Open Access iconOpen Access

ARTICLE

crossmark

Optimal Shape Factor and Fictitious Radius in the MQ-RBF: Solving Ill-Posed Laplacian Problems

Chein-Shan Liu1, Chung-Lun Kuo1, Chih-Wen Chang2,*

1 Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung, 202301, Taiwan
2 Department of Mechanical Engineering, National United University, Miaoli, 36063, Taiwan

* Corresponding Author: Chih-Wen Chang. Email: email

(This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)

Computer Modeling in Engineering & Sciences 2024, 139(3), 3189-3208. https://doi.org/10.32604/cmes.2023.046002

Abstract

To solve the Laplacian problems, we adopt a meshless method with the multiquadric radial basis function (MQ-RBF) as a basis whose center is distributed inside a circle with a fictitious radius. A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function. A sample function is interpolated by the MQ-RBF to provide a trial coefficient vector to compute the merit function. We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm. The novel method provides the optimal values of parameters and, hence, an optimal MQ-RBF; the performance of the method is validated in numerical examples. Moreover, nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition; this can overcome the problem of these problems being ill-posed. The optimal MQ-RBF is extremely accurate. We further propose a novel optimal polynomial method to solve the nonharmonic problems, which achieves high precision up to an order of 10−11.

Keywords


Cite This Article

APA Style
Liu, C., Kuo, C., Chang, C. (2024). Optimal shape factor and fictitious radius in the MQ-RBF: solving ill-posed laplacian problems. Computer Modeling in Engineering & Sciences, 139(3), 3189-3208. https://doi.org/10.32604/cmes.2023.046002
Vancouver Style
Liu C, Kuo C, Chang C. Optimal shape factor and fictitious radius in the MQ-RBF: solving ill-posed laplacian problems. Comput Model Eng Sci. 2024;139(3):3189-3208 https://doi.org/10.32604/cmes.2023.046002
IEEE Style
C. Liu, C. Kuo, and C. Chang, “Optimal Shape Factor and Fictitious Radius in the MQ-RBF: Solving Ill-Posed Laplacian Problems,” Comput. Model. Eng. Sci., vol. 139, no. 3, pp. 3189-3208, 2024. https://doi.org/10.32604/cmes.2023.046002



cc Copyright © 2024 The Author(s). Published by Tech Science Press.
This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • 825

    View

  • 343

    Download

  • 0

    Like

Share Link