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

Piaopiao Peng, Ning Wang and Yumin Cheng ()
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Piaopiao Peng: School of Electric Power, Civil Engineering and Architecture, Shanxi University, Taiyuan 030006, China
Ning Wang: School of Electric Power, Civil Engineering and Architecture, Shanxi University, Taiyuan 030006, China
Yumin Cheng: Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China

Mathematics, 2024, vol. 12, issue 12, 1-24

Abstract: 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.

Keywords: meshless method; reproducing kernel particle method; dimension splitting method; hybrid reproducing kernel particle method; Helmholtz equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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