Cette thèse s’intéresse à la simulation acoustique de problèmes de grandes tailles. La parallélisation des méthodes numériques d’acoustique est le sujet principal de cette étude. Le manuscrit est composé de trois parties : lancé de rayon, méthodes de décomposition de domaines et algorithmes asynchrones. / This thesis studies numerical methods for large-scale acoustic problems. The parallelization of the numerical acoustic methods is the main focus. The manuscript is composed of three parts: ray-tracing, optimized interface conditions for domain decomposition methods and asynchronous iterative algorithms.

Orientador: Valeria Abrão de Podesta / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-14T16:33:59Z (GMT). No. of bitstreams: 1 Passos_AdaoNascimentodos_M.pdf: 1173380 bytes, checksum: 9650e6a87755fbc73407fcb71aed15c1 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho é feito um estudo sobre Programação Linear e um texto sobre alguns de seus assuntos básicos, construído com uma linguagem didática, visando sua utilização em sala de aula. São apresentados alguns problemas lineares, os fundamentos matemáticos da Programação Linear e o método Simplex, finalizando com um estudo do princípio da decomposição de Dantzig-Wolfe, que é um procedimento para a resolução de problemas lineares de grande porte e com estrutura especial. / Abstract: In this work we have done a study on Linear Programming and a text with some basic issues, using a didactic language, and aiming its utilization in the classroom. Some linear problems are shown here, the mathematical background of Linear Programming and the Simplex method. Finaly, we have also presented a study on the principle of Dantzig-Wolfe's decomposition, which is a procedure for solving large linear problems with special structure. / Mestrado / Programação Linear / Mestre em Matemática

Cette thèse est consacrée à l'étude de l'influence d'une fine couche hétérogène sur le comportement élastique linéaire d'une structure tridimensionnelle.Deux types d'hétérogénéités sont pris en compte : des cavités et des inclusions élastiques. Une étude complémentaire, dans le cas d'inclusions de grande rigidité, a été réalisée en considérant un problème de conduction thermique.Une analyse formelle par la méthode des développements asymptotiques raccordés conduit à un problème d'interface qui caractérise le comportement macroscopique de la structure. Le comportement microscopique de la couche est lui déterminé sur une cellule de base. Le modèle asymptotique obtenu est ensuite implémenté dans un code éléments finis. Une étude numérique permet de valider les résultats de l'analyse asymptotique. / This thesis is devoted to the study of the influence of a thin heterogeneous layeron the linear elastic behavior of a three-dimensional structure. Two types of heterogeneties are considered : cavities and elastic inclusions. For inclusions of high rigidty a further study was performed in the case of a heat conduction problem.A formal analysis using the matched asymptotic expansions method leads to an interface problem which characterizes the macroscopic behavior of the structure. The microscopic behavior of the layer is determined in a basic cell.The asymptotic model obtained is then implemented in a finite element software.A numerical study is used to validate the results of the asymptotic analysis.

The present thesis is developed through four principal chapters. The first one provides a brief but rather exhaustive description of the context, with a global overview on the complex process of the embryogenesis in Drosophila Melanogaster. We amply focus on the three morphogenetic movements that will be numerically simulated, with particular emphasis on both the mechanical and the biological aspects that constitute the main peculiarity of each event. Also we propose a short review on the related previous works. The second chapter supplies the abstract tools for the analysis of the whole problem and points out the hypotheses that, for sake of simplicity, have been made. The gradient decomposition method is presented together with some interesting interpretations that better clarify the approach and put forward novel issues that have to be considered. By the Principle of the Virtual Power, we are able to write the mechanical equilibrium of the system which consists of the forces internal to the embryo domain and of the boundary conditions, such as the yolk pressure and contact with the vitelline membrane, that are essential for consistent results. A special concern is attributed to the choice of the constitutive law of the mesoderm that, from a biological point of view, may appear too simplistic. Here a Saint- Venant material is used in contrast with the Hyperelastic models found in literature; therefore a comparison between the two is proposed together with the advantages and the limitations of our study. Finally, we provide some simple examples that validate our model and support the exploited method. The third chapter can be divided into two parts. In the first one, by the parametrical description of the embryo geometry, we obtain the analytical formulations of the active deformation gradients for each morphogenetic movement according to the elementary forces introduced. Such expressions will be combined with the passive gradients in order to get the final deformation of the tissues. In the second part we interpret the results for each simulation. In particular, we provide a parametrical analysis for the simulation of the ventral furrow invagination, while for the germ band extension a comparison with experimental data is done. Furthermore we have been able to estimate the effects induced by the local deformations within the tissues; specifically, we have evaluated the magnitude of the pressure forces and the shear stress that may develop at long distance in the embryo when the active forces are applied in restraint regions. To conclude, we propose a collateral study on the influence of the global geometry of the embryo on the final results. Given the consistence of the results for the individual simulations, we have decided to test the concurrent simulation of the events, by two or three of them. In the last chapter, we show the results for a first essay for which we use the most intuitive method; it does not require in fact further manipulations of the analytical formulations previously obtained, but we simply couple together the active deformation gradients, following the chronological order of the movements. Although the method works well for the simulation of the two furrows, some drawbacks are detected when we introduce the germ band extension. Therefore we propose a new approach, more rigorous and appropriate, which allows to take into account some aspects so far put aside, but still significant for a realistic and complete reproduction of the different phases of the Drosophila gastrulation. / Ce travail de recherche a eu comme objectif principal la conception d'un modèle numérique aux éléments finis donnant une représentation réaliste des mouvements de l'embryon de la Drosophila Melanogaster. Les simulations de trois mouvements durant la phase de gastrulation de l'embryon ont été realisées soit individuelles soit simultanées, ce qui jusqu'à présent, n'a jamais été proposé, constituant ainsi une contribution originale de cette étude. La thèse est composée de quatre chapitres. Le premier fournit une brève mais assez complète description du contexte dans lequel ce travail se situe. Le processus complexe de l'embryogénèse chez la Drosophila Melanogaster est presenté en se focalisant sur les trois mouvements morphogénetiques qui seront ensuite simulés numériquement: l'invagination du sillon ventral, la formation du sillon céphalique et l'extension de la bande germinale. Chaque événement est décrit du point de vue biologique et mécanique, ce qui permet donc de mettre en avant les aspects les plus intéressants des différents mouvements. Une revue des plus récents travaux est aussi proposée. Dans le deuxième chapitre on présente les outils analytiques pour l'analyse du problème dans son intégrité. Etant donnée la complexité du système biologique, plusieurs hypothèses ont été introduites pour simplifier l'approche numérique utilisée. Seul le mésoderme est modélisé comme un milieu continu dans un espace tridimensionel par un ellipsoïde épais régulier de 500 µm de longueur. La méthode de la décomposition du gradient de déformation, dont quelques interprétations alternatives sont présentées, permet de coupler les déformations passives et actives subies par chaque point matériel du milieu. L'équilibre mécanique est écrit à partir du Principe des Puissances Virtuelles: les forces internes du système sont donc prises en compte avec les conditions aux limites. Dans notre cas particulier celles-ci sont fondamentales pour obtenir des configurations finales réalistes et comprennent le contact entre le mésoderme et la membrane vitelline externe et le pression exercée par le yolk sur la surface interne du mésoderme. Les propriétés mécaniques des tissus embryonnaires ne sont pas faciles à déterminer expérimentalement. Une approximation a été faite pour ce qui concerne la loi de comportement du mésoderme qui a été modélisé comme un matériau de Saint-Venant linéaire, élastique et isotrope. Notre choix étant en contraste avec le modèle hyperélastique qu'on retrouve souvent en littérature, une comparaison entre les deux matériaux est proposée tout en considérant les avantages et les limitations de notre démarche. La méthode de la décomposition du gradient de déformation a été auparavant testée sur des cas géométriquement très simples dont la solution analytique peut être facilement calculée et validée par les résultats obtenus à partir des simulations numériques. Le troisième chapitre peut être divisé en deux parties distinctes. Dans la première, grâce à la description paramétrique de l'ellipsoïde qui représente l'embryon, on calcule les expressions analytiques des positions intermédiaires où on voit apparaître les déformations actives responsables de chaque mouvement morphogénétique. Les gradients de déformation active sont donc couplés avec les gradients passifs pour obtenir la déformation finale. La deuxième partie du chapitre concerne l'analyse des résultats pour les simulations individuelles des événements. Pour la simulation de l'invagination du sillon ventral une étude paramétrique a été conduite pour évaluer l'influence de certains paramètres sur la configuration finale. Pour la simulation de l'extension de la bande germinale les résultats ont été comparés avec les données expérimentales. En particulier on s'est intéressé à l'analyse des contraintes mécaniques (les pressions et les contraintes de cisaillement) induites au niveau du pôle antérieur où un chemin de mécanotransduction aurait lieu et conduirait à l'expression du twist, un gène normalement exprimé seulement dans la partie ventrale de l'embryon. Pour conclure, d'autres géométries que celle de l'ellipsoïde ont été utilisées pour les simulations de l'invagination du sillon ventral et de l'extension de la bande germinale. Ces nouvelles représentations de l'embryon permettent de prendre en compte deux aspects intéressants: d'un côté l'arrondissement des deux pôles, de l'autre l'aplatissement de la partie dorsal par rapport à la partie ventrale. Le dernier chapitre du manuscrit introduit la simulation simultanée des trois mouvements qui a été mise en place pour deux raisons principales. Tout d'abord le fait que les événements analysés se produisent l'un après l'autre lors du développement de l'embryon. Deuxièmement, les résultats obtenus pour les simulations individuelles sont très encourageants et ont permis aussi de confirmer plusieurs hypothèses avancées par les biologistes; d'où l'intérêt de coupler les mouvements pour permettre une vision encore plus réaliste de cette phase importante de la gastrulation chez l'embryon de la Drosophila Melanogaster. Deux méthodes différentes ont été testées. La première, la plus intuitive et simple, permet de combiner les gradients de déformation active de chaque mouvement et ne requiert pas de manipulations supplémentaires des équations précédemment trouvées, tout en prenant en compte le déphasage réel entre les événements. Cette approche ne pose pas de problèmes quand seulement les deux sillons sont couplés, alors que l'introduction de l'extension de la bande germinale donne lieu à quelque limitations. Une nouvelle démarche est donc proposée, plus rigoureuse et précise, qui nous a permis de considérer certains aspects importants pas encore développés d'un point de vue théorique.

Thesis (M. Sc. (Applied Mathematics)) -- University of Limpopo, 2021 / Modelling with differential equations is of paramount importance as it provides pertinent insight into the dynamics of many engineering and technological devices and/or processes. Many such models, however, involve differential equations that are inherently nonlinear and difficult to solve. Many numerical methods have been developed to solve a variety of differential equations that cannot be solved analytically. Most numerical methods, however, require discretisation, linearisation of the nonlinear terms and other simplifying approximations that may inhibit the accuracy of the solution. Further, in some methods high computational complexity is involved. Due to the importance of differential equations in modelling real life phenomena and these stated shortfalls, continuous pursuit of more efficient solution techniques by the scientific community is ongoing. Industrial and technological advancement are to a larger extent dependent upon efficient and accurate solution techniques. In this work, we investigate the use of Adomian decomposition method in solving nonlinear ordinary and partial differential equations. One advantage of Adomian decomposition method that has been demonstrated in literature is that it achieves a rapidly convergent infinite series solution. The method is also advantageous in that it does not require one to linearise and discretise the equations as is done with other numerical methods. In our investigation, among other important examples, we will apply the Adomian decomposition method to solve selected fluid flow and heat transfer problems. Fluid flow and heat transfer models have pertinent applications in engineering and technology. The Adomian decomposition method will be compared with other series solution methods, namely the differential transform method and the homotopy analysis method. The desirable attributes of the Adomian decomposition method that are stated in literature have been ascertained in this work and it has also been demonstrated that the Adomian decomposition method compares favourably with the other series solution methods. It has also been demonstrated that in some cases nonlinear complexity results in slow convergence rate of the Adomian decomposition method.

No description available.

We are interested in numerical methods for the optimization constrained second order ordinary differential equations arising in biofilm modelling. This class of problems is challenging for several reasons. One of the reasons is that the underlying solution has a steep slope, making it difficult to resolve. We propose a new numerical method with techniques such as domain decomposition and asynchronous iterations for solving certain types of ordinary differential equations more efficiently. In fact, for our class of problems after applying the techniques of domain decomposition with overlap we are able to solve the ordinary differential equations with a steep slope on a larger domain than previously possible. After applying asynchronous iteration techniques, we are able to solve the problem with less time.~We provide theoretical conditions for the convergence of each of the techniques. The other reason is that the second order ordinary differential equations are coupled with an optimization problem, which can be viewed as the constraints. We propose a numerical method for solving the coupled problem and show that it converges under certain conditions. An application of the proposed methods on biofilm modeling is discussed. The numerical method proposed is adopted to solve the biofilm problem, and we are able to solve the problem with larger thickness of the biofilm than possible before as is shown in the numerical experiments. / Mathematics

The electrical activity of the heart is a well studied process. Mathematical modeling and computer simulations are used to study the cardiac electrical activity: several mathematical models exist, among them the microscopic model, which is based on the explicit representation of individual cells. The cardiac tissue is viewed as two separate domains: the intra-cellular and extra-cellular domains, Ωᵢ and Ωₑ, respectively, separated by cellular membranes Γ. The microscopic model consists of a set of Poisson equations, one for each sub-domain, Ωᵢ and Ωₑ, coupled on interfaces Γ with nonlinear transmission conditions involving a system of ODEs. The unusual transmission conditions on Γ make the model challenging to solve numerically. In this thesis, we first focus on the dimensional analysis of the microscopic model. We then reformulate the problem on the interface Γ using a Steklov-Poincaré operator. We discretize the model in space using finite element methods. We prove the existence of a semi-discrete solution using a reformulation of the model as an ODE system on the interface Γ. We derive stability and error estimates for the finite element method. Afterwards, we consider five numerical schemes including the Godunov splitting method, two implicit methods, (Backward Euler (BE) and second order Backward Differentiation Formula (BDF2)), and two semi-implicit methods (Forward Backward Euler (FBE), and second order Semi-implicit Backward Differentiation Formula (SBDF2)). A convergence analysis of the implicit and semi-implicit methods is performed and the results are compared with manufactured solutions that we have proposed. Numerical results are presented to compare the stability, accuracy and efficiency of the methods. CPU times needed to solve the problem over a single cell using FBE, SBDF2 and Godunov splitting methods are reported. The results show that FBE and Godunov splitting methods achieve better numerical accuracy and efficiency than implicit and SBDF2 schemes, for a given computational time. Finally, we solve the model using FBE and Domain Decomposition Method (DDM) for two cells connected to each other by a gap junction. We investigate the influence of the space discretization and we explore the differences between a conforming and nonconforming mesh on Γ. We compare the solutions obtained with both FBE and DDM methods. The results show that both methods give the same solution. Therefore, the DDM is capable of providing an accurate solution with a minimal number of sub-domain iterations.

Large scale optimization problems are tractable only if they are somehow decomposed. Hierarchical decompositions are inappropriate for some types of problems and do not parallelize well. Sobieszczanski-Sobieski has proposed a nonhierarchical decomposition strategy for nonlinear constrained optimization that is naturally parallel. Despite some successes on engineering problems, the algorithm as originally proposed fails on simple two dimensional quadratic programs. Here, the algorithm is carefully analyzed by testing it on simple quadratic programs, thereby recognizing the problems with the algorithm. Different modifications are made to improve its robustness and the best version is tested on a larger dimensional example. Some of the changes made are very fundamental, affecting the updating of the various tuning parameters present in the original algorithm. The algorithm involves solving a given problem by dividing it into subproblems and a final coordination phase. The results indicate good success with small problems. On testing it with a larger dimensional example, it was discovered that there is a basic flaw in the coordination phase which needs to be rectified. / Master of Science

Tendo em vista o aumento considerável da poltúção do ar provocado em grande parte pela industrialização e o aumento da emissão de poluentes resultantes da queima de combustíveis fósseis por veículos automotores, o presente trabalho tem como objetivo melhorar a previsão e o entendimento da dispersão turbulenta atmosférica. Para tanto, apresenta-se, pela primeira vez, uma representação analít ica para a equação de advecção-difusão-reação tridimensional transiente, com perfil de vento e coeficientes de difusão tmbulenta dependentes da altura, que modelam a dispersão de poluentes na atmosfera. A solução da equação é obtida pela combinação do método GILTT ( Generalized Integral Laplace Transform Technique) com o método da Decomposição de Adomian modificado. Consideram-se dois casos para a aplicação do modelo: no primeiro modela-se a dispersão de um poluente secundário formado por uma reação fotoquímica e no segundo caso, utiliza-se o modelo para determinar o campo de concentração de um poluente que sofre perdas e ganhos devido a influência da radiação solar. Para poder realizar essas análises propôs-se uma parametrização para o termo de reação fotoquímica. São apresentados os resultados numéricos e estatísticos, comparandose com os dados da campanha experimental da Usina Termelétrica de Candiota e com os dados de medições realizadas pela Fundação Estadual de Proteção Ambiental Henrique Luiz Roessler (FEPAM). / In view of the considerable increase of air pollution caused largely by industrialization and the increase of emission pollutants resulting from burning of fossil fuels by motor vehicles, the present work aims to improve the prediction and understanding of atmospheric turbu- lent dispersion. Therefore, is presented, for the rst time, an analytical representation to the transient three-dimensional advection-diffusion-reaction equation, with wind pro le and turbulent diffusion coefficients dependent of height, modeling the dispersion of pollutants in the atmosphere. The solution of the equation is obtained by combining of the GILTT method (Generalized Integral Laplace Transform Technique) with the modi ed Adomian Decomposition method. It is considered two cases for the application of the model: in the rst is modeled the dispersion of a secondary pollutant formed by a photochemical reaction, and in the second case the model is used to determine the concentration eld of a pollutant that suffers losses and gains due to the in uence of solar radiation. To realise these analisis a parameterization for the photochemical reaction term is proposed. Numerical and statistical results are presented, comparing with the experimental campaign data of the thermoelectric plant of Candiota and with data from measurements performed by the \Funda c~ao Estadual de Prote c~ao Ambiental Henrique Luiz Roessler" (FEPAM).