Geometry-based three-dimensional hp-finite element modelling and computations

Abstract

Efficient computational methodologies for hp-finite element modeling and computation for the numerical solution of partial differential equations defined on curved three-dimensional domains are presented.

Examples based on the solution of Poisson's Equation are presented to demonstrate applicability and efficiency of computational techniques developed in this thesis for the solution of partial differential equations in curved three-dimensional domains.

A general and efficient scheme for evaluation of higher order element level integrals is presented based on direct polynomial approximation of the integrand followed by use of precomputed exact values of the resulting polynomial integrals. It is shown that to preserve the rate of convergence of the finite element solution error for second order elliptic boundary value problems using a p-th order basis, the integrand approximation must be at least accurate to order p - 1. Blending schemes are used to construct mesh geometry mapping procedures that conform exactly to the domain boundary definition within a geometric modelling system. The concept of permutation of element parametric coordinates is developed to identify symmetries in the precomputed entries needed for elemental stiffness and mass matrices and elemental force vectors. Specific symmetries that reduce the number of precomputed entries which need to be stored are described.

A novel decomposition of hierarchic shape functions, based on mesh topological adjacency, is developed that allows for the unified and consistent specification and evaluation of variable p-order CO finite element approximations over conforming meshes with mixed topology entities including line, triangle, quadrilateral, tetrahedron, hexahedron, wedge and pyramid elements.

An algorithm to automatically detect and eliminate poorly shaped mesh entities resulting from extremely small geometric model features is described. The algorithm dramatically improves the quality of automatically generated meshes. An automated incremental algorithm based on local mesh modifications is developed to correct unacceptably distorted curvilinear mesh entities.