Open Access

ARTICLE

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

by Chein-Shan Liu¹, Chung-Lun Kuo¹, Chih-Wen Chang^2,*

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:

(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

Received 14 September 2023; Accepted 23 November 2023; Issue published 11 March 2024

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.

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Keywords

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