An effective overlapping finite element method : the method of finite spheres for three-dimensional linear elasticity problems

Abstract

The method of finite spheres is an effective overlapping finite element method developed to overcome challenges in mesh-based numerical methods. Commonly recognized challenges include mesh generation for geometrically complex domains, severe element distortion in nonlinear analysis with large strain effects, and modeling problems involving discontinuities and singularities which require mesh alignment and refinement. Substantial research efforts have been focused on addressing these issues, resulting in the introduction of numerous meshless methods. The ultimate purpose of the method of finite spheres is to be distinguished as a reliable and efficient meshless computational technique for the solution of boundary value problems on complex domains, to supplement the capabilities of the standard finite element method. The reliability of the method of finite spheres was previously verified for one- and two-dimensional linear static analysis of solids and fluids. The objective of this thesis is to demonstrate the reliability and effectiveness of the method of finite spheres for the solution of practical three-dimensional linear elasticity problems. An effective local approximation space, which is multiplied by the Shepard partition of unity function, is presented for the construction of three-dimensional interpolation functions. The piecewise Gauss-Legendre quadrature rule, a simple and efficient scheme for the integration of complicated non polynomial basis functions, is introduced for three-dimensional spherical domains. The three-dimensional formulation of the method of finite spheres is implemented in a commercial finite element analysis program with the user-element subroutine, in order to perform a computational efficiency comparison with the standard finite element method. A series of increasingly complex three-dimensional problems is studied (1) to establish the reliability of the method of finite spheres for practical three-dimensional linear elastic static problems and (2) to assess the effectiveness of the method of finite spheres as compared to the standard finite element method. Solution times indicate that the method of finite spheres is approximately an order of magnitude slower than the standard finite element method during the computation phase. However, this is still a promising result since there are significant time savings for the method of finite spheres during the pre-processing phase, particularly in the discretization of complicated three-dimensional geometries.