An ACA-SBM for some 2D steady-state heat conduction problems

Abstract

In this paper, an accelerated singular boundary method (SBM) incorporating adaptive cross approximation (ACA) is developed for the steady-state heat conduction problems. The SBM, a recently developed boundary collocation method, employs the fundamental solutions of the governing operators as the kernel functions, and desingularizes the source singularity with a concept of origin intensity factor. However, the SBM suffers fully-populated influence matrix which results in prohibitively expensive operation counts and memory requirements as the number of degrees of freedom increases. In this paper, the ACA is applied to accelerate the SBM meanwhile reducing the memory requirement. Furthermore, the ACA-SBM is robust to different fundamental solutions, which enables it to deal with different heat conduction problems. The effectiveness, feasibility and robustness of the proposed method are numerically tested on different heat conduction problems including isotropic homogeneous, anisotropic homogeneous and non-homogeneous media with quadratic material variation of thermal conductivity, highlighting the accuracy as well as the significant reduction in memory storage and analysis time in comparison with the traditional SBM.