A finite element method and the method of finite spheres enriched for analysis of wave propagations

Abstract

The objective of this thesis is to present a finite element method and the method of finite spheres enriched for the solution of various wave propagation problems. The first part of this thesis is to present an enriched finite element method which is an extension of the procedure introduced by Kohno, Bathe, and Wright for one dimensional problems. Specifically, the novelties are: two-dimensional problems are solved (and three-dimensional problems would be tackled similarly), a scheme is given to overcome ill-conditioning, the method is presented for time-dependent problems, and focus is on the solution of problems in solids and structures using real arithmetic only. The method combines advantages of finite element and spectral techniques, but an important point is that it preserves the fundamental properties of the finite element method. The second part of this thesis focuses on developing the method of finite spheres for the analysis of wave propagations. This method is a truly meshless technique developed for the solution of boundary value problems on geometrically complex domains. In the new development trigonometric functions are used to interpolate the solution of wave propagations. An effective numerical integration rule resulting in a significant reduction in computational cost is presented. Several numerical examples are provided demonstrating the effectiveness of the scheme.