Singular boundary method for heat conduction problems with certain spatially varying conductivity

The singular boundary method (SBM) is a recent boundary-type meshless collocation method, in which the solution of a given problem is expanded as a linear combination of the fundamental solutions in terms of the source points. The method circumvents the fictitious boundary long perplexing the method of fundamental solution (MFS) by the introduction of the concept of origin intensity factors (OIFs). This paper documents the first attempt to extend the method to heat conduction problems in nonhomogeneous materials. We derive the fundamental solutions of heat conduction problems with the thermal conductivity of the quadratic, exponential and trigonometric material variations in three directions. Furthermore, we firstly theoretically derive the value of the OIF for the natural logarithm function, which is later extended to the OIFs for a group of fundamental solutions. The feasibility, accuracy and stability of the presented SBM formulation are confirmed for both two- (2D) and three-dimensional (3D) examples.