An advanced fast multipole BEM for analyzing 2D heat conduction problems in multi-notched structures

A new fast multipole boundary element method (FMBEM) is developed for heat conduction in multi-notched structures. To address the heat flux singularity occurring at the tips of cracks or sharp notches, a novel variable-order asymptotic element (VAE) is proposed. The VAE offers the flexibility to represent various singular orders through a simple adjustment of the exponent, and it is designed to be compatible with conventional elements, whether they are discontinuous or continuous. Subsequently, the VAE is integrated into the FMBEM framework, and several algorithms are established to deal with the singularity problems of boundary integrals on the VAE. Compared to the conventional FMBEM with quadratic elements, the present method achieves more precise results with a very low computational cost, which proves to be accurate and efficient for heat analysis of porous structures containing non-conductive cracks and polygonal pores.