Multi-level boundary element method: Novel computational tool for large-scale heat conduction simulations

This paper concerns the development of a novel multi-level computational tool for simulations of very large-scale problems arising in science and technology. One of the particular applications can be numerical simulations of material properties such as effective thermal diffusivity and/or effective Young's moduli of nanocomposites reinforced by carbon nanotubes. Here we present the multi-level boundary element method (MLBEM) for solutions of very large thermal problems, and focus on efficient solutions of steady heat diffusion. First, we perform analyses of numerical error and computational complexity for the multi-level boundary element algorithm and show that the optimal complexity of the algorithm is O(N log N). Next, we consider a model problem of line multi-integral evaluation and investigate the performance of the MLBEM formulation using a single-patch approach. Then, we study the performance of the multi-level boundary element formulation on an example Neumann problem of steady heat diffusion leading to a boundary integral equation of the second kind. Here, we solve a problem involving four million degrees of freedom in less than one hour on a desk-top workstation. Next, we consider a model problem in a unit square with mixed boundary conditions and study the performance for the new MLBEM formulation. Finally, we consider an example problem of heat conduction in composite material with the heat conductivity ratio of 100:1 for fiber elements and a matrix, and study effective conductivity for volume fraction up to 3%.