Journal Article

A Hybrid Reproducing Kernel Particle Method for Three-Dimensional Helmholtz Equation

Wang, Ning, Peng, Piaopiao, Cheng, Yumin

2024

Overview

The reproducing kernel particle method (RKPM) is one of the most universal meshless methods. However, when solving three-dimensional (3D) problems, the computational efficiency is relatively low because of the complexity of the shape function. To overcome this disadvantage, in this study, we introduced the dimension splitting method into the RKPM to present a hybrid reproducing kernel particle method (HRKPM), and the 3D Helmholtz equation is solved. The 3D Helmholtz equation is transformed into a series of related two-dimensional (2D) ones, in which the 2D RKPM shape function is used, and the Galerkin weak form of these 2D problems is applied to obtain the discretized equations. In the dimension-splitting direction, the difference method is used to combine the discretized equations in all 2D domains. Three example problems are given to illustrate the performance of the HRKPM. Moreover, the numerical results show that the HRKPM can improve the computational efficiency of the RKPM significantly.

• Publisher: MDPI AG
• Subject: Accuracy / Approximation / Boundary conditions / Computational efficiency / dimension splitting method / Discretization / Efficiency / Eigenvalues / Finite element method / Helmholtz equation / Helmholtz equations / hybrid reproducing kernel particle method / Kernel functions / Laplacian operator / meshless method / Meshless methods / Methods / Partial differential equations / Particle methods (mathematics) / Propagation / reproducing kernel particle method / Shape functions / Splitting / Tests, problems and exercises