale). Pour des applications à haute fréquence, le grand nombre d'inconnues (de l'ordre du million) nous oblige à utiliser un solveur itératif pour résoudre l'équation intégrale. Le problème du conditionnement des systèmes linéaires est alors crucial. De récents développements ont permis de construire une équation intégrale performante (la GCSIE) et de conditionnement stable avec la montée en fréquence. Cependant, la présence d'une cavité large et résonnante dans l'objet diffractant (telle que la cavité moteur d'un avion) dégrade le conditionnement de cette équation. Nous proposons deux méthodes de décomposition de domaine (DDM) afin de découpler le problème de la cavité du problème extérieur. La première (DDM en Y) s'exprime en fonction des opérateurs Dirichlet-to-Neumann Y, qui sont synthétisés via la résolution de problèmes métalliques par équations intégrales dans chaque sous-domaine. La seconde (DDM en S) s'exprime en fonction des opérateurs de scattering S, synthétisés par résolution de problèmes de type métal-impédant, donc bien posés à toute fréquence. La DDM en S permet ainsi de se débarrasser des phénomènes de résonance dans les cavités. Nous proposons dans un premier temps un préconditionneur analytique pour la DDM en Y, basé sur l'opérateur électromagnétique de simple couche. Nous calculons ensuite les modes guidés le long d'un cylindre infini tangent à la cavité près de l'interface, et nous diagonalisons les opérateurs Dirichlet-to-Neumann et scattering dans la base des traces de modes guidés sur l'interface. On extrait de cette étude deux préconditionneurs spectraux respectivement pour la DDM en Y et la DDM en S. Les résultats numériques confirment l'efficacité des préconditionneurs proposés / This work is dedicated to the numerical solution of the tridimensional harmonic Maxwell equations, using domain decomposition methods coupling integral equations between them. To deal with scattering problems, integral equations methods are a precious tool. They allow to look for the electromagnetic field by parameterizing it with a source only defined on the boundary of the scattering object, solution of a new linear equation (the integral equation). For applications at high frequency, the great number of unknowns forces the use of iterative methods. To accelerate the solution of integral equations, one moreover has to ensure the good condition number of the linear systems, or to propose well-suited preconditioners. An efficient method, the GCSIE, was developed in Onera. It is an intrinsically well-conditioned integral equation whose condition number remains stable whith the frequency increase. However, the existence of large and resonant cavities (such as air intakes) deteriorates the condition number. In order to circumvent this problem, we propose two domain decomposition methods (DDM) allowing to decouple the exterior problem from the problem of the cavities. The first one (Y-DDM) is based on Dirichlet-to-Neumann operators Y, which are built through the solution of metallic problems using integral equations in each subdomain. The second one (S-DDM) is based on scattering operators S, built through the solution of problems of metallic-impedant type, which are well-posed at any frequency. The S-DDM allows to avoid the resonance phenomena inside the cavities. First, we propose an analytic preconditioner for the Y-DDM, based on the electromagnetic single layer operator. We then calculate the modes guided along an artificial infinite cylinder, that is tangent to the cavity near the interface. We diagonalize the Dirichlet-to-Neumann and scattering operators in the basis of the traces of the guided modes on the interface. We deduce from this study two spectral preconditioners for the Y-DDM and the S-DDM. The numerical results confirm the efficiency of the employed preconditioners.