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<p>This study employs the improved element-free Galerkin (IEFG) method to solve Helmholtz equations with variable wave number in both two and three dimensions. The approximation function is constructed based on improved moving least squares (IMLS) approach, which lacks interpolation properties, necessitates the implementation of essential boundary conditions through penalty function method. The solved equations are derived by introducing the Galerkin weak form of the integral formulation, thereby establishing the complete numerical framework for solving the Helmholtz equations. Section&#xa0;<InternalRef RefID="Sec3">3</InternalRef> presents a comprehensive analysis of three critical computational parameters: nodes distribution, penalty factor selection and scaling parameter selection, and also compares numerical solutions with analytical ones in different directions. Finally, it is concluded that the IEFG method is convergent and has the same computational accuracy as EFG method. Meanwhile, the IEFG method has higher computational efficiency.</p>

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Numerical solution of 2D and 3D variable wave number Helmholtz equations via improved element-free Galerkin method

  • 期刊论文

  • 发表日期：

    2025年10月15日

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  • Yichen Yang,
  • Heng Cheng

摘要

This study employs the improved element-free Galerkin (IEFG) method to solve Helmholtz equations with variable wave number in both two and three dimensions. The approximation function is constructed based on improved moving least squares (IMLS) approach, which lacks interpolation properties, necessitates the implementation of essential boundary conditions through penalty function method. The solved equations are derived by introducing the Galerkin weak form of the integral formulation, thereby establishing the complete numerical framework for solving the Helmholtz equations. Section 3 presents a comprehensive analysis of three critical computational parameters: nodes distribution, penalty factor selection and scaling parameter selection, and also compares numerical solutions with analytical ones in different directions. Finally, it is concluded that the IEFG method is convergent and has the same computational accuracy as EFG method. Meanwhile, the IEFG method has higher computational efficiency.