Analysis of Large-Scale Heat Conduction Problems Using a Modified Single-Level Fast Multipole Method

A Modified Single-Level Fast Multipole Method (MSLFMM) for large-scale heat conduction problems is presented. This method is obtained by embedding the far-field approximation (FFA) within the traditional single-level fast multipole method (SLFMM). The FFA is used to compute the influence coefficients of the far elements within adjacent cells, and also to determine the moments of the elements within far cells. This approximation not only reduces the difficulty of procedures and programming, but also causes a significant decrease in the CPU time. Several problems are considered to verify and evaluate the proposed method. The computational cost of the MSLFMM is demonstrated by comparing the Conventional Boundary Element Method (CBEM), the SLFMM, and the Multi-Level Fast Multipole Method (MLFMM). It is shown that the MSLFMM is much faster than the SLFMM, and comparable with the MLFMM due to its ease of use. Finally, to check the ability of the proposed method in modeling a complicated problem, steady-state heat conduction in an engine block is solved. The numerical results show a good agreement with those obtained by a Finite Volume Method (FVM), and its difference is less than 1.5%.