A Triple Reciprocity Boundary ElementMethod For Transient Heat ConductionAnalysis

A triple reciprocity boundary element method is discussed for analyzing heat conduction problems. This method is similar to the well-known dual reciprocity method in that the domain integral in the boundary integral equation is eliminated by approximating a "generalized forcing function" in the interior of the domain with the use of radial basis functions and then using additional reciprocities. The advantage of the triple reciprocity method over the dual reciprocity method is that the generalized forcing function is determinate, and hence, no additional unknowns are introduced by the interior approximation. Hence, in principle, the triple reciprocity method has the potential to be more efficient. However, because of the poor behavior of the approximate particular solution associated with the triple reciprocity method, this method is less reliable than the dual reciprocity method. INTRODUCTION The boundary element method (BEM) has become an attractive alternative technique to domain methods such as the finite difference and finite element methods for solving partial differential equations. Originally, the method was used primarily to analyze linear elliptic boundary value problems because of the inherent reduction in the dimensionality of the problem. However, subsequently, the BEM has been shown to be a viable technique for a variety of nonelliptic linear and nonlinear PDEs. In particular, several boundary element formulations have been developed to solve parabolic differential equations. These formulations generally fall into one of three categories, namely, Laplace transform boundary element methods (Rizzo and Shippy*; Moridis^), total Green's function methods (Wrobel and Brebbia^; Taigbenu and Liggett^), and discretization in time boundary element methods (Curran et a/.̂ ; Ingber and Phan-Thien^). For nonlinear parabolic Transactions on Modelling and Simulation vol 8, © 1994 WIT Press, www.witpress. com, ISSN 1743-355X 42 Boundary Element Technology equations, only the third category based on discretization in time has been used with much success (Wrobel and Brebbia^; Azevedo and Wrobel*). Position or functional dependent conductivities cause difficulties for Lapalce transform boundary element methods and time-dependent Green's functions simply do not exist for this class of problems. Boundary element methods based on discretization in time discretize the spatial derivatives using the boundary element method and the temporal derivatives using the finite difference method. After performing the finite difference approximation for the time derivative, the resulting equation can be analyzed as an elliptic boundary value problem at each time step. There are essentially two alternative formulations for discretization in time methods applied to transient heat conduction problems. In the first formulation, the finite difference approximation for the time derivative term and the forcing function are combined into a generalized forcing function. The generalized forcing function is approximated in the interior of the domain using radial basis functions. An approximate particular solution can then be constructed and one additional application of Green's second identity can be used to eliminate the domain integral in the resulting boundary integral equation. Hence, this method is designated as a dual reciprocity method since each application of Green's second identity represents one reciprocity. In the second formulation, only the discretized portion of the time derivative term containing values of the dependent variable at previous time steps is combined with the forcing function into a generalized forcing function. The domain integral can similarly be eliminated in this formulation by constructing an approximate particular solution, and then twice applying Green's second identity. Hence, this method is designated as a triple reciprocity method. In the dual reciprocity method, because of the introduction of interior unknowns into the generalized forcing function, additional collocation is required to either eliminate the interior unknowns (along with the inversion of a dense matrix) or to simply close the system of equations. No additional collocation is required for the triple reciprocity method. Because of this, the triple reciprocity method has the potential to be much more economical. The purpose of this paper is to present the formulation for the triple reciprocity method and discuss some of the benefits and limitations of the method. In fact, there are some severe mathematical problems associated with the construction of an approximate particular solution for the triple reciprocity method. However, if these problems can be overcome, this method may offer a more economical BEM approach for solving heat conduction problems compared to the dual reciprocity method.