Source-weighted domain integral approximation for linear transient heat conduction

In employing the Boundary Element Method (BEM) to solve linear transient heat conduction problems, domain integrals need to be calculated. These integrals are generated by initial or pseudo-initial conditions and can be calculated directly by discritizing the domain. The need for domain meshing undermines the elegance of the boundary element approach and so a number of techniques have been developed in an attempt to overcome this. The most recent of these being the Multiple and Dual Reciprocity methods. This paper is concerned with a new approach which involves the direct approximation of fundamental solutions using linear combinations of sources positioned at different points in time. The weighting associated with each source is determined by minimization of the maximum absolute error using a single point exchange algorithm. In this way it is possible to determine the domain integrals to a high degree of accuracy with minimal computational effort. Error bounds for the approximation are naturally provided by the error reduction procedure giving an indication of the number of sources required for accurate domain integrals. The procedure is developed in detail for two and three-dimensional parabolic integral equations. Accuracy and stability are examined and the results of numerical tests are presented.