On the development of reliable and efficient "overlapping finite elements" for the new paradigm of finite element solutions

Abstract

The finite element method has been used widely in the analysis of structures, fluids and multi-physics problems. However, despite its success, there are frequently still great difficulties in establishing an adequate finite element mesh. In engineering analyses of complex components, oftentimes, much more time is spent on reaching an adequate mesh than obtaining the solution of a finite element model. The objective of this thesis is to develop distortion insensitive and computationally efficient overlapping finite elements, including a scheme to couple these elements to traditional finite elements. These procedures are developed for a new paradigm of finite element solutions in which elements can overlap. The property that finite elements can overlap removes many of meshing difficulties, leads to an effective meshing procedure and an overall easy-to-use solution scheme for an analyst or a designer using CAD programs. The overlapping finite elements are designed to have two important fundamental properties: 1/ the numerical integration and solution effort are computationally efficient, and 2/ the overlapping elements are distortion insensitive. To achieve these goals, a new interpolation scheme is proposed based on a two-layer interpolation process: local and global interpolations. Solutions of basic numerical examples are employed to investigate robustness, efficiency and convergence of the new overlapping finite elements. The use of the overlapping finite elements in the meshing of the new paradigm of solutions is presented and discussed. Some problems are solved to illustrate the performance of the solution scheme. The results show that the overlapping elements are very promising, in particular in the new paradigm of analysis using finite elements in CAD.