Chemotaxis, the phenomenon through which cells respond to external chemical signals, is one of the most important and universally observable in nature. It has been the object of considerable modelling effort in the last decades. The models for chemotaxis available in the literature cannot reconcile the dynamics of external chemical signals and the intracellular signalling pathways leading to the response of the cells. The reason is that models used for cells do not contain the distinction between the extracellular and intracellular domains. The work presented in this dissertation intends to resolve this issue. We set up a numerical hybrid simulation framework containing such description and enabling the coupling of models for phenomena occurring at extracellular and intracellular levels. Mathematically, this is achieved by the use of the fictitious domain method for finite elements, allowing the simulation of partial differential equations on evolving domains. In order to make the modelling of the membrane binding of chemical signals possible, we derive a suitable fictitious domain method for Robin boundary elliptic problems. We also display ways to minimise the computational cost of such simulation by deriving a suitable preconditioner for the linear systems resulting from the Robin fictitious domain method, as well as an efficient algorithm to compute fictitious domain specific linear operators. Lastly, we discuss the use of a simpler cell model from the literature and match it with our own model. Our numerical experiments show the relevance of the matching, as well as the stability and accuracy of the numerical scheme presented in the thesis.

For the purpose of numerically simulating continuum mechanical structures, different types of material may be represented by the extreme values {<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />,1}, where 0&lt;<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /><img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cll" />1, of a varying coefficient <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> in the governing equations. The paramter <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> is not allowed to vanish in order for the equations to be solvable, which means that the exact conditions are approximated. For example, for linear elasticity problems, presence of material is represented by the value <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = 1, while <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> provides an approximation of void, meaning that material-free regions are approximated with a weak material. For acoustics applications, the value <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = 1 corresponds to air and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> to an approximation of sound-hard material using a dense fluid. Here we analyze the convergence properties of such material approximations as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />!0, and we employ this type of approximations to perform design optimization. In Paper I, we carry out boundary shape optimization of an acoustic horn. We suggest a shape parameterization based on a local, discrete curvature combined with a fixed mesh that does not conform to the generated shapes. The values of the coefficient <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" />, which enters in the governing equation, are obtained by projecting the generated shapes onto the underlying computational mesh. The optimized horns are smooth and exhibit good transmission properties. Due to the choice of parameterization, the smoothness of the designs is achieved without imposing severe restrictions on the design variables. In Paper II, we analyze the convergence properties of a linear elasticity problem in which void is approximated by a weak material. We show that the error introduced by the weak material approximation, after a finite element discretization, is bounded by terms that scale as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />1/2hs, where h is the mesh size and s depends on the order of the finite element basis functions. In addition, we show that the condition number of the system matrix scales inversely proportional to <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />, and we also construct a left preconditioner that yields a system matrix with a condition number independent of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />. In Paper III, we observe that the standard sound-hard material approximation with <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" /> = <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> gives rise to ill-conditioned system matrices at certain wavenumbers due to resonances within the approximated sound-hard material. To cure this defect, we propose a stabilization scheme that makes the condition number of the system matrix independent of the wavenumber. In addition, we demonstrate that the stabilized formulation performs well in the context of design optimization of an acoustic waveguide transmission device. In Paper IV, we analyze the convergence properties of a wave propagation problem in which sound-hard material is approximated by a dense fluid. To avoid the occurrence of internal resonances, we generalize the stabilization scheme presented in Paper III. We show that the error between the solution obtained using the stabilized soundhard material approximation and the solution to the problem with exactly modeled sound-hard material is bounded proportionally to <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" />.

[pt] Este trabalho apresenta uma nova formulação implícita e totalmente acoplada para o problema de escoamentos tridimensionais com corpos rígidos suspensos. Esta é a principal contribuição deste trabalho. A formulação foi implementada em C mais mais e testada para o problema de sedimentação de uma partícula esférica. Os resultados indicam comportamento físico plausível apesar de serem limitados por inacurácia de malha. O programa resolve numericamente as Equações de Navier-Stokes acopladas com as Equações da Dinâmica de Corpo Rígido usando o Método de Elementos Finitos. O acoplamento entre os domínios fluido e sólido é feito pela Técnica do Domínio Fictício, que evita a geração de malha a cada passo de tempo. O escoamento tridimensional sem partículas também é estudado neste trabalho e é a base para a formulação do escoamento com partículas. / [en] This work presents a novel implicit and fully coupled formulation for the problem of 3D flows with suspended rigid bodies. This is the main contribution of the work. The formulation was implemented in C plus plus and tested for the sedimentation problem of one spherical particle. The results indicate plausible physical behavior in spite of being limited by mesh accuracy. The software solves numerically the Navier-Stokes Equations coupled with Rigid Body Dynamics Equations using the Finite Elements Method. The coupling between fluid and solid domains is done by means of the Fictitious Domain Technique, which avoids mesh generation for every time step. The 3D flow of non particulate flow is also studied in this work and is the basis for the particulate flow formulation.

This thesis presents a finite element methodology for simulating fluid–solid interactions in the left ventricle (LV) under LVAD support. The developed model was utilised to study the passive and active characteristics of ventricular function in anatomically accurate LV geometries constructed from normal and patient image data. A non–conforming ALE Navier–Stokes/finite–elasticity fluid–solid coupling system formed the core of the numerical scheme, onto which several novel numerical additions were made. These included a fictitious domain (FD) Lagrange multiplier method to capture the interactions between immersed rigid bodies and encasing elastic solids (required for the LVAD cannula), as well as modifications to the Newton–Raphson/line search algorithm (which provided a 2 to 10 fold reduction in simulation time). Additional developments involved methods for extending the model to ventricular simulations. This required the creation of coupling methods, for both fluid and solid problems, to enable the integration of a lumped parameter representation of the systemic and pulmonary circulatory networks; the implementation and tuning of models of passive and active myocardial behaviour; as well as the testing of appropriate element types for coupling non–conforming fluid– solid finite element models under high interface tractions (finding that curvilinear spatial interpolations of the fluid geometry perform best). The behaviour of the resulting numerical scheme was investigated in a series of canonical test problems and found to be convergent and stable. The FD convergence studies also found that discontinuous pressure elements were better at capturing pressure gradients across FD boundaries. The ventricular simulations focused firstly on studying the passive diastolic behaviour of the LV both with and without LVAD support. Substantially different vortical flow features were observed when LVAD outflow was included. Additionally, a study of LVAD cannula lengths, using a particle tracking algorithm to determine recirculation rates of blood within the LV, found that shorter cannulas improved the recirculation of blood from the LV apex. Incorporating myocardial contraction, the model was extended to simulate the full cardiac cycle, converging on a repeating pressure–volume loop over 2 heart beats. Studies on the normal LV geometry found that LVAD implementation restricts the recirculation of early diastolic inflow, and that fluid–solid coupled models introduce greater heterogeneity of myocardial work than was observed in equivalent solid only models. A patient study was undertaken using a myocardial geometry constructed using image data from an LVAD implant recipient. A series of different LVAD flow regimes were tested. It was found that the opening of the aortic valve had a homogenising effect on the spatial variation of work, indicating that the synchronisation of LVAD outflow with the cardiac cycle is more important if the valve remains shut. Additionally, increasing LVAD outflow during systole and decreasing it during diastole led to improved mixing of blood in the ventricular cavity – compared with either the inverse, or holding outflow constant. Validation of these findings has the potential to impact the treatment protocols of LVAD patients.

Cette thèse concerne l'étude des interactions entre le plasma et la paroi d'un réacteur à fusion nucléaire de type tokamak. L'objectif est de proposer des méthodes de résolution des systèmes d'équations issus de modèles de plasma de bord. Nous nous sommes intéressés au traitement de deux difficultés qui apparaissent lors de la résolution numérique de ces modèles. La première difficulté est liée à la forme complexe de la paroi du tokamak. Pour cela, il a été choisi d'utiliser des méthodes de pénalisation volumique. Des tests numériques de plusieurs méthodes de pénalisation ont été réalisés sur un problème hyperbolique non linéaire avec un domaine 1D. Une de ces méthodes a été étendue à un système hyperbolique quasilinéaire avec bord non caractéristique et conditions aux limites maximales strictement dissipatives sur un domaine multidimensionnel : il est alors démontré que cette méthode de pénalisation ne génère pas de couche limite. La deuxième difficulté provient de la forte anisotropie du plasma, entre la direction parallèle aux lignes de champ magnétique et la direction radiale. Pour le potentiel électrique, cela se traduit par une résistivité parallèle très faible. Afin d'éviter les difficultés liées au fait que le problème devient mal posé quand la résistivité parallèle tend vers 0, nous avons utilisé des méthodes de type asymptotic-preserving (AP). Pour les problèmes non linéaires modélisant le potentiel électrique avec un domaine 1D et 2D, nous avons fait l'analyse théorique ainsi que des tests numériques pour deux méthodes AP. Des tests numériques sur le cas 1D ont permis une étude préliminaire du couplage entre les méthodes de pénalisation volumique et AP. / This thesis deals with the study of wall plasma interactions in a nuclear fusion reactor such as a tokamak. The goal is to propose methods to solve partial differential equations issued from edge plasma models. We focus on two difficulties for the numerical resolution of these models. The first issue concerns the complex shape of the tokamak wall: we choose volume penalty methods. Numerical tests on several penalization methods have been performed on a nonlinear hyperbolic problem. One of these methods has been extended to a quasilinear hyperbolic system with a non characteristic boundary and maximally strictly dissipative boundary conditions on a multidimensional domain: it is proven that this penalty method does not generate any boundary layer. The second question comes from the strong plasma anisotropy between the direction parallel to the magnetic field lines and the radial one. Concerning the electrical potential, this results in a very low parallel resistivity. In order to avoid the troubles due to the ill-posedness of the equations when the parallel resistivity tends to 0, we study asymptotic preserving (AP) methods. For 1D and 2D nonlinear models of the electrical potential, we performed the theoretical analysis and numerical simulations for two AP methods. A preliminary study of the coupling between volume penalty and AP methods is also presented.

A l'échelle de plusieurs dizaines de millions d'années, un système de subduction implique de grandes déformations de la plaque plongeante assimilée un solide viscoélastique, et du manteau supérieur assimilé à un fluide newtonien. L'objectif de ce travail est de développer une stratégie de couplage solide-fluide appliquée à l'étude de l'interaction lithosphère-asthénosphère. Cette stratégie est basée sur l'utilisation de maillages non-conformes aux interfaces et d'une méthode de domaines fictifs (MDF) pour la résolution du problème fluide. Pour l'efficience des modèles 3D, nous employons une formulation simplifiée de la méthode de domaines fictifs par multiplicateurs de Lagrange. La MDF développée est validée par des comparaisons avec des solutions analytiques qui montrent que la méthode est d'ordre 1. La stratégie de couplage est également validée par la comparaison avec d'autres méthodes de couplage solide-fluide. Une première étude est ensuite menée pour analyser l'influence de certains paramètres rhéologiques et cinématiques sur la dynamique d'une subduction contrôlée par les vitesses des plaques. Cette étude, en 2D, concerne plus spécifiquement le mécanisme de plissement périodique du slab lorsque celui-ci est ancré à 660 km de profondeur. Ce mécanisme induit des variations de pendage du slab générant des variations de l'état de contrainte de la plaque chevauchante. Un intérêt particulier est porté sur l'influence de la viscosité du manteau sur les plissements. Dans ce cadre, nous réalisons une application à la subduction andine. / Over the time scale of tens of millions of years, a subduction system involves large deformations of tectonics plates, as one plate sinks into the Earth's mantle. The aim of this work was to develop a soli-fluid coupling method applied to the lithosphere-asthenosphere interaction in the context of subduction zones. Plates were assumed to behave as viscoelastic bodies, while the upper mantle was assimilated to a newtonian fluid. The method developped here is based on the use of non-matching interface meshes and a fictitious domain method (FDM) for the fluid problem. To optimize the computational efficiency of 3D model, we used a simplified version of the Lagrange multipliers fictitious domain method. The developped FDM has been benchmarked with analytical solutions and we showed that this FDM is a first-order method. The coupling method has also been compared to other fluid-solid coupling methods using matching interfaces meshes. A first two-dimensional study was performed in order to evaluate the influence of some rheological and kinematic parameters on the dynamics of a subduction controlled by the velocity of the plates. This study aimed at investigating cyclic slab folding over a rigid 660 km depth transition zone. This folding mechanism induces variations in slab dip that generate variations in the stress state of the overriding plate. We focussed on the influence of the upper mantle viscosity on slab folding. We also applied this model to the Andean subduction zone. Several studies have determined a cyclic variation of the South-American tectonic regime (period of 30-40~Myrs) which may have been related to the slab dip evolution.

Lorsqu'une onde acoustique se propage dans un milieu granulaire, elle est susceptible de provoquer la mobilité des grains, aussi infime soit-elle. Inversement, la mobilité d'un grain dans une matrice fluide peut induire un champ acoustique et dans les deux cas, l'énergie acoustique peut être transférée à la fois au travers des pores et des contacts entre grains. Nous avons mis au point un modèle original permettant de considérer ces deux modes de transfert d'énergie pour simuler la propagation d'ondes acoustiques dans les milieux granulaires immergés. Dans le cas des milieux granulaires secs, l'inertie du fluide est telle que l'énergie transférée dans l'air peut être négligée et le milieu modélisé avec des algorithmes de type "dynamique moléculaire". Au contraire, dans le cas de milieux immergés, l'énergie portée par le fluide ne peut pas être négligée et nous montrons que la méthode des domaines fictifs basée sur les multiplicateurs de Lagrange distribués permet de coupler les équations de la dynamique et l'équation d'onde. Nous utilisons la méthode des éléments finis pour propager l'onde dans le fluide, les grains étant modélisés en 2D par des sphères rigides et incompressibles afin de satisfaire les hypothèses de l'algorithme de dynamique moléculaire. Les résultats du modèle sur des expériences numériques simples mais pour lesquelles existent des solutions analytiques de l'acoustique mettent en évidence la validité du nouveau modèle. Nous en donnons une illustration pour l'étude des interactions subies par un empilement réaliste de multiples grains mobiles soumis à un signal acoustique. / When an acoustic wave propagates through a granular medium, it causes the grains to move, usually very slightly. In the same way, the movement of a grain embedded in a fluid matrix generates an acoustic wave. In both cases, acoustic energy is transmitted by the fluid and by the inter-granular contacts. We have developed a new numerical model for simulating wave propagation in submerged granular media that takes into account these two modes of energy transport. For the case of dry granular media, the grains are embedded in air whose inertia is so low that the energy it carries can be neglected. These media can be modeled with "Molecular Dynamics" or related methods. On the contrary, when granular media are submerged in water, the energy carried by the fluid cannot be neglected, rendering their modelization much more difficult. We use the fictitious domain method with distributed Lagrange multipliers to couple the equation of motion of the grains to the wave equation of the fluid. We use finite elements to propagate the wave in the fluid, and the grains are modeled in 2D by rigid, incompressible spheres compatible with the hypotheses of Molecular Dynamics. To validate the model, we perform series of numerical experiments whose results are compared to analytic solutions from acoustics. We also perform a simulation with hundreds of grains under an incident wave to demonstrate the possibilities of the model.

Notre travail est consacré à la définition, l'analyse et l'implémentation de nouveaux algorithmes numériques pour l'approximation de la solution de problèmes à 2 dimensions du type problème de Stefan. Dans ce type de problèmes une équation aux dérivée partielle parabolique posée sur un ouvert omega quelconque est couplée avec une autre équation qui contrôle la frontière gamma du domaine lui même. Les difficultés classiquement associés à ce type de problèmes sont: la formulation en particulier de l'équation pour le bord du domaine, l'approximation de la solution liées à la forme quelconque du domaine, les difficultés associées à l'implication des opérateurs de trace (approximation, conditionnement), les difficultés liées aux de régularité fonds du domaine.De plus, de nombreuse situations d'intérêt physique par exemple demandent des approximations de haut degré. Notre travail s'appuie sur une formulation de type espaces de niveaux (level set) pour l'équation du domaine, et une formulation de type domaine fictif (Omega) pour l'équation initiale.Le contrôle des conditions aux limites est effectué à partir de multiplicateurs de Lagrange agissant sur une frontière (Gamma) dite de contrôle différente de frontière(gamma) du domaine (omega). L'approximation est faite à partir d'un schéma aux différences finies pour les dérivées temporelle et une discrétisation à l'aide d'ondelettes bi-dimensionelles pour l'équation initiale et une dimensionnelle pour les multiplicateurs de Lagrange. Des opérateurs de prolongement de omega à Omega sont également construits à partir d'analyse multiéchelle sur l'intervalle. Nous obtenons aussi: une formulation pour laquelle existence de la solution est démontrées, un algorithme convergent pour laquelle une estimation globale d'erreur (sur Omega) est établie, une estimation intérieure prouvant sur l'erreur à un domaine omega, overline omega subset Xi, des estimations sur les conditionnement associés a l'opérateur de trace, des algorithmes de prolongement régulier. Différentes expériences numériques en 1D ou 2D sont effectuées. Le manuscrit est organisé comme suit: Le premier chapitre rappelle la construction des analyses multirésolutions, les propriétés importantes des ondelettes et des algorithmes numériques liées à l'application d'opérateurs aux dérivées partielles. Le second chapitre donne un aperçu des méthodes de domaine fictif classiques, approchées par la méthode de Galerkin ou de Petrov-Galerkin. Nous y découvrons les limites de ces méthodes ce qui donne la direction de notre travail. Le chapitre trois présente notre nouvelle méthode de domaine fictif que l'on appelle méthode de domaine fictif lisse.L'approximation est grâce à une méthode d'ondelettes de type Petrov-Galerkin. Cette section contient l'analyse théorique et décrit la mise en œuvre numérique. Différents avantages de cette méthode sont démontrés. Le chapitre quatre introduit une technique de prolongement régulier. Nous l'appliquons à des problèmes elliptiques en 1D ou 2D.\par Le cinquième chapitre décrit quelques simulations numériques de problème de Stefan. Nous testons l'efficacité de notre méthode sur différents exemples dont le problème de Stefan à 2 phases avec conditions aux limites de Gibbs-Thomson. / Our work is devoted to the definition, analysis and implementation of a new algorithms for numerical approximation of the solution of 2 dimensional Stefan problem. In this type of problem a parabolic partial differential equation defined on an openset Omega is coupled with another equation which controls the boundary gamma of the domain itself. The difficulties traditionally associated with this type of problems are: the particular formulation of equation on the boundary of domain, the approximation of the solution defined on general domain, the difficulties associated with the involvement of trace operation (approximation, conditioning), the difficulties associated with the regularity of domain. Addition, many situations of physical interest, for example,require approximations of high degree. Our work is based on aformulation of type level set for the equation on the domain, and aformulation of type fictitious domain (Omega) for the initialequation. The control of boundary conditions is carried out throughLagrange multipliers on boundary (Gamma), called control boundary, which is different with boundary (gamma) of the domain (omega). The approximation is done by a finite difference scheme for time derivative and the discretization by bi-dimensional wave letfor the initial equation and one-dimensional wave let for the Lagrange multipliers. The extension operators from omega to Omega are also constructed from multiresolution analysis on theinterval. We also obtain: a formulation for which the existence of solution is demonstrated, a convergent algorithm for which a global estimate error (on Omega) is established, interior error estimate on domain omega, overline omega subset estimates on the conditioning related to the trace operator, algorithms of smooth extension. Different numerical experiments in 1D or 2D are implemented. The work is organized as follows:The first chapter recalls theconstruction of multiresolution analysis, important properties of wavelet and numerical algorithms. The second chapter gives an outline of classical fictitious domain method, using Galerkin or Petrov-Galerkin method. We also describe the limitation of this method and point out the direction of our work.\par The third chapter presents a smooth fictitious domain method. It is coupled with Petrov-Galerkin wavelet method for elliptic equations. This section contains the theoretical analysis and numerical implementation to embody the advantages of this new method. The fourth chapter introduces a smooth extension technique. We apply it to elliptic problem with smooth fictitious domain method in 1D and 2D. The fifth chapter is the numerical simulation of the Stefan problem. The property of B-spline render us to exactly calculate the curvature on the moving boundary. We use two examples to test the efficiency of our new method. Then it is used to resolve the two-phase Stefan problem with Gibbs-Thomson boundary condition as an experimental case.