The complex variable reproducing kernel particle method for three-dimensional problems

This study develops a complex variable reproducing kernel particle method (CVRKPM) for the numerical analysis of three-dimensional (3D) potential, transient heat conduction, and advection-diffusion problems. The CVRKPM approximation for 3D domains is systematically formulated, providing the basis for the subsequent derivation of the discretized governing equations. In this formulation, two spatial coordinates are combined into a complex coordinate, while the remaining coordinate is retained as a real coordinate, leading to a mixed complex-real basis vector that replaces the conventional 3D real-variable basis vector. This treatment reduces the number of unknown coefficients and enhances computational performance. The discretized equations of the CVRKPM are derived for these three kinds of 3D problems. The performance and error analysis of the proposed method are examined through four numerical examples. Numerical results demonstrate the superior accuracy of the CVRKPM over the standard RKPM in solving 3D problems.