本論文是以探討積分微分方程數值解的問題為主。此文中吾人皆先對問題本身做分析 ，討論其存在解，然後再用有限元素法，對連續性的問題做分解，使其變為一非線性 的方程組。而後藉由同倫（HOMOTOPY）法來解此非線性方程組。最後吾人可得到當區 間分割得愈小，真實解與數值解的誤差會愈小。也就是吾人所用之方法，為一個收斂 的方法。 本文共分兩部分，第一部分中，吾人討論一維的微分積分方程在有限區間的問題。於 此部分中，我們分了６個章節。第一節中，給了關於此問題的簡單介紹，並給序一些 必需的假設。第二節中，吾人可得到在第一節的假設下，假如原問題有真實解的話， 那麼此真實解絕對值的極大值（SUPREMUM）必不大於某個大於零的常數。第三節中， 吾人討論原方程的存在解，而證此存在解是經由LERAY-SCHAUDER DEGREE 定理得來的 。且在更強的條件下，會有存在唯一解。更而證明假如原來問題中函數不滿足所給予 的假設，那麼可經由修正（MODIFIED）原來的問題，也可得到原問題存在有解。第四 節中，對原來的方程，經由變分法（VARIATIONAL ）的方法，把它變成一非線性的方 程組，而在某些條件下，吾人亦可得到此方程組有解。第五節中，吾人討論此非線性 方程組的數值解。並可得知，當區間分割的愈小，此數值解會更趨近實實的解。第六 節中，吾人給予平滑的多項式子空間來逼近真實解，結果可得到假如每個區間以（ｋ ＋１）個點的LAGRANGE多項式來做內插（INTERPOLATION ），可知其收斂速度為O(Hk (big O),h 是分割區間的最大距離。 第二部分中，吾人所討論的是二維以上的積分微分方程在有界區域的問題，於此部分 中討論的與第一部分中類似，探討其存在，數值解等等問題。 最後吾人並給予一些例子，來加以印證我們所得到的結果。

Cette thèse est composée de deux parties indépendantes. La première est consacrée à l'étude de quelques problèmes elliptiques de type de Kirchhoff de la forme suivante : -M(ʃΩNul² dx) Δu = f(x, u) xЄΩ ; u(x) = o xЄƋΩ où Ω cRN, N ≥ 2, f une fonction de Carathéodory et M une fonction strictement positive et continue sur R+. Dans le cas où la fonction f est asymptotiquement linéaire à l’infini par rapport à l'inconnue u, on montre, en combinant une technique de troncature et la méthode variationnelle, que le problème admet au moins une solution positive quand la fonction M est non décroissante. Et si f(x, u) = |u|p-1 u + λg(x), où p >0, λ un paramètre réel et g une fonction de classe C1 et changeant de signe sur Ω, alors sous certaines hypothèses sur M, il existe deux réels positifs λ. et λ. tels que le problème admet des solutions positives si 0 < λ <λ. et n'admet pas de solutions positives si λ > λ.. Dans la deuxième partie, on étudie deux problèmes soulevés en dynamique des fluides. Le premier est une généralisation d'un modèle décrivant la propagation unidirectionnelle dispersive des ondes longues dans un milieu à deux fluides. En écrivant le problème sous la forme d'une équation de point fixe, on montre l'existence d'au moins une solution positive. On montre ensuite sa symétrie et son unicité. Le deuxième problème consiste à prouver l'existence de la vitesse, la pression et la température d'un fluide non newtonien, incompressible et non isotherme, occupant un domaine borné, en prenant en compte un terme de convection. L’originalité dans ce travail est que la viscosité du fluide ne dépend pas seulement de la vitesse mais aussi de la température et du module du tenseur des taux de déformations. En se basant sur la notion des opérateurs pseudo-monotones, le théorème de De Rham et celui de point fixe de Schauder, l'existence du triplet, (vitesse, pression, température) est démontré / This thesis consists of two independent parts. The first is devoted to the study of some elliptic problems of Kirchhoff-type in the following form : -M(ʃΩNul² dx) Δu = f(x, u) xЄΩ ; u(x) = o xЄƋΩ where Ω cRN, N ≥ 2, f is a Caratheodory function and M is a strictly positive and continuous function on R+. In the case where the function f is asymptotically linear at infinity with respect to the unknown u, we show, by combining a truncation technique and the variational method, that the problem admits a positive solution when the function M is nondecreasing. And if f(x, u) = |u|p-1 u + λg(x) where p> 0, λ a real parameter and g is a function of class C1 and changes the sign in Ω, then under some assumptions on M, there exist two positive real λ. and λ. such that the problem admits positive solutions if 0 < λ <λ., and no positive solutions if λ > λ.. In the second part, we study two problems arising in fluid dynamics. The first is a generalization of a model describing the unidirectional propagation of long waves in dispersive medium with two fluids. By writing the problem as a fixed point equation, we prove the existence of at least one positive solution. We then show its symmetry and uniqueness. The second problem is to prove the existence of the velocity, pressure and temperature of a non-Newtonian, incompressible and isothermal fluid, occupying a bounded domain, taking into account a convection term. The originality in this work is that the fluid viscosity depends not only on the velocity but also on the temperature and the modulus of deformation rate tensor. Based on the notion of pseudo-monotone operators, the De Rham theorem and the Schauder fixed point theorem, the existence of the triplet, (velocity, pressure, temperature) is shown

Le travail présenté dans cette thèse porte sur l'estimation du flot optique par méthodes variationnelles en petits et en grands déplacements. Nous proposons un modèle basé sur la combinaison locale-globale à laquelle nous ajoutons la prise en compte des variations de la luminosité. La particularité de ce manuscrit réside dans l'utilisation de la méthode des éléments finis pour la résolution des équations. En effet, cette méthode se fait pour le moment très rare dans le domaine du flot optique. Grâce à ce choix de résolution, nous proposons d'implémenter un contrôle local de la régularisation ainsi qu'une adaptation de maillage permettant d'affiner la solution au niveau des arêtes de l'image. Afin de réduire les temps de calcul, nous parallélisons les programmes. La première méthode implémentée est la méthode parallèle en temps appelée pararéel. En couplant un solveur grossier et un solveur fin, cet algorithme permet d'accélérer les calculs. Pour pouvoir obtenir un gain de temps encore plus important et également traiter les séquences en haute définition, nous utilisons ensuite une méthode de décomposition de domaine. Combinée au solveur massivement parallèle MUMPS, cette méthode permet un gain de temps de calcul significatif. Enfin, nous proposons de coupler la méthode de décomposition de domaine et le pararéel afin de profiter des avantages de chacune. Dans une seconde partie, nous appliquons tous ces modèles dans le cas de l'estimation du flot optique en grands déplacements. Nous proposons de nous servir du pararéel afin de traiter la non-linéarité de ce problème. Nous terminons par un exemple concret d'application du flot optique en restauration de films. / The work presented in this thesis focuses on the estimation of the optical flow through variational methods in small and large displacements. We propose a model based on the combined local-global strategy to which we add the consideration of brightness intensity variations. The particularity of this manuscript is the use of the finite element method to solve the equations. Indeed, for now, this method is really rare in the field of the optical flow. Thanks to this choice of resolution, we implement an adaptive control of the regularization and a mesh adaptation to refine the solution on the edges of the image. To reduce computation times, we parallelize the programs. The first method implemented is a parallel in time method called parareal. By combining a coarse and a fine solver, this algorithm speeds up the computations. To save even more time and to also be able to handle high resolution sequences, we then use a domain decomposition method. Combined with the massively parallel solver MUMPS, this method allows a significant reduction of computation times. Finally, we propose to couple the domain decomposition method and the parareal to have the benefits of both methods. In the second part, we apply all these models to the case of the optical flow estimation in large displacements. We use the parareal method to cope with the non-linearity of the problem. We end by a concrete example of application of the optical flow in film restoration.

Dans cette thèse ont été développés de nouveaux outils de calcul théorique de spectres moléculaires rovibrationnels qui permettent de mieux traiter les états vibrationnels très excités ainsi que les mouvements de grandes amplitudes avec la méthode d’interactions de configurations en champ moyen. Dans un premier temps, nous avons discuté la question du choix des bases modales et les différents compromis à trouver afin de pallier aux défauts possibles des surfaces d'énergie potentielle. Dans ce cadre nous avons également développé un critère de sélection visant à améliorer la qualité des fonctions d'ondes rovibrationnelles de base. Ces approches ont été appliquées avec succès à la molécule de méthane CH4.Dans un second temps, nous avons implémenté un algorithme de calcul formel des opérateurs d'énergie cinétique en coordonnées quelconques qui permet d'avoir des expressions exactes ainsi que leurs développements en série de Taylor ou Fourier, qui exploite au mieux les potentialités du logiciel MATHEMATICA et a permis d'obtenir des hamiltoniens rovibrationnels en coordonnées de valence de façon particulièrement efficace. Enfin, nous avons généralisé la méthode d’interactions de configurations en champ moyen en ajoutant de façon perturbative un champ effectif d'ordre deux. Nous avons appliqué cette généralisation à la molécule de péroxyde d'hydrogène HOOH, ce qui a permis de montrer son intérêt tant pour l'amélioration des niveaux d'énergie que des fonctions d'onde associées, lorsqu'on a affaire à des groupes de degrés de liberté bien séparés énergétiquement. / In this thesis we developed new theoretical tools for molecular rovibrational spectra for a better description of the excited vibrational states and movements with large amplitudes using mean field configuration interaction method. First, we discussed the choice of modal basis and different trade-off to overcome the possible shortcomings of potential energy surfaces. In this context we have also developed selection criteria to improve the quality of rovibrational wave functions. These approaches have been successfully applied to the methane molecule (CH4). Secondly, we have implemented a formal algorithm for calculating the kinetic energy operators in arbitrary coordinates that allows the derivation of exact expressions and their Taylor and Fourier series, using, in a very efficient way, the capabilities of the software MATHEMATICA which yield to the derivation of rovibrational Hamiltonians in valence coordinated. Finally, we have generalized the mean-field configuration interaction method by adding perturbatively a second order effective field. We applied this generalization to the hydrogen peroxide molecule (HOOH), which has shown an improvement for both energy levels and the associated wave functions, when dealing with groups of degrees of freedom that are energetically well separated.

L’essor récent de l’imagerie hybride combinant la Tomographie par Emission de Positons (TEP) à l’Imagerie par Résonance Magnétique (IRM) est une opportunité permettant d’exploiter des images d’un même territoire anatomo-pathologique obtenues simultanément et apportant des informations complémentaires. Cela représente aussi un véritable défi en raison de la différence de nature et de résolution spatiale des données acquises. Cette nouvelle technologie offre notamment des perspectives attrayantes en oncologie, et plus particulièrement en neuro-oncologie grâce au contraste qu’offre l’image IRM entre les tissus mous. Dans ce contexte et dans le cadre du projet PIM (Physique et Ingénierie pour la Médecine) de l’Université Paris-Saclay, l’objectif de cette thèse a été de développer un processus de segmentation multimodale adapté aux images TEP et IRM, comprenant une méthode de détection des volumes tumoraux en TEP et IRM, et une technique de segmentation précise du volume tumoral IRM. Ce processus doit être suffisamment générique pour s’appliquer à diverses pathologies cérébrales, différentes par leur nature même et par l’application clinique considérée. La première partie de la thèse aborde la détection de tumeurs par une approche hiérarchique. Plus précisément, la méthode de détection, réalisée sur les images IRM ou TEP, repose sur la création d’un nouveau critère de contexte spatial permettant de sélectionner les lésions potentielles par filtrage d’une représentation de l’image par max-tree. La deuxième partie de la thèse concerne la segmentation du volume tumoral sur les images IRM par une méthode variationnelle par ensembles de niveaux. La méthode de segmentation développée repose sur la minimisation d’une énergie globalement convexe associée à une partition d’une image RM en régions homogènes guidée par des informations de la TEP. Enfin, une dernière partie étend les méthodes proposées précédemment à l’imagerie multimodale IRM, notamment dans le cadre de suivi longitudinal. Les méthodes développées ont été testées sur plusieurs bases de données, chacune correspondant à une pathologie cérébrale et un radiotraceur TEP distincts. Les données TEP-IRM disponibles comprennent, d’une part, des examens de méningiomes et de gliomes acquis sur des machines séparées, et d’autre part, des examens réalisés sur le scanner hybride du Service Hospitalier Frédéric Joliot d’Orsay dans le cadre de recherches de tumeurs cérébrales. La méthode de détection développée a aussi été adaptée à l’imagerie multimodale IRM pour la recherche de lésions de sclérose en plaques ou le suivi longitudinal. Les résultats obtenus montrent que la méthode développée, reposant sur un socle générique, mais étant aussi modulable à travers le choix de paramètres, peut s’adapter à diverses applications cliniques. Par exemple, la qualité de la segmentation des images issues de la machine combinée a été mesurée par le coefficient de Dice, la distance de Hausdorff (DH) et la distance moyenne (DM), en prenant comme référence une segmentation manuelle de la tumeur validée par un expert médical. Les résultats expérimentaux sur ces données montrent que la méthode détecte les lésions visibles à la fois sur les images TEP et IRM, et que la segmentation contoure correctement la lésion (Dice, DH et DM valant respectivement 0, 85 ± 0, 09, 7, 28 ± 5, 42 mm et 0, 72 ± 0, 36mm). / The recent development of hybrid imaging combining Positron Emission Tomography (PET) and Magnetic Resonance Imaging (MRI) is an opportunity to exploit images of a same structure obtained simultaneously and providing complementary information. This also represents a real challenge due to the difference of nature and voxel size of the images. This new technology offers attractive prospects in oncology, and more precisely in neuro-oncology thanks to the contrast between the soft tissues provided by the MRI images. In this context, and as part of the PIM (Physics in Medicine) project of Paris-Saclay University, the goal of this thesis was to develop a multimodal segmentation pipeline adapted to PET and MRI images, including a tumor detection method in PET and MRI, and a segmentation method of the tumor in MRI. This process must be generic to be applied to multiple brain pathologies, of different nature, and for different clinical application. The first part of the thesis focuses on tumor detection using a hierarchical approach. More precisely, the detection method uses a new spatial context criterion applied on a max-tree representation of the MRI and PET images to select potential lesions. The second part presents a MRI tumor segmentation method using a variational approach. This method minimizes a globally convex energy function guided by PET information. Finally, the third part proposes an extension of the detection and segmentation methods developed previously to MRI multimodal segmentation, and also to longitudinal follow-up. The detection and segmentation methods were tested on images from several data bases, each of them standing for a specific brain pathology and PET radiotracer. The dataset used for PET-MRI detection and segmentation is composed of PET and MRI images of gliomas and meningiomas acquired from different systems, and images of brain lesions acquired on the hybrid PET-MRI system of Frédéric Joliot Hospital at Orsay. The detection method was also adapted to multimodal MRI imaging to detect multiple sclerosis lesions and follow-up studies. The results show that the proposed method, characterized by a generic approach using flexible parameters, can be adapted to multiple clinical applications. For example, the quality of the segmentation of images from the hybrid PET-MR system was assessed using the Dice coefficient, the Hausdorff distance (HD) and the average distance (AD) to a manual segmentation of the tumor validated by a medical expert. Experimental results on these datasets show that lesions visible on both PET and MR images are detected, and that the segmentation delineates precisely the tumor contours (Dice, HD and MD values of 0.85 ± 0.09, 7.28 ± 5.42 mm and 0.72 ± 0.36mm respectively).

L'hybridation des méthodes numériques est l'une des nombreuses pistes dans la recherche de la rapidité et de l'efficacité et de la précision d'une modélisation électromagnétique des structures complexes associant des parties de formes régulières de grandes dimensions électriques et des parties de formes complexes de dimensions plus modestes. Au lieu d'une seule formulation globale, on cherche à appliquer l'hybridation de plusieurs méthodes numériques notamment la méthode variationnelle multimodale (MVM), la méthode des éléments finis (FEM) et les réseaux de neurones artificiels. Un nouveau schéma hybride original qui combine la MVM et la FEM a été proposé pour caractériser une discontinuité complexe dans un guide d'onde rectangulaire. Les résultats obtenus tout en étant conformes aux résultats fournis par les simulateurs commerciaux et les résultats expérimentaux, apportent une amélioration sensible quant au temps de calcul. Le schéma hybride a été étendu pour la caractérisation des discontinuités complexes en cascade et appliqué à la conception de filtres micro-onde présentant des discontinuités complexes permettant ainsi un gain de temps très important. L'hybridation des réseaux de neurones artificiels et les méthodes modales a amélioré le temps de calcul pour l'analyse des discontinuités simples dans les guides d'onde rectangulaires ce qui a permis d'améliorer l'optimisation des filtres à guides d'ondes nervurés. / Hybridization of numerical methods is one inventive way in the research of the rapidity, the efficiency and the precision of the electromagnetic modeling of complex structures joining straight and large elements with complex and small ones. Instead of a global and unique formulation, we hybridize many numerical methods which are the modal methods, the finite element methods and the artificial neural networks. A novel computer- ided design (CAD) tool of complex passive microwave devices in rectangular waveguide technology is suggested. The multimodal variational method is applied to the full-wave description in the rectangular waveguides while the finite element analysis characterizes waves in the arbitrarily shaped discontinuities. The suggested hybrid approach is successfully applied to the full-wave analysis of complex discontinuities with great practical interest, thus improving CPU time and memory storage against several full-wave finite element method (FEM) based CAD tools. The proposed hybrid CAD tool is successfully extended to the design of filters with cascaded complex discontinuities. The hybridization of modal methods and the artificial neural networks improved the CPU time in the analysis of simple waveguide discontinuities which enhanced the optimization of rectangular ridged waveguide filters.

Estudamos a Eletrodinâmica de Wheeler-Feynman usando um princípio variacional para um funcional de ação finito acoplado a um problema de valor na fronteira. Para trajetórias C2 por trechos, a condição de ponto crítico desse funcional fornece as equações de movimento de Wheeler-Feynman mais uma condição de continuidade dos momentos parciais e energias parciais, conhecida como condição de quina de Weierstrass-Erdmann. Estudamos em detalhe um sub-caso mais simples, onde os dados de fronteira têm um comprimento mínimo. Nesse caso, mostramos que a condição de extremo se reduz a um problema de valor na chegada para uma equação diferencial com retardo misto dependente do estado e do tipo neutro. Resolvemos numericamente esse problema usando um método de shooting e um método de Runge-Kutta de quarta ordem. Para os casos em que as fronteiras mínimas têm velocidades descontínuas, elaboramos uma técnica para resolver as condições de quina de Weierstrass-Erdmann junto com o problema de valor na chegada. As trajetórias com velocidades descontínuas previstas pelo método variacional foram verificadas por experimentos numéricos. Em um segundo desenvolvimento, para o caso mais difícil de fronteiras de comprimento arbitrário, implementamos um método de minimização com gradiente fraco para o princípio variacional e problema de fronteira acima citado. Elaboramos dois métodos numéricos, ambos implementados em MATLAB, para encontrar soluções do problema eletromagnético de dois corpos. O primeiro combina o método de elementos finitos com o método de Newton para encontrar as soluções que anulam o gradiente fraco do funcional para fronteiras genéricas. O segundo usa o método do declive máximo para encontrar as soluções que minimizam a ação. Nesses dois métodos as trajetórias são aproximadas dentro de um espaço de dimensão finita gerado por uma Galerkiana que suporta velocidades descontínuas. Foram realizados diversos testes e experimentos numéricos para verificar a convergência das trajetórias calculada numericamente; também comparamos os valores do funcional calculados numericamente com alguns resultados analíticos sobre órbitas circulares. / We study the Wheeler-Feynman electrodynamics using a variational principle for an action functional coupled to a finite boundary value problem. For piecewise C2 trajectories, the critical point condition for this functional gives the Wheeler-Feynman equations of motion in addition to a continuity condition of partial moments and partial energies, known as the Weierstrass-Erdmann corner conditions. In the simplest case, for the boundary value problem of shortest length, we show that the critical point condition reduces to a two-point boundary value problem for a state-dependent mixed-type neutral differential-delay equation. We solve this special problem numerically using a shooting method and a fourth order Runge-Kutta. For the cases where the boundary segment has discontinuous velocities we developed a technique to solve the Weierstrass-Erdmann corner conditions and the two-point boundary value problem together. The trajectories with discontinuous velocities presupposed by the variational method were verified by numerical experiments. In a second development, for the harder case with boundaries of arbitrary length, we implemented a method of minimization with weak gradient for the variational principle quoted above. Two numerical methods were implemented in MATLAB to find solutions of the two-body electromagnetic problem. The first combines the finite element method with Newtons method to find the solutions that vanish the weak gradient. The second uses the method of steepest descent to find the solutions that minimize the action. In both methods the trajectories are approximated within a finite-dimensional space generated by a Galerkian that supports discontinuous velocities. Many tests and numerical experiments were performed to verify the convergence of the numerically calculated trajectories; also were compared the values of the functional computed numerically with some known analytical results on circular orbits.

<p>A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers enforcing constraints. Having finite energy, the multipliers are thus massless ghost particles. The main numerical stepping method produced with the framework is called SPOOK.</p><p>Variational integrators preserve physical invariants globally, exactly in some cases, approximately but within fixed global bounds for others. This allows to product realistic physical trajectories even with the low order methods. These are needed in the solution of nonsmooth problems such as dry frictional contacts and in addition, they are computationally inexpensive. The combination of strong stability, low order, and the global preservation of invariants allows for large integration time steps, but without loosing accuracy on the important and visible physical quantities. SPOOK is thus well-suited for interactive simulations, such as those commonly used in virtual environment applications, because it is fast, stable, and faithful to the physics.</p><p>New results include a stable discretization of highly oscillatory terms of constraint regularization; a linearly stable constraint stabilization scheme based on ghost potential and Rayleigh dissipation terms; a single-step, strictly dissipative, approximate impact model; a quasi-linear complementarity formulation of dry friction that is isotropic and solvable for any nonnegative value of friction coefficients; an analysis of a splitting scheme to solve frictional contact complementarity problems; a stable, quaternion-based rigid body stepping scheme and a stable linear approximation thereof. SPOOK includes all these elements. It is linearly implicit and linearly stable, it requires the solution of either one linear system of equations of one mixed linear complementarity problem per regular time step, and two of the same when an impact condition is detected. The changes in energy caused by constraints, impacts, and dry friction, are all shown to be strictly dissipative in comparison with the free system. Since all regularization and stabilization parameters are introduced in the physics, they map directly onto physical properties and thus allow modeling of a variety of phenomena, such as constraint compliance, for instance.</p><p>Tutorial material is included for continuous and discrete-time analytic mechanics, quaternion algebra, complementarity problems, rigid body dynamics, constraint kinematics, and special topics in numerical linear algebra needed in the solution of the stepping equations of SPOOK.</p><p>The qualitative and quantitative aspects of SPOOK are demonstrated by comparison with a variety of standard techniques on well known test cases which are analyzed in details. SPOOK compares favorably for all these examples. In particular, it handles ill-posed and degenerate problems seamlessly and systematically. An implementation suitable for large scale performance and accuracy testing is left for future work.</p>

A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers enforcing constraints. Having finite energy, the multipliers are thus massless ghost particles. The main numerical stepping method produced with the framework is called SPOOK. Variational integrators preserve physical invariants globally, exactly in some cases, approximately but within fixed global bounds for others. This allows to product realistic physical trajectories even with the low order methods. These are needed in the solution of nonsmooth problems such as dry frictional contacts and in addition, they are computationally inexpensive. The combination of strong stability, low order, and the global preservation of invariants allows for large integration time steps, but without loosing accuracy on the important and visible physical quantities. SPOOK is thus well-suited for interactive simulations, such as those commonly used in virtual environment applications, because it is fast, stable, and faithful to the physics. New results include a stable discretization of highly oscillatory terms of constraint regularization; a linearly stable constraint stabilization scheme based on ghost potential and Rayleigh dissipation terms; a single-step, strictly dissipative, approximate impact model; a quasi-linear complementarity formulation of dry friction that is isotropic and solvable for any nonnegative value of friction coefficients; an analysis of a splitting scheme to solve frictional contact complementarity problems; a stable, quaternion-based rigid body stepping scheme and a stable linear approximation thereof. SPOOK includes all these elements. It is linearly implicit and linearly stable, it requires the solution of either one linear system of equations of one mixed linear complementarity problem per regular time step, and two of the same when an impact condition is detected. The changes in energy caused by constraints, impacts, and dry friction, are all shown to be strictly dissipative in comparison with the free system. Since all regularization and stabilization parameters are introduced in the physics, they map directly onto physical properties and thus allow modeling of a variety of phenomena, such as constraint compliance, for instance. Tutorial material is included for continuous and discrete-time analytic mechanics, quaternion algebra, complementarity problems, rigid body dynamics, constraint kinematics, and special topics in numerical linear algebra needed in the solution of the stepping equations of SPOOK. The qualitative and quantitative aspects of SPOOK are demonstrated by comparison with a variety of standard techniques on well known test cases which are analyzed in details. SPOOK compares favorably for all these examples. In particular, it handles ill-posed and degenerate problems seamlessly and systematically. An implementation suitable for large scale performance and accuracy testing is left for future work.

Estudamos a Eletrodinâmica de Wheeler-Feynman usando um princípio variacional para um funcional de ação finito acoplado a um problema de valor na fronteira. Para trajetórias C2 por trechos, a condição de ponto crítico desse funcional fornece as equações de movimento de Wheeler-Feynman mais uma condição de continuidade dos momentos parciais e energias parciais, conhecida como condição de quina de Weierstrass-Erdmann. Estudamos em detalhe um sub-caso mais simples, onde os dados de fronteira têm um comprimento mínimo. Nesse caso, mostramos que a condição de extremo se reduz a um problema de valor na chegada para uma equação diferencial com retardo misto dependente do estado e do tipo neutro. Resolvemos numericamente esse problema usando um método de shooting e um método de Runge-Kutta de quarta ordem. Para os casos em que as fronteiras mínimas têm velocidades descontínuas, elaboramos uma técnica para resolver as condições de quina de Weierstrass-Erdmann junto com o problema de valor na chegada. As trajetórias com velocidades descontínuas previstas pelo método variacional foram verificadas por experimentos numéricos. Em um segundo desenvolvimento, para o caso mais difícil de fronteiras de comprimento arbitrário, implementamos um método de minimização com gradiente fraco para o princípio variacional e problema de fronteira acima citado. Elaboramos dois métodos numéricos, ambos implementados em MATLAB, para encontrar soluções do problema eletromagnético de dois corpos. O primeiro combina o método de elementos finitos com o método de Newton para encontrar as soluções que anulam o gradiente fraco do funcional para fronteiras genéricas. O segundo usa o método do declive máximo para encontrar as soluções que minimizam a ação. Nesses dois métodos as trajetórias são aproximadas dentro de um espaço de dimensão finita gerado por uma Galerkiana que suporta velocidades descontínuas. Foram realizados diversos testes e experimentos numéricos para verificar a convergência das trajetórias calculada numericamente; também comparamos os valores do funcional calculados numericamente com alguns resultados analíticos sobre órbitas circulares. / We study the Wheeler-Feynman electrodynamics using a variational principle for an action functional coupled to a finite boundary value problem. For piecewise C2 trajectories, the critical point condition for this functional gives the Wheeler-Feynman equations of motion in addition to a continuity condition of partial moments and partial energies, known as the Weierstrass-Erdmann corner conditions. In the simplest case, for the boundary value problem of shortest length, we show that the critical point condition reduces to a two-point boundary value problem for a state-dependent mixed-type neutral differential-delay equation. We solve this special problem numerically using a shooting method and a fourth order Runge-Kutta. For the cases where the boundary segment has discontinuous velocities we developed a technique to solve the Weierstrass-Erdmann corner conditions and the two-point boundary value problem together. The trajectories with discontinuous velocities presupposed by the variational method were verified by numerical experiments. In a second development, for the harder case with boundaries of arbitrary length, we implemented a method of minimization with weak gradient for the variational principle quoted above. Two numerical methods were implemented in MATLAB to find solutions of the two-body electromagnetic problem. The first combines the finite element method with Newtons method to find the solutions that vanish the weak gradient. The second uses the method of steepest descent to find the solutions that minimize the action. In both methods the trajectories are approximated within a finite-dimensional space generated by a Galerkian that supports discontinuous velocities. Many tests and numerical experiments were performed to verify the convergence of the numerically calculated trajectories; also were compared the values of the functional computed numerically with some known analytical results on circular orbits.