Numerical methods for all-speed flows in fluid-dynamics and non-linear elasticity

In this thesis we are concerned with the numerical simulation of compressible materials flows, including gases, liquids and elastic solids. These materials are described by a monolithic Eulerian model of conservation laws, closed by an hyperelastic state law that includes the different behaviours of the considered materials. A novel implicit relaxation scheme to solve compressible flows at all speeds is proposed, with Mach numbers ranging from very small to the order of unity. The scheme is general and has the same formulation for all the considered materials, since a direct dependence on the state law is avoided via the relaxation. It is based on a fully implicit time discretization, easily implemented thanks to the linearity of the transport operator in the relaxation system. The spatial discretization is obtained by a combination of upwind and centered schemes in order to recover the correct numerical viscosity in different Mach regimes. The scheme is validated with one and two dimensional simulations of fluid flows and of deformations of compressible solids. We exploit the domain discretization through Cartesian grids, allowing for massively parallel computations (HPC) that drastically reduce the computational times on 2D test cases. Moreover, the scheme is adapted to the resolution on adaptive grids based on quadtrees, implementing adaptive mesh refinement techinques. The last part of the thesis is devoted to the numerical simulation of heterogeneous multi-material flows. A novel sharp interface method is proposed, with the derivation of implicit equilibrium conditions. The aim of the implicit framework is the solution of weakly compressible and low Mach flows, thus the proposed multi-material conditions are coupled with the implicit relaxation scheme that is solved in the bulk of the flow. Dans cette thèse on s’intéresse à la simulation numérique d’écoulements des matériaux compressibles, voir fluides et solides élastiques. Les matériaux considérés sont décrits avec un modèle monolithique eulérian, fermé avec une loi d’état hyperélastique qui considère les différents comportéments des matériaux. On propose un nouveau schéma de relaxation qui résout les écoulements compressibles dans des différents régimes, avec des nombres de Mach très petits jusqu’à l’ordre 1. Le schéma a une formulation générale qui est la même pour tous le matériaux considérés, parce que il ne dépend pas directement de la loi d’état. Il se base sur une discrétization complétement implicite, facile à implémenter grâce à la linearité de l’opérateur de transport du système de relaxation. La discrétization en éspace est donnée par la combinaison de flux upwind et centrés, pour retrouver la correcte viscosité numérique dans les différents régimes. L’utilisation de mailles cartésiennes pour les cas 2D s’adapte bien à une parallélisation massive, qui permet de réduire drastiquement le temps de calcul. De plus, le schéma a été adapté pour la résolution sur des mailles quadtree, pour implémenter l’adaptivité de la maille avec des critères entropiques. La dernière partie de la thèse concerne la simulation numérique d’écoulements multi-matériaux. On a proposé une nouvelle méthode d’interface “sharp”, en dérivant les conditions d’équilibre en implicite. L’objectif est la résolution d’interfaces physiques dans des régimes faiblement compressibles et avec un nombre de Mach faible, donc les conditions multi-matériaux sont couplées au schéma implicite de relaxation

A geometric multigrid library for quadtree/octree AMR grids coupled to MPI-AMRVAC

We present an efficient MPI-parallel geometric multigrid library for quadtree (2D) or octree (3D) grids with adaptive refinement. Cartesian 2D/3D and cylindrical 2D geometries are supported, with second-order discretizations for the elliptic operators. Periodic, Dirichlet, and Neumann boundary conditions can be handled, as well as free-space boundary conditions for 3D Poisson problems, for which we use an FFT-based solver on the coarse grid. Scaling results up to 1792 cores are presented. The library can be used to extend adaptive mesh refinement frameworks with an elliptic solver, which we demonstrate by coupling it to MPI-AMRVAC. Several test cases are presented in which the multigrid routines are used to control the divergence of the magnetic field in magnetohydrodynamic simulations

A geometric multigrid library for quadtree/octree AMR grids coupled to MPI-AMRVAC

We present an efficient MPI-parallel geometric multigrid library for quadtree (2D) or octree (3D) grids with adaptive refinement. Cartesian 2D/3D and cylindrical 2D geometries are supported, with second-order discretizations for the elliptic operators. Periodic, Dirichlet, and Neumann boundary conditions can be handled, as well as free-space boundary conditions for 3D Poisson problems, for which we use an FFT-based solver on the coarse grid. Scaling results up to 1792 cores are presented. The library can be used to extend adaptive mesh refinement frameworks with an elliptic solver, which we demonstrate by coupling it to MPI-AMRVAC. Several test cases are presented in which the multigrid routines are used to control the divergence of the magnetic field in magnetohydrodynamic simulations

Level set and PDE methods for visualization

Notes from IEEE Visualization 2005 Course #6, Minneapolis, MN, October 25, 2005. Retrieved 3/16/2006 from http://www.cs.drexel.edu/~david/Papers/Viz05_Course6_Notes.pdf.Level set methods, an important class of partial differential equation (PDE) methods, define dynamic surfaces implicitly as the level set (isosurface) of a sampled, evolving nD function. This course is targeted for researchers interested in learning about level set and other PDE-based methods, and their application to visualization. The course material will be presented by several of the recognized experts in the field, and will include introductory concepts, practical considerations and extensive details on a variety of level set/PDE applications. The course will begin with preparatory material that introduces the concept of using partial differential equations to solve problems in visualization. This will include the structure and behavior of several different types of differential equations, e.g. the level set, heat and reaction-diffusion equations, as well as a general approach to developing PDE-based applications. The second stage of the course will describe the numerical methods and algorithms needed to implement the mathematics and methods presented in the first stage, including information on implementing the algorithms on GPUs. Throughout the course the technical material will be tied to applications, e.g. image processing, geometric modeling, dataset segmentation, model processing, surface reconstruction, anisotropic geometric diffusion, flow field post-processing and vector visualization. Prerequisites: Knowledge of calculus, linear algebra, computer graphics, visualization, geometric modeling and computer vision. Some familiarity with differential geometry, differential equations, numerical computing and image processing is strongly recommended, but not required

Solveur Parallèle pour l'Equation de Poisson sur Mailles Superposées et Hiérarchiques, dans le Cadre du Langage Python

Adaptive discretizations are important in compressible/incompressible flow problems since it is often necessary to resolve details on multiple levels, allowing large regions of space to be modeled using a reduced number of degrees of freedom (reducing the computational time). There are a wide variety of methods for adaptively discretizing space, but Cartesian grids have often outperformed them even at high resolutions due to their simple and accurate numerical stencils and their superior parallel performances. Such performance and simplicity are in general obtained applying a finite-difference scheme for the resolution of the problems involved, but this discretization approach does not present, by contrast, an easy adapting path. In a finite-volume scheme, instead, we can incorporate different types of grids, more suitable for adaptive refinements, increasing the complexity on the stencils and getting a greater flexibility. The Laplace operator is an essential building block of the Navier-Stokes equations, a model that governs fluid flows, but it occurs also in differential equations that describe many other physical phenomena, such as electric and gravitational potentials, and quantum mechanics. So, it is a very important differential operator, and all the studies carried out on it, prove its relevance. In this work will be presented 2D finite-difference and finite-volume approaches to solve the Laplacian operator, applying patches of overlapping grids where a more fined level is needed, leaving coarsermeshes in the rest of the computational domain. These overlapping grids will have generic quadrilateral shapes. Specifically, the topics covered will be: 1) introduction to the finite difference method, finite volume method, domain partitioning, solution approximation; 2)overview of different types of meshes to represent in a discrete way the geometry involved in a problem, with a focus on the octree data structure, presenting PABLO and PABLitO. The first one is an external library used to manage each single grid’s creation, load balancing andinternal communications, while the second one is the Python API of that library written ad hoc for the current project; 3) presentation of the algorithm used to communicate data between meshes (being all of them unaware of each other’s existence) using MPI inter-communicators and clarification of the monolithic approach applied building the final matrix for the system to solve, taking into account diagonal, restriction and prolongation blocks; 4) presentation of some results; conclusions, references. It is important to underline that everything is done under Python as programming framework, using Cython for the writing of PABLitO, MPI4Py for the communications between grids, PETSc4py for the assembling and resolution parts of the system of unknowns, NumPy for contiguous memory buffer objects. The choice of this programming language has been made because Python, easy to learn and understand, is today a significant contender for the numerical computing and HPC ecosystem, thanks to its clean style, its packages, its compilers and, why not, its specific architecture optimized versions.Les discrétisations adaptatives sont importantes dans les problèmes de flux compressible/incompressible puisqu'il est souvent nécessaire de résoudre des détails sur plusieurs niveaux, en permettant de modéliser de grandes régions d'espace en utilisant un nombre réduit de degrés de liberté (et en réduisant le temps de calcul). Il existe une grande variété de méthodes de discrétisation adaptative, mais les grilles cartésiennes sont les plus efficaces, grâce à leurs stencils numériques simples et précis et à leurs performances parallèles supérieures. Et telles performance et simplicité sont généralement obtenues en appliquant un schéma de différences finies pour la résolution des problèmes, mais cette approche de discrétisation ne présente pas, au contraire, un chemin facile d'adaptation.Dans un schéma de volumes finis, en revanche, nous pouvons incorporer différents types de maillages, plus appropriées aux raffinements adaptatifs, en augmentant la complexité sur les stencils et en obtenant une plus grande flexibilité. L'opérateur de Laplace est un élémentessentiel des équations de Navier-Stokes, un modèle qui gouverne les écoulements de fluides, mais il se produit également dans des équations différentielles qui décrivent de nombreux autres phénomènes physiques, tels que les potentiels électriques et gravitationnels. Il s'agit donc d'un opérateur différentiel très important, et toutes les études qui ont été effectuées sur celui-ci, prouvent sa pertinence. Dans ce travail seront présentés des approches de différences finies et de volumes finis 2D pour résoudre l'opérateur laplacien, en appliquant des patchs de grilles superposées où un niveau plus fin est nécessaire, en laissant des maillages plus grossiers dans le reste du domaine de calcul. Ces grilles superposées auront des formes quadrilatérales génériques. Plus précisément, les sujets abordés seront les suivants: 1) introduction à la méthode des différences finies, méthode des volumes finis, partitionnement des domaines, approximation de la solution; 2)récapitulatif des différents types de maillages pour représenter de façon discrète la géométrie impliquée dans un problème, avec un focus sur la structure de données octree, présentant PABLO et PABLitO. Le premier est une bibliothèque externe utilisée pour gérer la création de chaque grille, l'équilibrage de charge et les communications internes, tandis que la seconde est l'API Python de cette bibliothèque, écrite ad hoc pour le projet en cours; 3) la présentation de l'algorithme utilisé pour communiquer les données entre les maillages (en ignorant chacune l'existence de l'autre) en utilisant les intercommunicateurs MPI et la clarification de l'approche monolithique appliquée à la construction finale de la matrice pour résoudre le système, en tenant compte des blocs diagonaux, de restriction et de prolongement; 4) la présentation de certains résultats; conclusions, références. Il est important de souligner que tout est fait sous Python comme framework de programmation, en utilisant Cython pour l'écriture de PABLitO, MPI4Py pour les communications entre grilles, PETSc4py pour les parties assemblage et résolution du système d'inconnues, NumPy pour les objets à mémoire continue. Le choix de ce langage de programmation a été fait car Python, facile à apprendre et à comprendre, est aujourd'hui un concurrent significatif pour l'informatique numérique et l'écosystème HPC, grâce à son style épuré, ses packages, ses compilateurs et pourquoi pas ses versions optimisées pour des architectures spécifiques

Numerical Simulations of the Two-phase flow and Fluid-Structure Interaction Problems with Adaptive Mesh Refinement

Numerical simulations of two-phase flow and fluid structure interaction problems are of great interest in many environmental problems and engineering applications. To capture the complex physical processes involved in these problems, a high grid resolution is usually needed. However, one does not need or maybe cannot afford a fine grid of uniformly high resolution across the whole domain. The need to resolve local fine features can be addressed by the adaptive mesh refinement (AMR) method, which increases the grid resolution in regions of interest as needed during the simulation while leaving general estimates in other regions. In this work, we propose a block-structured adaptive mesh refinement (BSAMR) framework to simulate two-phase flows using the level set (LS) function with both the subcycling and non-subcycling methods on a collocated grid. To the best of our knowledge, this is the first framework that unifies the subcycling and non-subcycling methods to simulate two-phase flows. The use of the collocated grid is also the first among the two-phase BSAMR framework, which significantly simplifies the implementation of multi-level differential operators and interpolation schemes. We design the synchronization operations, including the averaging, refluxing, and synchronization projection, which ensures that the flow field is divergence-free on the multi-level grid. It is shown that the present multi-level scheme can accurately resolve the interfaces of the two-phase flows with gravitational and surface tension effects while having good momentum and energy conservation.Comment: 178 page