Partition of Unity Methods

8 eXtended Boundary Element Method (XBEM) for Fracture Mechanics and Wave Problems

Jon Trevelyan

Durham University, UK

8.1 Introduction

The boundary element method (BEM) is rooted in the classical theories of potential flow and integral equations, so that its development can be traced back at least as far as finite element methods (FEMs). Indeed, it can be viewed as a discretized form of the integral equations of Fredholm. As the name of the method suggests, the BEM requires the discretization of only the boundary of the object or domain to be analyzed, and not its volume as required in the FEM. Thus, a BEM model of a two-dimensional (2D) object will consist of elements that are line segments describing portions of the perimeter contour, while boundary elements for a 3D object will describe portions of the object’s surface area.

This reduction of dimensionality is a major benefit over the FEM, for a number of reasons: the reduction in data preparation effort and the improved robustness of automatically meshed models are two fairly obvious benefits. However, on further reflection we find other advantages, such as the fact that we can reverse all the element normals so that the analysis domain comprises the infinite region surrounding an internal boundary. This is used to great effect for problems of wave scattering, for example, in which the analysis domain might be the region surrounding a scattering obstacle. Such problems can be very efficiently solved using just a few elements. ...