Domain Integrals

The domain integrals arise in the boundary element formulation for the potential, elastostatic, elastodynamic, fluid flow and other problems which involve body forces or or nonlinearities. Thus, for example, the domain integral appears in the boundary integral equation in (6.5.23) in elastostatic problems; in (7.1.10) for elastodynamic problems; in (8.1.11) for fluid flow problems; in (9.2.18) in piezoelectric problems; and in (10.3.18) for the Poisson problem. The boundary integral equation for the Helmholtz equation (∇2 + k 2 )u = b in the case of forced oscillations, where b is a function of space that measures the yield of the sources that are continuously disturbed or concentrated in a point, leads to a domain integral of the type (10.3.18). This integral is encountered in fracture problems, piezoelectrics,and also in aerodynamic flows around lifting bodies and in flows through porous media. If the body forces are functions of space coordinates only, the domain integrals do not introduce any unknowns. But the boundary element method loses its advantage over the boundary-only formulation. The interior cell method developed in §10.3 is not a powerful method to numerically compute the domain integrals since it involves an interior discretization and evaluation of the double integral over each interior cell, which in turn increases the numerical data considerably. In order to avoid this situation there are three different methods available at present to transform the domain integrals into boundary integrals. These methods have historically evolved as follows: (i) Dual reciprocity method (DRM), (ii) Fourier series expansion method (FSM), and (iii) Multiple reciprocity method (MRM). The DRM was developed by Nardini and Brebbia (1982, 1985); it is also known as the particular integral method (Pape and Banerjee, 1987). The FSM was developed by Tang (1988), and the MRM by Nowak and Brebbia (1989).