A Comparison of Several Techniques of Coupling the Method of Finite Spheres to the Finite Element Method

摘要

The capability to automatically solve boundary value problems on complex domains has led to the widespread use of the finite element techniques. However, problems associated with the generation of a "good quality" mesh has resulted in the development of so-called 'meshfree methods', which offer an alterative to the finite element methods by circumventing the problem of mesh generation. Several of these techniques have been reviewed in reference [1]. The method of finite spheres (MFS) [1] is a truly mesh free method in which the computational domain is discretized using spheres. The partition of unity paradigm is used to generate the approximation functions and specialized numerical integration schemes have been developed to efficiently compute the terms in the Galerkin weak form [2].To reduce computational costs and still benefit from the advantages of the meshfree nature of our scheme, we have developed several techniques of coupling the method of finite spheres with the finite element technique [2]. The use of finite elements in parts of the domain, especially at the Dirichlet boundary has an additional advantage of facilitating the imposition of boundary conditions. We present different techniques of coupling the method of finite spheres with the finite element method based on Lagrange multipliers, penalty formulation, and several techniques of replacing the Lagrange multipliers with their physical significance. Comparing these techniques using numerical examples and some rough theoretical estimates, we show that the use of Lagrange multipliers results in greatest computational efficiency for the same accuracy of solution.