A Finite-Difference Method for the Variable Coefficient Poisson Equation on Hierarchical Cartesian Meshes

We consider problems governed by a linear elliptic equation with varying coefficients across internal interfaces. The solution and its normal derivative can undergo significant variations through these internal boundaries. We present a compact finite-difference scheme on a tree-based adaptive grid that can be efficiently solved using a natively parallel data structure. The main idea is to optimize the truncation error of the discretization scheme as a function of the local grid configuration to achieve second-order accuracy. Numerical illustrations are presented in two and three-dimensional configurations. Finite difference method; Hierarchical Cartesian grid; Octree/Quadtree; Variable coefficient Poisson equation

Simulation of the Ferrofluid Interface

Ferrofluids were initially invented as an additive for rocket fuels. The commercial applications of ferrofluids have since expanded, and have become popularized as desktop toys as well as an art form. Since 1987, ferrofluid simulations have been developed for engineering and physics applications. The Rosensweig instability, a visually appealing behaviour of ferrofluids, has been one of the focuses for ferrofluid simulation. While some simulations were successful, they have either placed restrictive assumptions on the problem, used non-physical models, or failed to reproduce this phenomenon due to high computational expense. One recent exception was a concurrent work from 2019 by Huang et al. that used a particle based fluid simulation method. They successfully reproduced this phenomenon without these issues, but adopted a different computational approach. We present a methodology for simulating ferrofluid with its accompanying Rosensweig instability using finite difference schemes within a grid based simulation. This is the first simulator to use a grid based methodology to approximately reproduce the Rosensweig instability. After Huang et al., our simulator is the second to approximately reproduce the Rosensweig instability with a nonrestrictive physically faithful model. The simulator accommodates any magnetic field and initial configuration of the fluid. Due to the high level of interface detail required by the Rosensweig instability we developed improved curvature estimation and surface tracking methods. We use a normal-aligned height function curvature stencil paired with a modified version of the particle level set. Instead of using particles for error detection, they are directly seeded on the interface to track it. These particles can then be used directly to determine the interface location for curvature estimation. The new particle level set is also able run on a GPU for a twenty to thirty times performance improvement compared to its CPU counterpart. This coupling of methods produces curvature estimates that are two to five times more accurate than when operating independently of each other. After verifying the curvature and surface tracking methods, the ferrofluid simulator is demonstrated by inducing motion into a ferrofluid droplet using an applied magnetic field. Lastly, the Rosensweig instability is produced for a pool of ferrofluid sitting in a dish above a dipole magnet

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