Dans cette thèse, nous présentons des méthodes asymptotiques <br />mathématiquement justifiées permettant de connaître les champs <br />électromagnétiques dans des milieux à couches minces hétérogènes. <br />La motivation de ce travail est le calcul du champ électrique dans des <br />cellules biologiques composées d'un cytoplasme conducteur entouré <br />d'une fine membrane très isolante. <br />Nous remplaçons la membrane, lorsque son épaisseur est infiniment <br />petite, par des conditions de transmission ou des conditions aux <br />limites appropriées et nous estimons l'erreur commise par ces <br />approximations.<br /> Pour les basses fréquences, nous considérons l'équation quasistatique<br />donnant le potentiel dont dérive le champ. A l'aide d'un <br />calcul en géométrie circulaire nous obtenons les expressions explicites<br /> du potentiel et nous en déduisons les asymptotiques du champ <br />électrique, en fonction de l'épaisseur de la couche mince, avec des <br />estimations de l'erreur. Nous estimons ensuite la différence entre le <br />champ réel et le champ statique. Puis nous généralisons notre <br />développement asymptotique à une géométrie quelconque. <br /> La deuxième partie de cette thèse traite des moyennes fréquences : <br />nous donnons le développement asymptotique de la solution de <br />l'équation de Helmholtz lorsque l'épaisseur de la membrane tend vers <br />0. Tous ces précédents résultats sont illustrés par des calculs par <br />éléments finis.<br /> Enfin, pour les hautes fréquences, nous construisons une condition <br />d'impédance pseudodifférentielle permettant de concentrer l'effet de <br />la couche sur son bord intérieur. Nous concluons cette thèse par un <br />problème de diffraction à haute fréquence d'une onde incidente par <br />un disque de petite taille. A l'aide d'une analyse pseudodifférentielle, <br />nous bornons la norme de la trace du champ diffracté à distance fixe <br />de l'inhomogénéité en fonction de la taille de l'objet et de l'onde <br />incidente.

The main objective of this research is the study of the magnetization of ferromagnetic spheres in the presence of external magnetic fields. The exact analytical solutions derived in this thesis are benchmark solutions, valuable in testing the correctness and accuracy of various approximate models and numerical methods. The total scalar magnetic potential outside the spheres, related to the magnetic field intensity, is obtained by the superposition of the potentials due to all spheres and the potential corresponding to the external field. The translational addition theorems for scalar Laplacian functions are used to solve boundary value by imposing exact boundary conditions. The scalar magnetic potential inside each sphere, related to the magnetic flux density, also satisfies the Laplace equation, which is solved by imposing the boundary conditions known from the solution of the outside field. Finally, the expressions derived are used to generate numerical results of controllable accuracy for field quantities.

In this thesis we study boundary value problems for the Laplace equation on do mains with smooth boundary. Central to our analysis is a relation, known as the global relation, that couples the boundary data for a given BVP. Previously, the global re lation has primarily been applied to elliptic PDEs defined on polygonal domains. In this thesis we extend the use of the global relation to domains with smooth boundary. This is done by introducing a new transform, denoted by F_p, that is an analogue of the Fourier transform on smooth convex curves. We show that the F_p-transform is a bounded and invertible integral operator. Following this, we show that the F_p-transform naturally arises in the global relation for the Laplace equation on domains with smooth boundary. Using properties of the F_p-transform, we show that the global relation defines a continuously invertible map between the Dirichlet and Neumann data for a given BVP for the Laplace equation. Following this, we construct a numerical method that uses the global relation to find the Neumann data, given the Dirichlet data, for a given BVP for the Laplace equation on a domain with smooth boundary.

No description available.

This thesis presents the numerical algorithms to carry out the face transformation. The main efforts are denoted to the finite volume method (FVM) with the Delaunay triangulation to solve the Laplace equations in the harmonic transformation undergone in face images. The advantages of the FVM with the Delaunay triangulation are: (1) Easy to formulate the linear algebraic equations, (2) Good to retain the geometric and physical properties, (3) less CPU time needed. The numerical and graphical experiments are reported for the face transformations from a female to a male, and vice versa. The computed sequential and absolute errors are and , where N is division number of a pixel into subpixels. Such computed errors coincide with the analysis on the splitting-shooting method (SSM) with piecewise constant interpolation in [Li and Bui, 1998c].

A method for trimming surfaces generated as solutions to Partial Differential Equations (PDEs) is presented. The work we present here utilises the 2D parameter space on which the trim curves are defined whose projection on the parametrically represented PDE surface is then trimmed out. To do this we define the trim curves to be a set of boundary conditions which enable us to solve a low order elliptic PDE on the parameter space. The chosen elliptic PDE is solved analytically, even in the case of a very general complex trim, allowing the design process to be carried out interactively in real time. To demonstrate the capability for this technique we discuss a series of examples where trimmed PDE surfaces may be applicable.

O principal objetivo deste trabalho é estudar as singularidades e hipersingularidades existentes nas formulações: singular - clássica - e hipersingular no Método dos Elementos de Contorno (MEC). Também é proposto um esquema residual h-adaptativo para a solução numérica do problema físico governado pela equação de Laplace. Usa-se malha poligonal, juntamente, com funções de interpolação - distribuição - de forma, dos tipos: constantes e lineares. Para controlar o erro a posteriori, é considerado o valor do resíduo, fora dos pontos de colocação. Também é testada uma técnica de quadratura numérica chamada adaptativa, específica para subelementos, no sentido de verificar se a precisão no cálculo das integrais com singularidades é melhorada. O uso de funções hierárquicas é discutido na forma de um algoritmo para atualização da matriz principal do sistema linear. / The main purpose of this work is to study the existing singularities and hypersingularities in the Boundary Element Method (BEM) with singular - classical - and hypersingular formulations. Also, an h-adaptive residual scheme for the numerical solution of the physical problem, driven by Laplace equation, is proposed. Boundary polygonal mesh, with constant and linear interpolation - distribution - shape functions together are used. To control the a posteriori error, is considered the residue value outside the collocation points. Also, a sub-element specific adaptive numerical quadrature technique, in an effort to verify if the precision when dealing with integrals possessing singularities is increased, is tested. The use of hierarchical functions is discussed, as an algorithm to update the linear system main matrix.

<p>Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(<i>u</i>) ≡ <i>u</i>²_x<i>u</i>_xx+ 2<i>u</i>_x<i>u</i>_y<i>u</i>_xy+<i>u</i>²_y<i>u</i>_yy= 0. For numerical reasons ∆<i>q</i>(<i>u</i>) = div (|▼<i>u</i>|<i>q</i>▼<i>u</i>)<i> = </i>0, which (formally) approaches as ∆∞(<i>u</i>) = 0 as <i>q</i> → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(<i>u</i>) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.</p>

Using the finite element software FEMLAB® solutions are computed to Dirichlet problems for the Infinity-Laplace equation ∆∞(u) ≡ u2xuxx + 2uxuyuxy + u2yuyy = 0. For numerical reasons ∆q(u) = div (|▼u|q▼u) = 0, which (formally) approaches as ∆∞(u) = 0 as q → ∞, is used in computation. A parametric nonlinear solver is used, which employs a variant of the damped Newton-Gauss method. The analysis of the experiments is done using the known theory of solutions to Dirichlet problems for ∆∞(u) = 0, which includes AMLEs (Absolutely Minimizing Lipschitz Extensions), sets of uniqueness, critical segments and lines of singularity. From the experiments one main conjecture is formulated: For Dirichlet problems, which have a non-constant boundary function, it is possible to predict the structure of the lines of singularity in solutions in the Infinity-Laplace case by examining the corresponding Laplace case.

O principal objetivo deste trabalho é estudar as singularidades e hipersingularidades existentes nas formulações: singular - clássica - e hipersingular no Método dos Elementos de Contorno (MEC). Também é proposto um esquema residual h-adaptativo para a solução numérica do problema físico governado pela equação de Laplace. Usa-se malha poligonal, juntamente, com funções de interpolação - distribuição - de forma, dos tipos: constantes e lineares. Para controlar o erro a posteriori, é considerado o valor do resíduo, fora dos pontos de colocação. Também é testada uma técnica de quadratura numérica chamada adaptativa, específica para subelementos, no sentido de verificar se a precisão no cálculo das integrais com singularidades é melhorada. O uso de funções hierárquicas é discutido na forma de um algoritmo para atualização da matriz principal do sistema linear. / The main purpose of this work is to study the existing singularities and hypersingularities in the Boundary Element Method (BEM) with singular - classical - and hypersingular formulations. Also, an h-adaptive residual scheme for the numerical solution of the physical problem, driven by Laplace equation, is proposed. Boundary polygonal mesh, with constant and linear interpolation - distribution - shape functions together are used. To control the a posteriori error, is considered the residue value outside the collocation points. Also, a sub-element specific adaptive numerical quadrature technique, in an effort to verify if the precision when dealing with integrals possessing singularities is increased, is tested. The use of hierarchical functions is discussed, as an algorithm to update the linear system main matrix.