Analytical evaluation of the origin intensity factor of time-dependent diffusion fundamental solution for a matrix-free singular boundary method formulation

The singular boundary method (SBM) with the empirical formulas of the origin intensity factors (OIFs) can be effectively used to simulate one- and two-dimensional time-dependent diffusion problems. However, there is no such empirical formula available for determining the OIFs in three-dimensional problems so that the traditional inverse interpolation technique (IIT) has to be employed in three-dimensional case. This paper presents the analytical evaluation formulas to derive the OIFs and thereby overcome the above shortcomings. The proposed new formulation not only has clear theoretical foundations, but also ensures good stability compared with the IIT. Moreover, the present method can effectively simulate three-dimensional diffusion problems. Consequently, our new formulation, most importantly, is matrix-free and fully explicit due to completely avoiding the IIT. As a result, the proposed SBM formulation is mathematically simple, computationally fast and stable, and requiring very low memory since it does not need to solve any algebraic equations. In stark contrast to the boundary element method, the present SBM only requires integration and background grid to calculate the OIFs, while remaining free of integration and mesh for the rest of the calculation. Five benchmark problems are tested to verify the feasibility and accuracy of the new formulation. Numerical results clearly demonstrate the applicability and accuracy of the proposed SBM for solving three-dimensional transient diffusion problems.