The radial point interpolation mixed collocation method for the solution of transient diffusion problems

Abstract

The Radial Point Interpolation Mixed Collocation (RPIMC) method is proposed in this paper for transient analysis of diffusion problems. RPIMC is an efficient purely meshfree method where the solution of the field variable is obtained through collocation. The field function and its gradient are both interpolated (mixed collocation approach) leading to reduced C-continuity requirement compared to strong-form collocation schemes. The accuracy of RPIMC was evaluated in heat conduction benchmark problems and compared against the Meshless Local Petrov–Galerkin Mixed Collocation (MLPG-MC) method and the Finite Element Method (FEM). These three methods were subsequently applied to solve electrical propagation in a cardiac tissue slab and a biventricular geometry. In benchmark problems, RPIMC achieved high accuracy, of the same order as FEM, and always higher than MLPG-MC due to the delta Kronecker property of RPIMC. In solving cardiac electrical propagation, RPIMC rendered activation time maps similar to those of FEM, with an improved performance compared to MLPG-MC. As a conclusion, RPIMC is shown to be a promising meshfree alternative to FEM for transient diffusion problems.