Developments in the method of finite spheres : efficiency and coupling to the traditional finite element method

Abstract

In this thesis we develop some advances in the method of finite spheres which is a truly meshless numerical technique for the solution of boundary value problems on geometrically complex domains. We present the development of a preprocessor for the auto-generation of finite spheres on two-dimensional computational domains. The techniques enable to determine the radii of the spheres as well as to detect the boundary of the analysis domain. The numerical integration for the calculation of stiffness matrices is expensive. However, by utilizing the compact support characteristic it is possible to transform the integral equations into more efficient expressions. The improved equations reduce the effort of integration because for most terms, only line integrations are used. We also propose a new coupling scheme to couple finite element discretizations with finite spheres. The idea is that we can use finite elements and finite spheres simultaneously to utilize their mutual advantages. Hence, we can employ finite spheres only in areas where their use is efficient. In addition, we propose an enriching scheme which makes it possible to superpose spheres on conventional finite element topologies to reach a higher order of convergence in the numerical solution of problems.