Solving multi-domain 2D heat conduction problems by the least squares collocation method with RBF interpolation on virtual boundary

Abstract

Abstract This paper is concerned with the virtual boundary meshless least squares collocation method of two dimensional steady-state heat conduction problems in multi-domain. In the proposed method, each region with different heat transfer properties is considered as a piecewise homogeneous in a heterogeneous system. Being different from the conventional virtual boundary element method (VBEM), this method incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Consequently, this method has the advantages of boundary-type meshless methods. In addition, it does not have to deal with singular integral and has the symmetric coefficient matrix, and the pre-processing operation is also very simple. Since the mathematical model for steady-state heat conduction can be represented by a generalized Poisson equation, this problem is solved by finding an approximate particular solution to the Poisson equation and making use of the virtual boundary meshless least squares collocation method to solve the resulting Laplace equation. This method can be used to analyze multi-domain composite structures with each subdomain having different heat transfer properties or materials or geometries. Since the configuration of virtual boundary has a certain preparability, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2D problems) to reduce the complicity and difficulty of calculus without loss of accuracy, while “Vertex Question” existing in BEM can be avoided. In the end, several numerical examples are analyzed using the proposed method and some other commonly used methods for verification and comparison purposes.