La simulation numérique des systèmes multicorps en présence d'interactions complexes, dont le contact frottant, pose de nombreux défis, tant en terme de modélisation que de temps de calcul. Dans ce manuscrit de thèse, nous étudions deux familles de décomposition de domaine adaptées au formalisme de la dynamique non régulière des contacts (NSCD). Cette méthode d'intégration implicite en temps de l'évolution d'une collection de corps en interaction a pour caractéristique de prendre en compte le caractère discret et non régulier d'un tel milieu. Les techniques de décomposition de domaine classiques ne peuvent de ce fait être directement transposées. Deux méthodes de décomposition de domaine proches des formalismes des méthodes de Schwarz et de complément de Schur sont présentées. Ces méthodes se révèlent être de puissants outils pour la parallélisation en mémoire distribuée des simulations granulaires 2D et 3D sur un centre de calcul haute performance. Le comportement de structure des milieux granulaires denses est de plus exploité afin de propager rapidement l'information sur l'ensemble des sous-domaines via un schéma semi-implicite d'intégration en temps. / Numerical simulations of the dynamics of discrete structures in presence of numerous impacts and frictional contacts leads to CPU-intensive large time computations. To deal with such realistic assemblies, numerical tools have been developed, in particular the method called nonsmooth contact dynamics (NSCD). Such modeling has to deal with discreteness and nonsmoothness, such that domain decomposition approaches for regular continuum media has to be rethought. We present further two domain decomposition method linked to Schwarz and Schur formalism. Scalability and numerical performances of the methods for 2D and 3D granular media is studied, showing good parallel behavior on a supercomputer platform. The structural behavior of dense granular packing is herein used to introduce a spacial multilevel preconditioner with a coarse problem to improve convergence in a space-time approach.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-07-12T12:23:47Z No. of bitstreams: 1 hernandojoserochafranco.pdf: 1674623 bytes, checksum: f6df7317dd6a8e94e51045dbf75e8241 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-07-17T11:56:13Z (GMT) No. of bitstreams: 1 hernandojoserochafranco.pdf: 1674623 bytes, checksum: f6df7317dd6a8e94e51045dbf75e8241 (MD5) / Made available in DSpace on 2018-07-17T11:56:13Z (GMT). No. of bitstreams: 1 hernandojoserochafranco.pdf: 1674623 bytes, checksum: f6df7317dd6a8e94e51045dbf75e8241 (MD5) Previous issue date: 2018-06-08 / A teoria clássica de otimização presume a existência de certas condições, por exemplo, que as funções envolvidas em um problema desta natureza sejam pelo menos uma vez continuamente diferenciáveis. Entretanto, em muitas aplicações práticas que requerem o emprego de métodos de otimização, essa característica não se encontra presente. Problemas de otimização não diferenciáveis são considerados mais difíceis de lidar. Nesta classe, aqueles que envolvem funções não convexas são ainda mais complexos. O Interior Epigraph Directions (IED) é um método de otimização que se baseia na teoria da Dualidade Lagrangeana e se aplica à resolução de problemas não diferenciáveis, não convexos e com restrições. Neste estudo, apresentamos duas novas versões para o referido método a partir de implementações computacionais de outros algoritmos. A primeira versão, denominada IED+NFDNA, recebeu a incorporação de uma implementação do algoritmo Nonsmooth Feasible Direction Nonconvex Algorithm (NFDNA). Esta versão, ao ser aplicada em experimentos numéricos com problemas teste da literatura, apresentou desempenho satisfatório quando comparada ao IED original e a outros solvers de otimização. Com o objetivo de aperfeiçoar mais o método, reduzindo sua dependência de parâmetros iniciais e também do cálculo de subgradientes, uma segunda versão, IED+GA, foi desenvolvida com a utilização de algoritmos genéticos. Além da resolução de problemas teste, o IED-FGA obteve bons resultados quando aplicado a problemas de engenharia. / The classical theory of optimization assumes the existence of certain conditions, for example, that the functions involved in a problem of this nature are at least once continuously differentiable. However, in many practical applications that require the use of optimization methods, this characteristic is not present. Non-differentiable optimization problems are considered more difficult to deal with. In this class, those involving nonconvex functions are even more complex. Interior Epigraph Directions (IED) is an optimization method that is based on Lagrangean duality theory and applies to the resolution of non-differentiable, non-convex and constrained problems. In this study, we present two new versions for this method from computational implementations of other algorithms. The first version, called IED + NFDNA, received the incorporation of an implementation of the Nonsmooth Feasible Direction Nonconvex Algorithm (NFDNA) algorithm. This version, when applied in numerical experiments with problems in the literature, presented satisfactory performance when compared to the original IED and other optimization solvers. A second version, IED + GA, was developed with the use of genetic algorithms in order to further refine the method, reducing its dependence on initial parameters and also on the calculation of subgradients. In addition to solving test problems, IED + GA achieved good results when applied to engineering problems.

The task of modeling and controlling the evolution of dynamical sys- tems is one of the main objectives in mechatronics engineering. When approaching the problem of controlling physical or digital systems, the dynamical models have been historically divided into continuous-time, described by differential equations, and discrete-time, described by difference equations. In the last decade, a new class of models, known as hybrid dynamical systems, has gained popularity in the control community because of its high versatility. This framework combines continuous-time and discrete- time evolution, thus allowing for both the description of a broader class of systems and the achievement of better-performing controllers, compared to the traditional continuous-time alternatives. After the first rigorous introduction of the framework, several Lyapunov-based results were published in the literature, and numerous application areas were shown to benefit from the introduction of a hybrid dynamics, like systems involving impacts or physical systems connected to digital controllers (cyber-physical systems). In this thesis, we use the hybrid framework to study different mechatronics-inspired control problems. The applications we consider are diverse, so we split the presentation into three parts. In the first part we further analyze a particular hybrid control strategy, known as reset control, providing some new theoretical guarantees, together with an application to adaptive control. In the second part we consider two applications of the hybrid framework to the network dynamics field, specifically we analyze the problems of distributed state estimation and of uniform synchronization of nonlinear oscillators. In the third part, we use a hybrid approach to study two applications where this framework has been rarely employed, or not at all, namely smart agriculture and trajectory tracking for a bipedal walking robot. We study these application-inspired problems from a theoretical point of view, giving robust Lyapunov-based stability guarantees. We complement the theoretical analysis with numerical results, obtained from simulations or from experiments.

We present efficient and globally convergent solvers for several classes of plasticity models. The models in this work are formulated in the primal form as energetic rate-independent systems with an elastic energy potential and a plastic dissipation component. Different hardening rules are considered, as well as different flow rules. The time discretization leads to a sequence of nonsmooth minimization problems. For small strains, the unknowns live in vector spaces while for finite strains we have to deal with manifold-valued quantities. For the latter, a reformulation in tangent space is performed to end up with the same dissipation functional as in the small-strain case. We present the Newton-type TNNMG solver for convex and nonsmooth minimization problems and a newly developed Proximal Newton (PN) method that can also handle nonconvex problems. The PN method generates a sequence of penalized convex, coercive but nonsmooth subproblems. These subproblems are in the form of block-separable small-strain plasticity problems, to which TNNMG can be applied. Global convergence theorems are available for both methods. In several numerical experiments, both the efficiency and the flexibility of the methods for small-strain and finite-strain models are tested.

Les problèmes de complémentarité interviennent dans de nombreux domaines scientifiques : économie, mécanique des solides, mécanique des fluides. Ce n’est que récemment qu’ils ont commencé d’intéresser les chercheurs étudiant les écoulements et le transport en milieu poreux. Les problèmes de complémentarité sont un cas particulier des inéquations variationnelles. Dans cette thèse, on offre plusieurs contributions aux méthodes numériques pour résoudre les problèmes de complémentarité. Dans la première partie de cette thèse, on étudie les problèmes de complémentarité linéaires 0 6 x ⊥ (Mx+q) > 0 où, x l’inconnue est dans Rn et où les données sont q, un vecteur de Rn, et M, une matrice d’ordre n. L’existence et l’unicité de ce problème est obtenue quand la matrice M est une P-matrice. Une méthode très efficace pour résoudre les problèmes de complémentarité est la méthode de Newton-min, une extension de la méthode de Newton aux problèmes non lisses.Dans cette thèse on montre d’abord, en construisant deux familles de contre-exemples, que la méthode de Newton-min ne converge pas pour la classe des P-matrices, sauf si n= 1 ou 2. Ensuite on caractérise algorithmiquement la classe des P-matrices : c’est la classe des matrices qui sont telles que quel que, soit le vecteur q, l’algorithme de Newton-min ne fait pas de cycle de deux points. Enfin ces résultats de non-convergence nous ont conduit à construire une méthode de globalisation de l’algorithme de Newton-min dont nous avons démontré la convergence globale pour les P-matrices. Des résultats numériques montrent l’efficacité de cet algorithme et sa convergence polynomiale pour les cas considérés. Dans la deuxième partie de cette thèse, nous nous sommes intéressés à un exemple de problème de complémentarité non linéaire concernant les écoulements en milieu poreux. Il s’agit d’un écoulement liquide-gaz à deux composants eau-hydrogène que l’on rencontre dans le cadre de l’étude du stockage des déchets radioactifs en milieu géologique. Nous présentons un modèle mathématique utilisant des conditions de complémentarité non linéaires décrivant ces écoulements. D’une part, nous proposons une méthode de résolution et un solveur pour ce problème. D’autre part, nous présentons les résultats numériques que nous avons obtenus suite à la simulation des cas-tests proposés par l’ANDRA (Agence Nationale pour la gestion des Déchets Radioactifs) et le GNR MoMaS. En particulier, ces résultats montrent l’efficacité de l’algorithme proposé et sa convergence quadratique pour ces cas-tests / This manuscript deals with numerical methods for linear and nonlinear complementarity problems,and, more specifically, with solving gas phase appearance and disappearance modeled as a complementarity problem. In the first part of this manuscript, we focused on the plain Newton-min method to solve the linear complementarity problem (LCP for short) 0 6 x ⊥ (Mx+q) > 0 that can be viewed as a nonsmooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x,Mx+q) = 0, which is equivalent to the LCP. When M is an M-matrix of order n, the algorithm was known to converge in at most n iterations. We show that this resultno longer holds when M is a P-matrix of order > 3. On the one hand, we offer counter-examplesshowing that the algorithm may cycle in those cases. P-matrices are interesting since they are thoseensuring the existence and uniqueness of the solution to the LCP for an arbitrary q. Incidentally,convergence occurs for a P-matrix of order 1 or 2. On the other hand, we provide a new algorithmic characterization of P-matricity : we show that a nondegenerate square real matrix M is a P-matrixif and only if, whatever is the real vector q, the Newton-min algorithm does not cycle between twopoints. In order to force the convergence of the Newton-min algorithm with P-matrices, we havederived a new method, which is robust, easy to describe, and simple to implement. It is globallyconvergent and the numerical results reported in this manuscript show that it outperforms a methodof Harker and Pang. In the second part of this manuscript, we consider the modeling of migration of hydrogen produced by the corrosion of the nuclear waste packages in an underground storage including the dissolution of hydrogen. It results in a set of nonlinear partial differential equations with nonlinear complementarity constraints. We show how to apply a robust and efficient solution strategy, the Newton-min method considered for LCP in the first part, to this geoscience problem and investigates its applicability and efficiency on this difficult problem. The practical interest of this solution technique is corroborated by numerical experiments from the Couplex Gas benchmark proposed by Andra and GNR MoMas. In particular, numerical results show that the Newton-min method is quadratically convergent for these problems

im Postscript-Format / PostScript

Dans une première partie, nous nous intéressons aux systèmes dynamiques gradients gouvernés par des fonctions non lisses, mais aussi non convexes, satisfaisant l'inégalité de Kurdyka-Lojasiewicz. Après avoir obtenu quelques résultats préliminaires pour la dynamique de la plus grande pente continue, nous étudions un algorithme de descente général. Nous prouvons, sous une hypothèse de compacité, que tout suite générée par ce schéma général converge vers un point critique de la fonction. Nous obtenons aussi de nouveaux résultats sur la vitesse de convergence, tant pour les valeurs que pour les itérés. Ce schéma général couvre en particulier des versions parallélisées de la méthode forward-backward, autorisant une métrique variable et des erreurs relatives. Cela nous permet par exemple de proposer une version non convexe non lisse de l'algorithme Levenberg-Marquardt. Enfin, nous proposons quelques applications de ces algorithmes aux problèmes de faisabilité, et aux problèmes inverses. Dans une seconde partie, cette thèse développe une dynamique de descente associée à des problèmes d'optimisation vectoriels sous contrainte. Pour cela, nous adaptons la dynamique de la plus grande pente usuelle aux fonctions à valeurs dans un espace ordonné par un cône convexe fermé solide. Cette dynamique peut être vue comme l'analogue continu de nombreux algorithmes développés ces dernières années. Nous avons un intérêt particulier pour les problèmes de décision multi-objectifs, pour lesquels cette dynamique de descente fait décroitre toutes les fonctions objectif au cours du temps. Nous prouvons l'existence de trajectoires pour cette dynamique continue, ainsi que leur convergence vers des points faiblement efficients. Finalement, nous explorons une nouvelle dynamique inertielle pour les problèmes multi-objectif, avec l'ambition de développer des méthodes rapides convergeant vers des équilibres de Pareto. / In a first part, we focus on gradient dynamical systems governed by non-smooth but also non-convex functions, satisfying the so-called Kurdyka-Lojasiewicz inequality.After obtaining preliminary results for a continuous steepest descent dynamic, we study a general descent algorithm. We prove, under a compactness assumption, that any sequence generated by this general scheme converges to a critical point of the function.We also obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method, with variable metric and relative errors. As an example, a non-smooth and non-convex version of the Levenberg-Marquardt algorithm is detailed.Applications to non-convex feasibility problems, and to sparse inverse problems are discussed.In a second part, the thesis explores descent dynamics associated to constrained vector optimization problems. For this, we adapt the classic steepest descent dynamic to functions with values in a vector space ordered by a solid closed convex cone. It can be seen as the continuous analogue of various descent algorithms developed in the last years.We have a particular interest for multi-objective decision problems, for which the dynamic make decrease all the objective functions along time.We prove the existence of trajectories for this continuous dynamic, and show their convergence to weak efficient points.Then, we explore an inertial dynamic for multi-objective problems, with the aim to provide fast methods converging to Pareto points.

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study.

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study.

We consider the eigenvalue problem associated to the 1-Laplace operator. We define higher eigensolutions by means of weak slope and establish existence of a sequence of eigensolutions by using nonsmooth critical point theory. In addition, we deduce a new necessary condition for the first eigenvalue of the 1-Laplace operator by means of inner variations. / Wir betrachten das zum 1-Laplace-Operator gehörige Eigenwertproblem. Wir definieren höhere Eigenlösungen mittels weak slope und weisen die Existenz einer Folge von Eigenlösungen nach, indem wir die nichtglatte Theorie kritischer Punkte anwenden. Zusätzlich leiten wir eine neue notwendige Bedingung für den ersten Eigenwert des 1-Laplace-Operators mittels innerer Variationen her.