La thèse constitue une juxtaposition de plusieurs contributions originales à lélaboration et lanalyse de modèles numériques capables de décrire une vaste gamme découlements à surface libre ainsi que les phénomènes de transport associés. Deux axes principaux sous-tendent les recherches entreprises. Il sagit, dune part, dune contribution à létude de modèles visant à reproduire adéquatement les interactions du fluide avec divers constituants transportés, tels que de lair entraîné, un polluant ou des sédiments. Cette phase du travail inclut également la caractérisation et la prédiction du comportement de lécoulement en présence dune topographie mobile ou érodable, y compris dans le cas dun barrage en remblai subissant une surverse. Dautre part, partant du constat quune modélisation fidèle des processus de transport, notamment hydrosédimentaires, passe inévitablement par un raffinement du calcul hydrodynamique proprement dit, une partie des travaux effectués dans le cadre de cette thèse est spécifiquement orientée vers un enrichissement de la connaissance des champs hydrodynamiques au sein du modèle.

Diffusion Tensor MRI can be used to depict the anisotropy of tissue. Translation of this technique to moving objects such as the beating heart has recently become feasible, but remains a challenging task, often leading to high noise levels and limited accuracy. Ultimately, knowledge of the 3D fibre architecture of the myocardium invivo should allow for a better understanding of the cardiac function both in healthy and pathological situations. The main goal of the work presented in this thesis is to overcome the difficulties that such technology presents, by introducing a combination of image processing and analysis approaches. In particular, the characteristic ellipsoidal shape of the left ventricular chamber is used to introduce a shape-based prolate spheroidal coordinate frame that allows for comprehensive, robust and dedicated analysis of diffusion tensor data within the myocardial wall. It is shown that the description of this information is more compact in this coordinate frame. Furthermore, it is demonstrated that the acquisition limitations can be overcome by introducing an approximation scheme based on this coordinate frame. These techniques are tested on ex-vivo datasets to assess their fidelity and sensitivity. Finally, these techniques are applied in-vivo on a group of healthy volunteers, where 2D DTI slices of the LV were acquired at end diastole and end systole, using cardiac dedicated diffusion MR acquisition. Results demonstrate the advantages of using curvilinear coordinates both for the analysis and the approximation of cardiac DTI information. Resulting in-vivo fibre architectures were found to agree with previously reported studies on ex-vivo specimens. The outcome of this work can open the door for clinical applications and cardiac electrophysiology modelling, and improve the understanding of the left ventricular structure and dynamics.

In this thesis we look at acoustic wave propagation under water. We look in particular at waves generated by a point source and what happens with the propagation when we model the bottom as flat or as curvilinear. We assume the source to be working at a certain frequency and therefore we model this problem by solving the Helmholtz equation. Since Helmholtz equation has some unwanted numerical properties we are interested in finding new numerical methods that could accelerate the solver. In this thesis we use the Waveholtz iteration, which solves Helmholtz equation by connecting it to the time-dependent wave equation. We use finite differences and the SBP-SAT method to approximate the spatial problem numerically and for modelling the sea bottom we use curvilinear coordinates.  To compare the Waveholtz iteration we also solve Helmholtz equation with a naive solver. The naive solver consists of approximating the equation with finite differences and then solving the linear system of equation by some iterative solver, which for our tests will be GMRES. The results show that the Waveholtz iteration converges in less iterations than our naive solver. It also shows that the number of iterations stays unchanged when changing our discretization, which otherwise is a big problem for our naive solver. This allows us to increase the accuracy of our numerical solution without changing the computation time too much.  We show that the number of iterations increases according to theory for an increasing frequency, and that for open problems we even see a smaller increase. For certain resonant frequencies in Helmholtz equation we do not expect the Waveholtz iteration to converge. In the neighbourhood of these frequencies the convergence becomes slow and we need many iterations for a solution of a certain accuracy. By reformulating the Waveholtz iteration as a Krylov solution we can see that resonances in Helmholtz equation have a smaller impact of the convergence. / I detta examensarbete undersöker vi akustisk vågutbredning i vatten. Vi kollar specifikt på vågor som genereras av en punktkälla och vad som sker när vi modellerar botten som plan eller som kurvlinjär. Då vi antar att punktkällan arbetar vid en bestämd frekvens, kommer vi modellera det fysikaliska problemet genom att lösa Helmholtz ekvation. Helmholtz ekvation har dock några numeriska egenskaper som är oönskade, och därför finns ett intresse av att hitta nya numeriska metoder som löser ekvationen. I detta examensarbete undersöker vi Waveholtz iteration, som löser Helmholtz ekvation genom att koppla den till den tidsberoende vågekvationen. Vi använder finita differenser och SBP-SAT metoden för att approximera det rumsliga problemet numeriskt. För att ge en detaljerad beskrivning av botten använder vi kurvlinjära koordinater. För att jämföra Waveholtz iterationen med något löser vi även Helmholtz med hjälp av en naiv lösare. Den naiva lösaren består av att approximera problemet med finita differenser och sedan lösa det linjära systemet rakt av med en iterativ lösare (vilket för våra fall kommer vara GMRES). Resultatet visar att Waveholtz iteration konvergerar på ett lägre antal iterationer än vår naiva lösare. Det visar även att antalet iterationer inte förändras när vi ändrar diskretisering, vilket annars är ett problem för vår naiva lösare. Detta innebär att vi kan få en högre noggrannhet utan att förlänga beräkningstiden alltför mycket.  Vi visar även att antalet iterationer ökar som förväntat med en ökad frekvens, samt att för öppna problem så ökar antalet iteration mindre än enligt teorin. Vid vissa resonanta frekvenser i Helmholtz ekvation förväntar vi oss att Waveholtz iteration inte kommer konvergerar. I närheten av dessa frekvenser blir konvergensen långsam och vi behöver många iterationer för att lösa problemet. Genom att formulera Waveholtz iteration som en Krylov lösning kommer resonanser i Helmholtz ekvation ge en mindre negativ effekt på konvergensen än om den är formulerad som en fixpunkts iteration.

This dissertation supplies the methodology of the measure (area) in the Earth model adopted for Geodesy. This model is the ellipsoid of revolution, in which the system of Cartesian coordinates, the curvilinear coordinate system and the polar coordinate system are described. The coordinate nature in the development of the surface measure calculation is discussed. The following demonstrations are illustrated: the ellipse equation, the eccentricity of the ellipse, the meridian section curvature radius equation, the meridian transversal section curvature radius equation and elliptic integral. It define algebraic geodesic line and geometrically. The juridical basis are the article 3º of Brazilian Federal Law 10.267/2001, which modify article 176 of the Brazilian Federal Law 6.015/1973 (Public Record Law) and adds to this article the paragraphs 3º and 4º, the paragraph 3º of article 225 of the Brazilian Law 6.015/1973 and the article 971 of the Código de Processo Civil (CPC), which require the coordinates of the corners of the real property on the Brazilian Geodetic System (SGB). The partitioning of the regular ellipsoid quadrilateral and the partitioning of the irregular ellipsoid quadrilateral located in the real property Pó de Serra is presented. To become this partitioning, it was used surface of the Lambert azimuthal equal-area projection, i. e., the curvilinear geodetic coordinates in plane coordinates has been transformed. The surface partitioning was determined using the method area equation of the Gauss trapezes connected with the equation of the straight line. The direct problem of the Lambert azimuthal equal-area projection and the inverse problem supply the methodology that become feasible the juridical exigence (articles 176 and 225 of Brazilian Federal Law 6.015/1973 and article 971 of the CPC). The methodology to the geodetic coordinates system with the purpose to calculate the partitioned areas of surface on the ellipsoid can be applied. The calculation of surface measure supplies the effective practice of the mencioned juridical basis. / O propósito deste trabalho é fornecer os fundamentos de cálculo de medida de superfície (área) no modelo de Terra adotado pela Geodésia. Esse modelo é o elipsóide de revolução ao qual se vincula o sistema de coordenadas cartesianas, o sistema de coordenadas curvilíneas e o sistema de coordenadas polares. Discute a natureza das coordenadas no desenvolvimento do cálculo da medida de superfície. Efetuam-se as seguintes demonstrações: equação da elipse, equação da excentricidade da elipse, equação do raio de curvatura da seção meridiana, equação do raio de curvatura da seção transversal meridiana e integral elíptica. Define linha geodésica algébrica e geometricamente. Apresentam-se os instrumentos legais que são o artigo 3º da Lei 10.267/2001, o qual altera o artigo 176, inciso II da Lei 6.015/1973 (Lei de Registros Públicos) e acrescenta a este artigo os parágrafos 3º e 4º, o parágrafo 3º do artigo 225 da Lei 6.015/1973 e o artigo 971 do Código de Processo Civil (CPC), os quais vinculam as coordenadas dos vértices do imóvel ao Sistema Geodésico Brasileiro (SGB). Efetuam-se a divisão do quadrilátero elipsóidico regular e também do quadrilátero elipsóidico irregular localizado na gleba Pó de Serra. Para se fazer esta divisão, usou-se a superfície da projeção azimutal equivalente de Lambert, ou seja, as coordenadas elipsóidicas curvilíneas foram transformadas em coordenadas planas desse sistema de projeção. A divisão destas superfícies foi efetuada pelo método da equação da área dos trapézios de Gauss em conjunto com a equação da reta. Os problemas direto e inverso da projeção azimutal equivalente de Lambert fornecem a metodologia que tornam exeqüíveis os dispositivos legais (artigos 176 e 225 da Lei 6.015/1973 e artigo 971 do CPC). A metodologia de cálculo proposto pode ser aplicada ao sistema de coordenadas geodésicas com a finalidade de calcular as áreas de uma divisão de superfície no elipsóide. Os fundamentos do cálculo de medida de superfície instrumentalizam o efetivo cumprimento dos dispositivos legais retrocitados.