ABSTRACT
Boundary integral equations, which form the basis of the boundary element method, have been in existence for a long time. The boundary element method as it is known today, however, has been developed largely in the last four decades. The rapid development of the method may have drawn its motivation from the limitations of the finite element method. A major step in performing the finite element analysis is the discretization of the domain into finite elements, called meshing. For arbitrarily shaped three-dimensional objects, meshing is an extremely tedious job. Often, this step may take weeks, even months, to accomplish, whereas the rest of the analysis may require only a few days. There are situations where the mesh for the object may need to be defined anew several times for the same analysis. In metal-forming operations, the metal workpiece undergoes very large deformations, including large rotations. The individual elements in the mesh may become severely deformed and possess unduly large aspect ratios. Unless the mesh for the deformed workpiece is redefined several times during the analysis, the original mesh may lead to erroneous results. In the simulation of crack propagation in a solid object, the mesh is first defined with respect to the initial geometry of the crack. If the crack does not propagate along element boundaries, a rare occurrence, the crack will intersect one or more elements. To account for the new location of the crack, the mesh may need to be redefined for each advance of the crack front. In shape optimization problems (Saigal and Kane, 1990), the geometry of the object is revised continually in each calculation step to proceed toward its optimal configuration. For each revision of the geometry, a new mesh needs to be defined. The applications mentioned earlier and several others make use of the finite element method undesirable. There have been, and continue to be, several attempts in the literature to obviate the problems associated with meshing. Through the use of appropriate mathematical theorems, the boundary element method reduces the dimensions of the problem by one degree. Thus, for a three-dimensional object, a two-dimensional discretization—that of the surfaces bounding the object—is required. Similarly, for two-dimensional analyses, a one-dimensional discretization of the lines enclosing the object is required. This reduction in the dimensions of the problem by one degree leads to significant advantages in terms of ease of discretization of the domain and of reducing the overall time to perform the analysis of objects with complex geometries. Consider the plate with centrally located hole under uniaxial loading shown in Figure 6.1a. For this problem, the boundary element meshes using quadratic boundary elements are shown in Figure 6.1b and c, respectively. No boundary elements are shown on the axes of symmetry of the plate in the discretized model shown in Figure 6.1c. The effects due to symmetry can be included within the analytical formulation of the boundary elements (Kaljevic and Saigal, 1995b; Saigal et al., 1990a). This eliminates the need for providing boundary elements on the axes of symmetry (Kaljevic and Saigal, 1995a). In view of the meshes shown in Figure 6.1b and c, respectively, the advantage offered by boundary elements in terms of meshing requirements is quite apparent.
