Localized boundary knot method and its application to large-scale acoustic problems

Abstract

We present a new localized boundary knot method (LBKM) for the solution of large-scale partial differential equations based on the non-singular general solutions. The proposed local algorithm only requires several discrete nodes inside the physical domain and along its boundary without any mesh. For every node, a local subdomain with a simple geometry can be firstly determined via Euclidean distance between nodes. And then the unknown variable at every node can be expressed as a linear combination of function values at nodes inside its corresponding local subdomain. Finally, a sparse linear system can be formed by using the governing equation and the corresponding boundary conditions. In our computations, two different-types of LBKM formulations have been proposed for deriving the final sparse linear system. One is based on the inverse matrix calculation and the other is based on the moving least square (MLS) technique. Unlike the traditional boundary knot method (BKM) with the “global” discretization, the present LBKM is a local discretization method and requires less computer time and storage due to its feature of sparse and banded matrix, which makes it more suitable for solving large-scale problems. Preliminary numerical experiments are carried out in the large-scale acoustics models and bimaterial problems. The comparisons of proposed two types of LBKM formulations are also investigated. The results show the accuracy, validity and great promising applications of the proposed LBKM for the simulation of large-scale problems in complicated geometries.