Ces dernières années, les techniques d'imagerie ont fait l'objet de beaucoup d'améliorations. Elles permettent de fournir des images numériques 2D ou 3D précises de zones parfois invisibles à l’œil nu. Ces techniques s'appliquent dans de nombreux domaines comme l'industrie cinématographique, la photographie ou l'imagerie médicale... Dans cette thèse, l'imagerie sera utilisée pour effectuer des simulations numériques en la couplant avec un solveur éléments finis. Nous présenterons, en premier lieu, la morphologie mathématique et la méthode d'immersion d'image. Elles permettront l'extraction d'informations permettant la transformation d'une image dans un maillage exploitable. Puis, une méthode itérative d'adaptation de maillage basée sur un estimateur d'erreur sera utilisée afin de construire un maillage optimal. Ainsi, un maillage sera construit uniquement avec les données d'une image. Nous proposerons également une nouvelle méthodologie pour construire une fonction régulière a l'aide d'une méthode de réinitialisation de la distance signée. Deux avantages sont à noter : l'utilisation de la fonction régularisée permet une bonne adaptation de maillage. De plus, elle est directement utilisable par le solveur éléments finis. Les simulations numériques sont donc réalisées en couplant éléments finis stabilisés, adaptation de maillage anisotrope et réinitialisation. L'objectif de cette thèse est donc de simplifier le calcul numérique à partir d'image, d'améliorer la précision numérique, la construction d'un maillage automatique et de réaliser des calculs numériques parallèles efficaces. Les applications envisagées peuvent être dans le domaine médical, de la physique des matériaux ou du design industriel. / Imaging techniques have well improved in the last decades. They may accurately provide numerical descriptions from 2D or 3D images, opening perspectives towards inner information, not seen otherwise, with applications in different fields, like medicine studies, material science or urban environments. In this work, a technique to build a numerical description under the mesh format has been implemented and used in numerical simulations when coupled to finite element solvers. Firstly, mathematical morphology techniques have been introduced to handle image information, providing the specific features of interest for the simulation. The immersed image method was then proposed to interpolate the image information on a mesh. Then, an iterative anisotropic mesh adaptation operator was developed to construct the optimal mesh, based on the estimated error concerning the image interpolation. The mesh is thus directly constructed from the image information. We have also proposed a new methodology to build a regularized phase function, corresponding to the objects we wish to distinguish from the image, using a redistancing method. Two main advantages of having such function are: the gradient of the regularized function performs better for mesh adaptation; the regularized function may be directly used for the finite element solver. Stabilized finite element flow and advection solvers were coupled to the constructed anisotropic mesh and the redistancing function, allowing its application to multiphase flow numerical simulations. All these developments have been extended in a massively parallel context. An important objective of this work is the simplification of the image based computations, through a modified way to segment the image and by coupling all to an automatic way to construct the mesh used in the finite element simulations.

In this thesis we develop 2D parallel unstructured mesh adaptation methods for the solution of partial differential equations (PDEs) by the finite element method (FEM). Additionally, we develop a novel block preconditioner for the iterative solution of the linear systems arising from the finite element discretisation of the Föppl-von Kàrmàn equations. Two of the problems arising in the numerical solution of PDEs by FEM are the memory constraints that limit the solution of large problems, and the inefficiency of solving the associated linear systems by direct or iterative solvers. We initially focus on mesh adaptation, which is a memory demanding task of the FEM. The size of the problem increases by adding more elements and nodes to the mesh during mesh refinement. In problems involving a large number of elements, the problem size is limited by the memory available on a single processor. In order to be able to work with large problems, we use a domain decomposition approach to distribute the problem over multiple processors. One of the main objectives of this thesis is the development of 2D parallel unstructured mesh adaptation methods for the solution of PDEs by the FEM in a variety of problems; including domains with curved boundaries, holes and internal boundaries. Our newly developed methods are implemented in the software library oomph-lib, an open-source object oriented multi-physics software library implementing the FEM. We validate and demonstrate their utility in a set of increasingly complex problems ranging from scalar PDEs to fully coupled multi-physics problems. Having implemented and validated the infrastructure which facilitates the finite-element-based discretisation of PDEs in a distributed environment, we shift our focus to the second problem concerning this thesis and one of the major challenges in the computational solution of PDEs: the solution of the large linear systems arising from their discretisation. For sufficiently large problems, the solution of their associated linear system by direct solvers becomes impossible or inefficient, typically because of memory and time constraints. We therefore focus on preconditioned Krylov subspace methods whose efficiency depends crucially on the provision of a good preconditioner. These preconditioners are invariably problem dependent. We illustrate their application and development in the solution of two elasticity problems which give rise to relatively large problems. First we consider the solution of a linear elasticity problem and compute the stress distribution near a crack tip where strong local mesh refinement is required. We then consider the deformation of thin plates which are described by the nonlinear Föppl-von Kàrmàn equations. A key contribution of this work is the development of a novel block preconditioner for the iterative solution of these equations, we present the development of the preconditioner and demonstrate its practical performance.

Dynamic response of materials to high-speed and high-intensity loading conditions is important in several applications including high-speed flows with droplets, bubbles and particles, and hyper-velocity impact and penetration processes. In such high-pressure physics problems, simulations encounter challenges associated with the treatment of material interfaces, particularly when strong nonlinear waves like shock and detonation waves impinge upon them. To simulate such complicated interfacial dynamics problems, a fixed Cartesian grid approach in conjunction with levelset interface tracking is attractive. In this regard, a sharp interface Cartesian grid-based, Ghost Fluid Method (GFM) is developed for resolving embedded fluid, elasto-plastic solid and rigid (solid) objects in hyper-velocity impact and high-intensity shock loaded environment. The embedded boundaries are tracked and represented by virtue of the level set interface tracking technique. The evolving multi-material interface and the flow are coupled by meticulously enforcing the boundary conditions and jump relations at the interface. In addition, a tree-based Local Mesh Refinement scheme is employed to efficiently resolve the desired physics. The framework developed is generic and is applicable to interfaces separating a wide range of materials and for a broad spectrum of speeds of interaction (O(km/s)). The wide repertoire of problems solved in this work demonstrates the flexibility, stability and robustness of the method in accurately capturing the dynamics of the embedded interface. Shocks interacting with large ensembles of particles are also computed.

Simulating transient compressible flows involving shock waves presents challenges to the CFD practitioner in terms of the mesh quality required to resolve discontinuities and prevent smearing. This document discusses a novel two-dimensional Cartesian anisotropic mesh adaptation technique implemented for transient compressible flow. This technique, originally developed for laminar incompressible flow, is efficient because it refines and coarsens cells using criteria that consider the solution in each of the cardinal directions separately. In this document the method will be applied to compressible flow. The procedure shows promise in its ability to deliver good quality solutions while achieving computational savings. Transient shock wave diffraction over a backward step and shock reflection over a forward step are considered as test cases because they demonstrate that the quality of the solution can be maintained as the mesh is refined and coarsened in time. The data structure is explained in relation to the computational mesh, and the object-oriented design and implementation of the code is presented. Refinement and coarsening algorithms are outlined. Computational savings over uniform and isotropic mesh approaches are shown to be significant.

In this work, we propose, implement and test two fully automated mesh adaptation methods for 1-D multigroup eigenproblems. The first method is the standard hp-adaptive refinement strategy and the second technique is a goal-oriented hp-adaptive refinement strategy. The hp-strategies deliver optimal guaranteed solutions obtained with exponential convergence rates with respect to the number of unknowns. The goal-oriented method combines the standard hp-adaptation technique with a goal-oriented adaptivity based on the simultaneous solution of an adjoint problem in order to compute quantities of interest, such as reaction rates in a sub-domain or point-wise fluxes or currents. These algorithms are tested for various multigroup 1-D diffusion problems and the numerical results confirm the optimal, exponential convergence rates predicted theoretically.

Accurate numerical simulation of reaction-diffusion systems can come with a high cost. A system may be stiff, and solutions may exhibit sharp localized features that require fine grids and small time steps to properly resolve the physical phenomena they represent. The development of efficient methods is crucial to cut down the demands of computational resources. In this thesis we consider the use of adaptive space and time methods driven by a posteriori error estimation. The error estimators for the spatial discretization are built from a variety of sources: the residual of the partial differential equation (PDE) system, gradient recovery operators and interpolation estimates. The interpolation estimates are anisotropic, not relying on classical mesh regularity assumptions. The adapted mesh is therefore allowed to include elements elongated in specified directions, as dictated by the type of solution being approximated. This thesis proposes an element-based adaptation method to be used for a residual estimator. This method avoids the usual conversion of the estimator to a metric, and instead applies the estimator to directly control the local mesh modifications. We derive a new error estimator for the L^2-norm in the same anisotropic setting and adjust the element-based adaptation algorithm to the new estimator. This thesis considers two new adaptive finite element settings for reaction-diffusion problems. The first is the extension to a PDE setting of an estimator for the time discretization with the backward difference formula of order 2 (BDF2), based on an estimator for ordinary differential equation (ODE) problems. Coupled with the residual estimator, we apply a space-time adaptation method. The second is the derivation of anisotropic error estimates for the monodomain model from cardiac electrophysiology. This model couples a nonlinear parabolic PDE with an ODE and this setting presents challenges theoretically as well as numerically. In addition to theoretical considerations, numerical tests are performed throughout to assess the reliability and efficiency of the proposed error estimators and numerical methods.

Cette thèse contribue à un ensemble de travaux consacrés à l’étude d’un schéma ENO centré-sommet (CENO) d’ordre élevé ainsi qu’à l’adaptation de maillage anisotrope pour des calculs de Mécaniques des Fluides précis à l’ordre 3. La première partie des travaux de cette thèse est consacré à une analyse approfondie de la précision du schéma CENO et à la création de termes correcteurs pour améliorer ses propriétés dispersives et dissipatives en une et deux dimensions. On propose un schéma CENO quadratique précis à l’ordre 3, puis cubique précis à l’ordre 4, pour les équations d’Euler des gaz compressibles, ainsi qu’ une première version du schéma avec capture de choc monotone. La deuxième partie des travaux est consacrée à la mise au point d’une plateforme numérique d’adaptation de maillage anisotrope multi-échelle et basée fonctionnelle intégrant le schéma CENO. Nous proposons un nouvel estimateur d’ordre 3 du schéma quadratique basé sur une reconstruction de hessien équivalent et son application à des simulations d’acoustiques instationnaire et de Scramjet stationnaire utilisant nos limiteurs. / This thesis presents to an assembly of work dedicated to the study of high order vertex-centred ENO scheme (CENO) and to anisotropic mesh adaptation for third-order accurate Fluid Mecanics problems. The thesis is structured in two parts. The first part is devoted to a thorough analysis of the CENO scheme accuracy and to the constuction of some corrector terms meant for improving the dissipative and dispersive properties for 1D and 2D numerical problems. We proposed a quadratique third-order accurate CENO scheme, then a cubic fourth-order accurate one, applied to Euler equations for compressible flows. A first monotone, shock capturing version of these scheme is also introduced in the first part. The second part of the thesis focuses on the implementation of a numerical platform for anisotropic multi-scale and goal-oriented mesh adaptivity involving the CENO scheme. A new third-order error estimator for the quadratic scheme is proposed, here based on a reconstuction of the Hessian. Numerical exemples for unsteady acoustic problems and a steady Scramjet problem computed with monotony preserving limiters are presented for validation of the theoretical results.

Code verification is the process of ensuring, to the degree possible, that there are no algorithm deficiencies and coding mistakes (bugs) in a scientific computing simulation. In this work, techniques are presented for performing code verification of boundary conditions commonly used in compressible and incompressible Computational Fluid Dynamics (CFD) codes. Using a compressible CFD code, this study assesses the subsonic inflow (isentropic and fixed-mass), subsonic outflow, supersonic outflow, no-slip wall (adiabatic and isothermal), and inviscid slip-wall. The use of simplified curved surfaces is proposed for easier generation of manufactured solutions during the verification of certain boundary conditions involving many constraints. To perform rigorous code verification, general grids with mixed cell types at the verified boundary are used. A novel approach is introduced to determine manufactured solutions for boundary condition verification when the velocity-field is constrained to be divergence-free during the simulation in an incompressible CFD code. Order of accuracy testing using the Method of Manufactured Solutions (MMS) is employed here for code verification of the major components of an open-source, multiphase flow code - MFIX. The presence of two-phase governing equations and a modified SIMPLE-based algorithm requiring divergence-free flows makes the selection of manufactured solutions more involved than for single-phase, compressible flows. Code verification is performed here on 2D and 3D, uniform and stretched meshes for incompressible, steady and unsteady, single-phase and two-phase flows using the two-fluid model of MFIX. In a CFD simulation, truncation error (TE) is the difference between the continuous governing equation and its discrete approximation. Since TE can be shown to be the local source term for the discretization error, TE is proposed as the criterion for determining which regions of the computational mesh should be refined/coarsened. For mesh modification, an error equidistribution strategy to perform r-refinement (i.e., mesh node relocation) is employed. This technique is applied to 1D and 2D inviscid flow problems where the exact (i.e., analytic) solution is available. For mesh adaptation based upon TE, about an order of magnitude improvement in discretization error levels is observed when compared with the uniform mesh. / Ph. D.

In the field of Computational Fluid Dynamics, several types of mesh adaptation strategies are used to enhance a mesh’s quality, thereby improving simulation speed and accuracy. Mesh smoothing (r-refinement) is a simple and effective technique, where nodes are repositioned to increase or decrease local mesh resolution. Mesh partitioning divides a mesh into sections, for use on distributed-memory parallel machines. As a more abstract form of modeling, graph theory can be used to simulate many real-world problems, and has applications in the fields of computer science, sociology, engineering and transportation, to name a few. One of the more important graph analysis tasks involves moving through the graph to evaluate and calculate nodal connectivity. The basic structures of meshes and graphs are the same, as both rely heavily on connectivity information, representing the relationships between constituent nodes and edges. This research examines the parallelization of these algorithms using commodity graphics hardware; a low-cost tool readily available to the computing community. Not only does this research look at the benefits of the fine-grained parallelism of an individual graphics processor, but the use of Message Passing Interface (MPI) on large-scale GPU-based supercomputers is also studied.

This paper gives an overview of the mathematical and practical tools that can be used in turbo-machinery RANS simulation to realize unstructured mesh adaptation. It first presents the concept of metric and recalls that the hessian of the physical flow properties can become, thanks to small modifications, both a metric and a upper bound of the P1 projection error. The resulting metric is then studied on a simple 2D case. In a second part, the industrial application of this concept is addressed and the tools used to overcome the turbo-machinery specificities are explained. Finally, some 2D and 3D results are presented.