Les courbes de Bezier et les courbes splines ont trouve un cadre de présentation simple et naturel avec la notion de floraison d'une fonction polynomiale, telle qu'elle a été présentée dans les travaux de Lyle Ramshaw (1987). Notre but a consiste a étendre cette présentation au cas des surfaces et aussi des variétés de dimension supérieure. Les splines simpliciales sont une généralisation naturelle des b-splines au cas de plusieurs variables. Nous présentons leurs principales propriétés ainsi qu'une définition de différences divisées pour fonctions de plusieurs variables. Un algorithme d'évaluation d'une spline simpliciale est propose et teste. Floraisons et splines simpliciales sont les éléments essentiels d'un nouveau schéma de b-splines introduit par Dahmen, Micchelli et Seidel (1992). Ce schéma est étudié et ses principales propriétés sont présentées. Une grande similarité avec l'étude des courbes est retrouvée

De récentes révolutions dans la technologie des lasers infrarouges permettent d'exposer la matière à des champs laser pulsés ultra intenses ( de 1013 W cm-2 à 1016 W cm-2 ) et ultra courts (de l'ordre de quelques femtosecondes). Nous nous intéressons dans cette étude théorique à la dynamique de l'interaction entre de tels champs lasers basse fréquence et l'hydrogène atomique modélisant un système à un électron actif. Nous étudions les deux processus en compétition lors d'une telle interaction. D'une part, la génération d'harmoniques d'ordre élevé et particulièrement les effets de la phase absolue du champ laser sur les spectres d'émission ainsi que le rôle joué par la structure atomique sont analysés. D'autre part, la dynamique de l'ionisation dans le régime tunnel ou au-dessus de la barrière fait l'objet d'une étude détaillée dans le cadre de l'approximation du champ fort en sondant le rôle du potentiel Coulombien dans le mécanisme d'ionisation. Théoriquement, deux démarches sont envisagées. L'équation de Schrödinger dépendante du temps peut être résolue exactement numériquement. D'autre part, une solution (analytique) approchée peut être déterminée en s'appuyant sur la méthode aux états atomiques essentiels ainsi que sur les rôles effectifs du champ électrique et du potentiel Coulombien. Cette approximation du champ fort, introduite par Keldysh en 1965, dans laquelle l'effet du champ électrique domine la dynamique de l'interaction a permis d'expliquer l'allure d'une partie des spectres des harmoniques émises par l'atome et des spectres des électrons éjectés. Elle postule que le mécanisme d'ionisation, aux basses fréquences considérées est l'éjection d'électrons par effet tunnel suite à laquelle l'analyse du mouvement de l'électron fait abstraction de la présence du potentiel Coulombien. L'électron peut être considéré comme un électron libre oscillant classiquement dans le champ laser. Ce mouvement quasi-classique lui permet éventuellement de revenir vers le noyau résiduel et de se recombiner radiativement (émission d'une harmonique) avec l'état fondamental ou d'être rediffusé par le potentiel. Ces mécanismes permettent effectivement de comprendre qualitativement l'allure des spectres. Toutefois les prédictions des taux d'ionisation ou de l'amplitude des harmoniques émises ne coïncident pas quantitativement avec les mesures expérimentales. Dans un premier temps, nous tirons profit de l'accord qualitatif entre les deux méthodes en ce qui concerne la génération d'harmoniques d'ordre élevé. Dans le cadre d'impulsions ultra-courtes, l'interaction ayant lieu pendant quelques périodes laser uniquement, la phase absolue du champ laser modifie sensiblement la dynamique de l'interaction d'un point de vue énergétique et par conséquent les spectres d'harmoniques émises. Une analyse temps-fréquence du signal harmonique émis par un seul atome nous permet de montrer que l'influence de la phase peut être comprise classiquement. Nous suggérons une méthode de diagnostic de ce paramètre non-adiabatique qui, jusqu'à présent, ne fait l'objet que d'une stabilisation et non d'un contrôle à l'échelle expérimentale. Nous considérons d'autre part, le point de vue macroscopique, c'est-à-dire la propagation des champs harmoniques dans le milieu gazeux partiellement ionisé en résolvant les équations de propagation de Maxwell. Nous constatons une survie de l'influence de la phase absolue pour des interactions inférieures à une dizaine de cycles optiques. Par ailleurs, l'approximation du champ fort, que nous avons étudiée dans le cadre de la génération d'harmoniques d'ordre élevé par un atome soumis à une impulsion laser de quelques femtosecondes, nous permet de comprendre l'importance du moment canonique dans la dynamique de l'interaction. La représentation des processus atomiques en terme de moment que nous effectuons s'avère être une remarquable sonde des effets réels du potentiel Coulombien sur la dynamique du mouvement des électrons. Nous développons une méthode de résolution de l'équation de Schrödinger dans l'espace des moments; nos résultats démontrent que, du point de vue de l'ionisation, les contributions dominantes ne sont pas celles décrites par l'approximation du champ fort mais qu'en revanche, la présence du potentiel Coulombien ne peut être négligée lorsque nous voulons décrire le mécanisme d'ionisation ; et ce même si l'on s'approche de l'intensité de saturation au-delà de laquelle l'atome s'ionise en moins d'une période laser. Notre étude replace en quelque sorte le potentiel Coulombien au centre du processus d'ionisation malgré l'idée consensuelle selon laquelle aux basses fréquences considérées, l'ionisation par le champ (effet tunnel ou ionisation au-dessus de la barrière) est dominante.

abstract: Current trends in the Computer Aided Engineering (CAE) involve the integration of legacy mesh-based finite element software with newer solid-modeling kernels or full CAD systems in order to simplify laborious or highly specialized tasks in engineering analysis. In particular, mesh generation is becoming increasingly automated. In addition, emphasis is increasingly placed on full assembly (multi-part) models, which in turn necessitates an automated approach to contact analysis. This task is challenging due to increases in algebraic system size, as well as increases in the number of distorted elements - both of which necessitate manual intervention to maintain accuracy and conserve computer resources. In this investigation, it is demonstrated that the use of a mesh-free B-Spline finite element basis for structural contact problems results in significantly smaller algebraic systems than mesh-based approaches for similar grid spacings. The relative error in calculated contact pressure is evaluated for simple two dimensional smooth domains at discrete points within the contact zone and compared to the analytical Hertz solution, as well as traditional mesh-based finite element solutions for similar grid spacings. For smooth curved domains, the relative error in contact pressure is shown to be less than for bi-quadratic Serendipity elements. The finite element formulation draws on some recent innovations, in which the domain to be analyzed is integrated with the use of transformed Gauss points within the domain, and boundary conditions are applied via distance functions (R-functions). However, the basis is stabilized through a novel selective normalization procedure. In addition, a novel contact algorithm is presented in which the B-Spline support grid is re-used for contact detection. The algorithm is demonstrated for two simple 2-dimensional assemblies. Finally, a modified Penalty Method is demonstrated for connecting elements with incompatible bases. / Dissertation/Thesis / Ph.D. Mechanical Engineering 2010

Apresenta-se uma formulação de adaptação de malha para problemas com grandes deformações. A formulação Arbitrária Lagrangeana-Euleraina (ALE) permite manter a qualidade dos elementos finitos durante o processo de cálculo através de rearranjo ou movimento de malha independente do movimento material. Nas formulações Lagrangeanas a malha fica “colada” ao corpo durante toda a análise, logo quando este sofre grandes deformações diferenciais o mesmo se reproduz numa malha distorcida. A formulação ALE desacoplada consta de dois passos: O passo Lagrangeano onde são aplicados os incrementos de carga, a malha permanece “colada” à matéria durante a análise. E cada certo “tempo” o passo Euleriano onde “descola-se” a malha da matéria e efetua-se o movimento de malha que se ajusta melhor ao corpo deformado. São apresentados assim métodos de realocação da malha e transferência ou atualização das variáveis necessárias para, depois do passo Euleriano, continuar a análise com a nova malha sem grandes distorções dos elementos. Os problemas de grandes deformações e deslocamentos são acompanhados de não linearidades físicas e geométricas, assim, são abordados os métodos para o tratamento destas não linearidades. Trabalha-se com o elemento hexaédrico tri-linear com integração reduzida e controle dos modos espúrios que tem demostrado um bom comportamento frente a grandes não linearidades geométricas assim como para as não linearidades físicas. A formulação ALE tem ganhando seu espaço na mecânica dos sólidos, em problemas de conformação mecânica e impacto, devido às grandes deformações e na última década está abrindo-se passo na área da geomecânica tratando problemas recalque e penetração de fundações em solos. / This work presents remeshing techniques for finite element simulation and investigates their performance for large deformation problems. Lagrangian formulation generally results in excessive mesh distortion owing to its attachment to the material. Meanwhile, the Lagrangian- Eulerian (ALE) formulation alouds to keep the finite element quality through the arbitrarily rearrangement or movement of the mesh, to optimize the element’s shape. The decoupled Arbitrary Lagrangian-Eulerian approach consists in a sequence of Lagrangian and Eulerian steps. The mesh is “coupled” to the material during the Lagrangian steps. From step to step, mesh is decoupled from the system material (Eulerian step), the nodes corresponding to free boundaries are relocated using an analytical approach, remeshing is performed and finally the state variables are remapped. Rearrangements methods for the element’s node are presented, as well as the variables remapping algorithms at the new quadrature points, in order to continue with the finite element analysis without altering the element topology of the original mesh. Special attention is given to methods dealing with geometric and physical nonlinearities. A trilinear hexahedral element is used with reduced integration and hourglass control. This combination has shown well behavior in front of large geometric and physical nonlinearities. ALE formulation field has considerably grown in geotechnical research, especially in impact and mechanical extrusion problems. Over the last decade, geomechanic is dealing with settlement problems and foundation penetration in soils.

Apresenta-se uma formulação de adaptação de malha para problemas com grandes deformações. A formulação Arbitrária Lagrangeana-Euleraina (ALE) permite manter a qualidade dos elementos finitos durante o processo de cálculo através de rearranjo ou movimento de malha independente do movimento material. Nas formulações Lagrangeanas a malha fica “colada” ao corpo durante toda a análise, logo quando este sofre grandes deformações diferenciais o mesmo se reproduz numa malha distorcida. A formulação ALE desacoplada consta de dois passos: O passo Lagrangeano onde são aplicados os incrementos de carga, a malha permanece “colada” à matéria durante a análise. E cada certo “tempo” o passo Euleriano onde “descola-se” a malha da matéria e efetua-se o movimento de malha que se ajusta melhor ao corpo deformado. São apresentados assim métodos de realocação da malha e transferência ou atualização das variáveis necessárias para, depois do passo Euleriano, continuar a análise com a nova malha sem grandes distorções dos elementos. Os problemas de grandes deformações e deslocamentos são acompanhados de não linearidades físicas e geométricas, assim, são abordados os métodos para o tratamento destas não linearidades. Trabalha-se com o elemento hexaédrico tri-linear com integração reduzida e controle dos modos espúrios que tem demostrado um bom comportamento frente a grandes não linearidades geométricas assim como para as não linearidades físicas. A formulação ALE tem ganhando seu espaço na mecânica dos sólidos, em problemas de conformação mecânica e impacto, devido às grandes deformações e na última década está abrindo-se passo na área da geomecânica tratando problemas recalque e penetração de fundações em solos. / This work presents remeshing techniques for finite element simulation and investigates their performance for large deformation problems. Lagrangian formulation generally results in excessive mesh distortion owing to its attachment to the material. Meanwhile, the Lagrangian- Eulerian (ALE) formulation alouds to keep the finite element quality through the arbitrarily rearrangement or movement of the mesh, to optimize the element’s shape. The decoupled Arbitrary Lagrangian-Eulerian approach consists in a sequence of Lagrangian and Eulerian steps. The mesh is “coupled” to the material during the Lagrangian steps. From step to step, mesh is decoupled from the system material (Eulerian step), the nodes corresponding to free boundaries are relocated using an analytical approach, remeshing is performed and finally the state variables are remapped. Rearrangements methods for the element’s node are presented, as well as the variables remapping algorithms at the new quadrature points, in order to continue with the finite element analysis without altering the element topology of the original mesh. Special attention is given to methods dealing with geometric and physical nonlinearities. A trilinear hexahedral element is used with reduced integration and hourglass control. This combination has shown well behavior in front of large geometric and physical nonlinearities. ALE formulation field has considerably grown in geotechnical research, especially in impact and mechanical extrusion problems. Over the last decade, geomechanic is dealing with settlement problems and foundation penetration in soils.

Devido à constante necessidade de construções de novas embarcações, quer seja pela demanda do mercado, quer seja pela renovação da frota, o desenvolvimento de programas computacionais que auxiliem na fase inicial de projeto torna-se bastante útil. Assim, o desenvolvimento de um procedimento de análise que permita obter formas de melhor desempenho vem a agregar valor nesta etapa de conceituação da geometria do navio. O trabalho aqui apresentado tem como objetivo discorrer sobre um método capaz de otimizar a geometria de um casco de deslocamento conhecido em relação a sua resistência ao avanço, sem perder, porém, as suas características principais, como corpo paralelo médio, por exemplo. Para tanto, dentro deste processo de otimização já estão inseridas algumas restrições que garantem a viabilidade da solução final, tais como variação máxima no comprimento, no volume total e na estabilidade do navio. A modelagem da embarcação pode ser feita através de funções B-Splines cúbicas de superfície, cujos pontos de controle (parâmetros inerentes à função) podem ser modificados de tal sorte a atingir um valor ótimo para a resistência ao avanço. Esta, por sua vez, será obtida através da soma de duas parcelas, sendo uma referente ao atrito e outra à geração de ondas pelo casco. Como a maior parte da resistência provém desta segunda parcela para a velocidade de projeto a ser considerada (alto número de Froude), a redução da resistência total pode ser assumida como conseqüência da diminuição da resistência devido à geração de ondas, a qual pode ser obtida através da formulação apresentada por Michell, em 1898. O cálculo das propriedades hidrostáticas como deslocamento, estabilidade ( KM transversal) e superfície molhada, usada para cálculo da resistência ao avanço, pode ser encontrado fazendo-se uso do cálculo vetorial. O procedimento a ser descrito foi desenvolvido em linguagem C++ (modelagem do casco) e com o auxílio do MATLAB® (método de otimização). Este trabalho foi realizado no Dep. de Eng. Naval e Oceânica da USP. / Due to an increasing necessity of building new vessels, whether by new orders or fleet renewal, the development of computational programs that could allow optimization of hull shapes is always helpful, saving project time and ensuring better performance at sea. Thus, the development of a synthesis procedure that allows obtaining shapes with better performance adds value to the initial phase of the ship geometry concept. The work to be presented herein objectives the presentation of a methodology to achieve optimal shapes for displacement hulls in relation to the total resistance, starting from an initial geometry given, describing hull form and applying specific constraints to optimization problem with the purpose of guarantee a reliable solution. Therefore, inside this optimization process there are included some constraints that ensure a feasible final solution, as maximum variation of ship length, total volume and stability. Hull geometry is described by using B-Spline surface functions and the ship wave resistance is calculated using Michell\'s formulation as a first approximation of the total resistance for high Froude numbers. Once vessel surface is well defined, B-Spline parameters are varied until an optimal form is attained and the minimum resistance is achieved. It can take a little time to calculate, depending on ship definition (number of buttocks and waterlines) and the problem complexity (number of constraints and variables). Ship displacement and other hydrostatic properties as stability, given by transversal KM , wetted surface, used for calculating ship resistance, can be obtained using the vectorial calculus. This work has been developed using C++ language, except the optimization process which makes use of a MATLAB® function called fmincon. This study has been held at the Department of Naval and Ocean Engineering of the University of São Paulo, Brazil.

Apresenta-se uma formulação de adaptação de malha para problemas com grandes deformações. A formulação Arbitrária Lagrangeana-Euleraina (ALE) permite manter a qualidade dos elementos finitos durante o processo de cálculo através de rearranjo ou movimento de malha independente do movimento material. Nas formulações Lagrangeanas a malha fica “colada” ao corpo durante toda a análise, logo quando este sofre grandes deformações diferenciais o mesmo se reproduz numa malha distorcida. A formulação ALE desacoplada consta de dois passos: O passo Lagrangeano onde são aplicados os incrementos de carga, a malha permanece “colada” à matéria durante a análise. E cada certo “tempo” o passo Euleriano onde “descola-se” a malha da matéria e efetua-se o movimento de malha que se ajusta melhor ao corpo deformado. São apresentados assim métodos de realocação da malha e transferência ou atualização das variáveis necessárias para, depois do passo Euleriano, continuar a análise com a nova malha sem grandes distorções dos elementos. Os problemas de grandes deformações e deslocamentos são acompanhados de não linearidades físicas e geométricas, assim, são abordados os métodos para o tratamento destas não linearidades. Trabalha-se com o elemento hexaédrico tri-linear com integração reduzida e controle dos modos espúrios que tem demostrado um bom comportamento frente a grandes não linearidades geométricas assim como para as não linearidades físicas. A formulação ALE tem ganhando seu espaço na mecânica dos sólidos, em problemas de conformação mecânica e impacto, devido às grandes deformações e na última década está abrindo-se passo na área da geomecânica tratando problemas recalque e penetração de fundações em solos. / This work presents remeshing techniques for finite element simulation and investigates their performance for large deformation problems. Lagrangian formulation generally results in excessive mesh distortion owing to its attachment to the material. Meanwhile, the Lagrangian- Eulerian (ALE) formulation alouds to keep the finite element quality through the arbitrarily rearrangement or movement of the mesh, to optimize the element’s shape. The decoupled Arbitrary Lagrangian-Eulerian approach consists in a sequence of Lagrangian and Eulerian steps. The mesh is “coupled” to the material during the Lagrangian steps. From step to step, mesh is decoupled from the system material (Eulerian step), the nodes corresponding to free boundaries are relocated using an analytical approach, remeshing is performed and finally the state variables are remapped. Rearrangements methods for the element’s node are presented, as well as the variables remapping algorithms at the new quadrature points, in order to continue with the finite element analysis without altering the element topology of the original mesh. Special attention is given to methods dealing with geometric and physical nonlinearities. A trilinear hexahedral element is used with reduced integration and hourglass control. This combination has shown well behavior in front of large geometric and physical nonlinearities. ALE formulation field has considerably grown in geotechnical research, especially in impact and mechanical extrusion problems. Over the last decade, geomechanic is dealing with settlement problems and foundation penetration in soils.

The goal of this thesis is to nd types of curves which would allow for the construction of a path that could be traversed by a vehicle. It seems that a minimal constraint for such a path is the continuity of curve's curvature. This leads to a closer look at the three types of curves: Clothoids, which are able to smoothly connect straights with arcs of a constant curvature, interpolation quintic splines, which are C2 smooth in the interpolation nodes and -splines, these belong to the family of quintic polynomial curves too, however, they are characterised by the vector of parameters which modies the shape of the curve. The thesis is accompanied by an application allowing for manual construction of the path composed of spline curves.

In dem aktuellen Beitrag wird eine kurze Einführung in die Mischung bzw. Entmischung hochviskoser inkompressibler Fluide gegeben. Hierzu wird die Methode der Phasenfeldmodellierung und ihre numerische Diskretisierung vorgestellt. Anhand eines konkreten Beispiels wird die technische Relevanz des vorgestellten Ansatzes demonstriert.

<div>Many practical applications benefit from the reconstruction of a smooth multivariate function from discrete data for purposes such as reducing file size or improving analytic and visualization performance. Among the different reconstruction methods, tensor product B-spline has a number of advantageous properties over alternative data representation. However, the problem of constructing a best-fit B-spline approximation effectively contains many roadblocks. Within the many free parameters in the B-spline model, the choice of the knot vectors, which defines the separation of each piecewise polynomial patch in a B-spline construction, has a major influence on the resulting reconstruction quality. Yet existing knot placement methods are still ineffective, computationally expensive, or impose limitations on the dataset format or the B-spline order. Moving beyond the 1D cases (curves) and onto higher dimensional datasets (surfaces, volumes, hypervolumes) introduces additional computational challenges as well. Further complications also arise in the case of undersampled data points where the approximation problem can become ill-posed and existing regularization proves unsatisfactory.</div><div><br></div><div>This dissertation is concerned with improving the efficiency and accuracy of the construction of a B-spline approximation on discrete data. Specifically, we present a novel B-splines knot placement approach for accurate reconstruction of discretely sampled data, first in 1D, then extended to higher dimensions for both structured and unstructured formats. Our knot placement methods take into account the feature or complexity of the input data by estimating its high-order derivatives such that the resulting approximation is highly accurate with a low number of control points. We demonstrate our method on various 1D to 3D structured and unstructured datasets, including synthetic, simulation, and captured data. We compare our method with state-of-the-art knot placement methods and show that our approach achieves higher accuracy while requiring fewer B-spline control points. We discuss a regression approach to the selection of the number of knots for multivariate data given a target error threshold. In the case of the reconstruction of irregularly sampled data, where the linear system often becomes ill-posed, we propose a locally varying regularization scheme to address cases for which a straightforward regularization fails to produce a satisfactory reconstruction.</div>