A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems

Abstract

A node-based smoothed point interpolation method (NS-PIM) is formulated for three-dimensional (3D) heat transfer problems with complex geometries and complicated boundary conditions. Shape functions constructed here through PIM possess the delta function property and hence allow the straightforward enforcement of essential boundary conditions. The smoothed Galerkin weak form is employed to create discretized system equations, and the node-based smoothing domains are used to perform the smoothing operation and the numerical integration. The accuracy and efficiency of the NS-PIM solutions are studied through detailed analyses of actual 3D heat transfer problems. It is found that the NS-PIM can provide higher accuracy in temperature and its gradient than the reference approach does, in which very fine meshes are used in standard FEM code available with homogeneous essential boundary conditions. More importantly, the upper bound property of the NS-PIM is obtained using the same tetrahedral mesh. Together with the FEM, we now have a simple means to obtain both upper and lower bounds of the exact solution to heat transfer using the same type of mesh.