Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method

This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain. The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities. The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy. In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost. To reach this goal, we first need to answer the question: ''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?'' We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications. This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh. Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation. Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements. The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios

On pixel-exact rendering for high-order mesh and solution

International audienceWith the increasing use of high-order methods and high-order meshes, scientific visualization software need to adapt themselves to reliably render the associated meshes and numerical solutions. In this paper, a novel approach, based on OpenGL 4 framework, enables a GPU-based rendering of high-order meshes as well as an almost pixel-exact rendering of high-order solutions. Several aspects of the OpenGL Shading Language and in particular the use of dedicated shaders (GPU programs) allows to answer this visualization challenge. Fragment shaders are used to compute the exact solution for each pixel, made possible by the transfer of degrees of freedom and shape functions to the GPU with textures. Tessellation shaders, combined with geometric error estimates, allow us to render high-order curved meshes by providing an adaptive subdivision of elements on the GPU directly. A convenient way to compute bounds for high-order solutions is described. The interest of using Bézier basis instead of Lagrange functions lies in the existence of fast and robust evaluation of polynomial functions with de Casteljau algorithm. A technique to plot highly nonlinear isolines and wire frames with a desired thickness is derived. It is based on a finite difference scheme performed on GPU. In comparison with standard techniques, we remove the use of any linear interpolation step and the need to generate a priori a fixed subdivided mesh. This reduces the memory footprint, improves the accuracy and the speed of the rendering. Finally, the method is illustrated with various 3D examples

5-6-7 meshes

In this thesis, we introduce a new type of meshes called 5-6-7 meshes. For many mesh processing tasks, low- or high-valence vertices are undesirable. At the same time, it is not always possible to achieve complete vertex valence regularity, i.e., to only have valence-6 vertices. A 5-6-7 mesh is a closed triangle mesh where each vertex has valence 5, 6, or 7. An intriguing question is whether it is always possible to convert an arbitrary mesh into a 5-6-7 mesh. In this thesis, we answer this question in the positive. We present a 5-6-7 remeshing algorithm which converts any closed triangle mesh with arbitrary genus into a 5-6-7 mesh, which a) closely approximates the original mesh geometrically, e.g., in terms of feature preservation, and b) has a comparable vertex count as the original mesh. We demonstrate the results of our remeshing algorithm on meshes with sharp features and different topology and complexity