Developing Relationships for Determining the Optimal Shape Parameter in the Multiquadric Meshless Method

Document Type : Original Article

Authors

Department of Civil Engineering, Faculty of Engineering, University of Qom, Qom, Iran

10.22091/cer.2025.12181.1598

Abstract

The accuracy and efficiency of the Multiquadric Radial Basis Function (MQ-RBF) method are highly sensitive to the choice of the shape parameter. This study aims to identify optimal relationships for determining the shape parameter in solving common partial differential equations (PDEs) encountered in water engineering. To this end, the procedure for reconstructing and solving PDEs using the MQ-RBF method is first outlined. Then, optimal values for the shape parameter are derived based on domain length and the number of computational centers. Based on these findings, empirical formulas for the optimal shape parameter are proposed, which significantly reduce computational cost. The performance of the proposed formulas is compared with exact solutions and existing empirical relations. Results show that, unlike previous approaches, the new formulas offer high accuracy, with negligible errors across different examples-sometimes approaching zero. Moreover, compared to optimization-based techniques, the proposed method dramatically improves computational speed by eliminating the need for iterative algorithms. The study also investigates stability conditions, showing that for diffusion, advection-diffusion, and Burgers’ equations, the maximum allowable time step depends on the diffusion coefficient, flow velocity, and Reynolds number.

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References

[1] Kansa EJ. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers & mathematics with applications. 1990;19(8-9):147-61. doi:10.1016/0898-1221(90)90271-K
[2] Seydaoğlu M. A meshless method for Burgers’ equation using multiquadric radial basis functions with a Lie-group integrator. Mathematics. 2019;7(2):113. doi:10.3390/math7020113  
[3] Zheng K, Li C, Wang F, editors. Gaussian Radial Basis Function for Unsteady Groundwater Flow. IOP Conference Series: Earth and Environmental Science; 2019: IOP Publishing. doi:10.1088/1755-1315/304/2/022052
[4] Mirabi MH, Jabbari E, Rajaee T. Numerical Solution of Steady Incompressible Turbulent Navier–Stokes Equations using Multiquadric Radial Basis Function (MQ-RBF) Method. Amirkabir Journal of Civil Engineering. 2022;53(12):5325-56. doi:10.22060/ceej.2021.18788.6964 [In Persian]
[5] Babaee R, Jabbari E, Eskandari Ghadi M. Development of the Multiquadric mesh-less method for analyzing the dynamic interaction of dam-reservoir-foundation problems. Sharif Journal of Civil Engineering. 2022;38(3.1):65-76. doi:10.24200/J30.2022.60113.3085 [In Persian]
[6] Babaee R, Jabbari E, Eskandari-Ghadi M. Application of Multiquadric Radial Basis Function method for Helmholtz equation in seismic wave analysis for reservoir of rigid dams. Amirkabir Journal of Civil Engineering. 2019;52(12):3015-30. doi:10.22060/ceej.2019.16443.6230 [In Persian]
[7] Chen C, Karageorghis A, Amuzu L. Kansa RBF collocation method with auxiliary boundary centres for high order BVPs. Journal of Computational and Applied Mathematics. 2021;398:113680. doi: 10.1016/j.cam.2021.113680  
[8] Golbabai A, Mohebianfar E, Rabiei H. On the new variable shape parameter strategies for radial basis functions. Computational and Applied Mathematics. 2015;34(2):691-704. doi: 10.1007/s40314-014-0132-0  
[9] Zhang H, Guo C, Su X, Chen L. Shape parameter selection for multi-quadrics function method in solving electromagnetic boundary value problems. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering. 2016;35(1):64-79. doi: 10.1108/COMPEL-12-2014-0350  
[10] Biazar J, Hosami M. Selection of an interval for variable shape parameter in approximation by radial basis functions. Advances in numerical analysis. 2016;2016(1):1397849. doi:10.1155/2016/1397849  
[11] Yaghouti M, Ramezannezhad Azarboni H. Determining optimal value of the shape parameter $ c $ in RBF for unequal distances topographical points by Cross-Validation algorithm. Journal of Mathematical Modeling. 2017;5(1):53-60. doi:10.22124/JMM.2017.2225  
[12] Chen W, Hong Y, Lin J. The sample solution approach for determination of the optimal shape parameter in the Multiquadric function of the Kansa method. Computers & Mathematics with Applications. 2018;75(8):2942-54. doi:10.1016/j.camwa.2018.01.023  
[13] Azarboni HR, Keyanpour M, Yaghouti M. Leave-Two-Out Cross Validation to optimal shape parameter in radial basis functions. Engineering Analysis with Boundary Elements. 2019;100:204-10. doi:10.1016/j.enganabound.2018.06.011  
[14] Luh L-T. The choice of the shape parameter–A friendly approach. Engineering Analysis with Boundary Elements. 2019;98:103-9. doi:10.1016/j.enganabound.2018.10.011  
[15] Koushki M, Babaee R, Jabbari E. Application of MQ-RBF method for solving seepage problems with a new algorithm for optimization of the shape parameter. Amirkabir Journal of Civil Engineering. 2020;52(4):1009-24. doi: 10.22060/CEEJ.2019.15155.5840  [In Persian]
[16] Fallah A, Jabbari E, Babaee R. Development of the Kansa method for solving seepage problems using a new algorithm for the shape parameter optimization. Computers & Mathematics with Applications. 2019;77(3):815-29. doi:10.1016/j.camwa.2018.10.021  
[17] Kahid Basiri H, Babaee R, Fallah A, Jabbari E. Development of multiquadric method for solving dam break problem. Journal of Hydraulics. 2020;14(4):83-98. doi:10.30482/JHYD.2020.105500 [In Persian]
[18] Sun J, Wang L, Gong D. Model for Choosing the Shape Parameter in the Multiquadratic Radial Basis Function Interpolation of an Arbitrary Sine Wave and Its Application. Mathematics. 2023;11(8):1856. doi:10.3390/math11081856  
[19] Sun J, Wang L, Gong D. An adaptive selection method for shape parameters in mq-rbf interpolation for two-dimensional scattered data and its application to integral equation solving. Fractal and Fractional. 2023;7(6):448. doi:10.3390/fractalfract7060448  
[20] Pekmen B, Kayabasi M. Machine Learning Modeling for Shape Parameter c in MQ-RBF Applied to Burgers’ Equations. doi:10.1007/978-3-031-70018-7_32  
[21] Pekmen Geridonmez B, Kayabasi M, editors. Machine Learning Modeling for Shape Parameter c in MQ-RBF Applied to Burgers’ Equations. International Conference on Intelligent and Fuzzy Systems; 2024: Springer. doi:10.1007/9
[22] Sun J, Wang W. Adaptive selection of shape parameters for MQRBF in arbitrary scattered data: enhancing finite difference solutions for complex PDEs. Computational and Applied Mathematics. 2024;43(8):1-41. doi:10.1007/s40314-024-02970-6  
[23] Taleni-Kalan H. Investigation of optimal shape parameter determination in multiquadric method for solving linear equation in water engineering. MSc Thesis, Faculty of Engineering, University of Qom, 2020. [In Persian]
[24] Mohsenzadeh Golafzani S. Investigation of Optimal Shape Parameter determination in Multiquadric Method for Solving nonlinear Equations in Water Engineering. MSc Thesis, Faculty of Engineering, University of Qom, 2020. [In Persian]
[25] Pourlak M, Jabbari E, Akbari H. The Effect of Initial Particles Distribution by in Smoothed Particle Hydrodynamic Method in Wave Generation Modeling Based on Laboratory Model. Civil Infrastructure Researches. 2023;9(2):35-50. doi:10.22091/CER.2023.9003.1451 [In Persian]
[26] MohammadAlian S, Babaee R, Jabbari E. A New Adaptive Algorithm for the Optimal Distribution of Computational Centers in the Meshless Multiquadric Method. Civil Infrastructure Researches. 2023;9(1):161-73. doi:10.22091/CER.2023.8470.1419 [In Persian]
[27] Gurarslan G, Karahan H, Alkaya D, Sari M, Yasar M. Numerical Solution of Advection‐Diffusion Equation Using a Sixth‐Order Compact Finite Difference Method. Mathematical Problems in Engineering. 2013;2013(1):672936. doi:10.1155/2013/67293610.1155/2013/672936  
[28] Hon Y-C, Mao X. An efficient numerical scheme for Burgers' equation. Applied Mathematics and Computation. 1998;95(1):37-50. doi:10.1016/S0096-3003(97)10060-1  
[29] Khaksarfard M, Ordokhani Y, Hashemi MS, Karimi K. Space-time radial basis function collocation method for one-dimensional advection-diffusion problem. Computational Methods for Differential Equations. 2018;6(4):426-37. doi:20.1001.1.23453982.2018.6.4.3.4  
 
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