An energy based macro-element method via a coupled finite element and boundary integral formulation

Abstract

An energy-based, systematic method for the coupling of the finite element (FE) method and the boundary integral equation (BIE) method is described in this paper. This method allows the use of the BIE method to represent those subdomains of a structure that are best suited to BIE method, and the use of the FE method to represent the rest of the structure. Different subdomains and their associated FE or BIE representations are coupled naturally through the total potential energy functional of the system. The associated discretized problem from the proposed method consists of a linear system of equations with a symmetric and blockwise banded matrix. As in regular FE methods, the derivation is started from the total potential energy principle. However, in those subdomains that are best suited to the BIE method, the exact solution of the associated free field equation is used to represent the true solution. The size and shape of each BIE subdomain, called “macro-element” in this paper, may be designed freely to meet various practical requirements concerning, for example, numerical efficiency, machine storage limitations, mesh generations, etc. Most significantly, unlike the existing BIE techniques, the present method does not seem to require special treatments for corner effects, thus reducing the computational complexity. Numerical experiments have been performed for generalized Poisson's equation as a prototype situation. The extension to 2D-3D elastic problems is straightforward.