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Analyzing Three-Dimensional Laplace Equations Using the Dimension Coupling Method

Author

Listed:
  • Fengbin Liu

    (College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China)

  • Mingmei Zuo

    (School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China)

  • Heng Cheng

    (School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China)

  • Ji Ma

    (College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China)

Abstract

Due to the low computational efficiency of the Improved Element-Free Galerkin (IEFG) method, efficiently solving three-dimensional (3D) Laplace problems using meshless methods has been a longstanding research direction. In this study, we propose the Dimension Coupling Method (DCM) as a promising alternative approach to address this challenge. Based on the Dimensional Splitting Method (DSM), the DCM divides the 3D problem domain into a coupling of multiple two-dimensional (2D) problems which are handled via the IEFG method. We use the Finite Element Method (FEM) in the third direction to combine the 2D discretized equations, which has advantages over the Finite Difference Method (FDM) used in traditional methods. Our numerical verification demonstrates the DCM’s convergence and enhancement of computational speed without losing computational accuracy compared to the IEFG method. Therefore, this proposed method significantly reduces computational time and costs when solving 3D Laplace equations with natural or mixed boundary conditions in a dimensional splitting direction, and expands the applicability of the dimension splitting EFG method.

Suggested Citation

  • Fengbin Liu & Mingmei Zuo & Heng Cheng & Ji Ma, 2023. "Analyzing Three-Dimensional Laplace Equations Using the Dimension Coupling Method," Mathematics, MDPI, vol. 11(17), pages 1-20, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3717-:d:1228212
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    References listed on IDEAS

    as
    1. Bagh Ali & Anum Shafiq & Meznah M. Alanazi & Awatif A. Hendi & Ahmed Kadhim Hussein & Nehad Ali Shah, 2023. "Significance of Nanoparticle Radius and Gravity Modulation on Dynamics of Nanofluid over Stretched Surface via Finite Element Simulation: The Case of Water-Based Copper Nanoparticles," Mathematics, MDPI, vol. 11(5), pages 1-15, March.
    2. Heng Cheng & Guodong Zheng, 2020. "Analyzing 3D Advection-Diffusion Problems by Using the Improved Element-Free Galerkin Method," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-13, August.
    3. Heng Cheng & Zebin Xing & Yan Liu, 2023. "The Improved Element-Free Galerkin Method for 3D Steady Convection-Diffusion-Reaction Problems with Variable Coefficients," Mathematics, MDPI, vol. 11(3), pages 1-19, February.
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    1. Heng Cheng & Zebin Xing & Yan Liu, 2023. "The Improved Element-Free Galerkin Method for 3D Steady Convection-Diffusion-Reaction Problems with Variable Coefficients," Mathematics, MDPI, vol. 11(3), pages 1-19, February.

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