Centroidal Voronoi Tessellations for Mesh Generation: From Uniform to Anisotropic Adaptive Triangulations

Dr. Hoa V. Nguyen

Florida State University

Abstract

  Mesh generation in regions in Euclidean space is a central task in computational science, especially for commonly used numerical methods for the solution of partial differential equations (PDEs), e.g., finite element and finite volume methods. Mesh generation can be classified into several categories depending on the element sizes (uniform or non-uniform) and shapes (isotropic or anisotropic). Uniform meshes have been well studied and still find application in a wide variety of problems. However, when solving certain types of partial differential equations for which the solution variations are large in some small regions of the domain, non-uniform meshes result in more efficient calculations. This motivates us to develop an adaptive non-uniform anisotropic mesh refinement based on the centroidal Voronoi tessellation (CVT) concept. In the context of solving the convection-diffusion equation with emphasis on convection-dominated problems, our method not only refines the underlying grid at these small regions but also stretches the elements according to the solution variation. Three main ingredients are incorporated to improve the accuracy of numerical solutions and increase the algorithm's robustness and efficiency. First, a streamline upwind Petrov Galerkin method is used to produce a stabilized solution. Second, an adapted metric tensor is computed from the approximate solution. Third, optimized anisotropic meshes are generated from the computed metric tensor. Our algorithm has been tested on a variety of 2-dimensional examples. It is robust in detecting layers and efficient in resolving non-physical oscillations in the numerical approximation.

Tuesday, April 29th, 3:00pm
101 Stanley Thomas Hall
Tulane University, Uptown.

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