Despite the linearity of the Navier equations, solutions to complex boundaryvalue problems require substantial computing resources, especially in the so-called exterior problems, where the deformation field in the space between the inclusions to infinity must be calculated. In the traditional spatial methods, such as finite difference, finite element, or finite volume, this space must be discretized, perhaps with the help of "infinite" elements or a truncation scheme at a finite but large distance from the inclusions (Beer and Watson, Zienkiewicz and Morgan). There are two important limitations of spatial methods. The first is the mesh generation problem. To be numerically efficient, we must use unstructured mesh and concentrate our effort on where it is needed. Efficient two-dimensional, unstructured, automatic mesh generation schemes exist-one good example is Jin and Wiberg-but unstructured three-dimensional mesh generation is still an active area of research. The second limitation is much more severe: even a moderately complicated problem requires the use of supercomputers (e.g., Graham et al.). Since we are concerned with the large-scaled simulations of particulate composites, with the aim of furnishing constitutive information for modeling purposes, our system will possibly have tens of thousands of particles, and therefore the spatial methods are out of the question. We have seen how the deformation field can be represented by a boundary integral equations, either by a direct method, which deals directly with primitive variables (displacement and trciction) on the surface of the domain, or by the indirect method, where the unknowns are the fictitious densities on the surface of the domain. When the field point is allowed to reside on the surface of the domain, then a set of boundary integral equations results that relates only to the variables on the boundary (displacement and traction, or fictitious densities), and this is the basis of the boundary element method. The boundary is then discretized, and the integrals are evaluated by suitable quadratures; this then leads to a set of algebraic equations to be solved for the unknown surface variables.
Numerical solution of boundary integral equations by means of attenuation factors
