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Generalized finite difference method for three-dimensional eigenproblems of Helmholtz equation

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  • Zhang, Juan
  • Shuy, Rong-Juin
  • Chu, Chiung-Lin
  • Fan, Chia-Ming

Abstract

In this paper, a meshless numerical procedure, based on the generalized finite difference method (GFDM) is proposed to efficiently and accurately solve the three-dimensional eigenproblems of the Helmholtz equation. The eigenvalues and eigenvectors are very important to various engineering applications in three-dimensional acoustics, optics and electromagnetics, so it is essential to develop an efficient numerical model to analyze the three-dimensional eigenproblems in irregular domains. In the GFDM, the Taylor series and the moving-least squares method are used to derive the expressions at every node. By enforcing the satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes, the resultant system of linear algebraic equations can be expressed as the eigenproblems of matrix and then the eigenvalues and eigenvectors can be efficiently acquired. In this paper, four numerical examples are provided to validate the accuracy and simplicity of the proposed numerical scheme. Furthermore, the numerical results are compared with analytical solutions and other numerical results to verify the merits of the proposed method.

Suggested Citation

  • Zhang, Juan & Shuy, Rong-Juin & Chu, Chiung-Lin & Fan, Chia-Ming, 2022. "Generalized finite difference method for three-dimensional eigenproblems of Helmholtz equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 45-67.
    • Handle: RePEc:eee:matcom:v:196:y:2022:i:c:p:45-67
    DOI: 10.1016/j.matcom.2022.01.007
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    References listed on IDEAS

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    1. Diecidue, Enrico & Schmidt, Ulrich & Zank, Horst, 2009. "Parametric weighting functions," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1102-1118, May.
    2. Drazen Prelec, 1998. "The Probability Weighting Function," Econometrica, Econometric Society, vol. 66(3), pages 497-528, May.
    3. Mohammed Abdellaoui & Olivier l’Haridon & Horst Zank, 2009. "Separating Curvature and Elevation: A Parametric Weighting Function," Economics Discussion Paper Series 0901, Economics, The University of Manchester.
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    Cited by:

    1. Clain, S. & Figueiredo, J., 2024. "Stencil and kernel optimisation for mesh-free very high-order generalised finite difference method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 49-78.

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