On the supporting nodes in the localized method of fundamental solutions for 2D potential problems with Dirichlet boundary condition

Zengtao Chen; Fajie Wang; Zengtao Chen; Fajie Wang

doi:10.3934/math.2021414

AIMS Mathematics

2021, Volume 6, Issue 7: 7056-7069. doi: 10.3934/math.2021414

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On the supporting nodes in the localized method of fundamental solutions for 2D potential problems with Dirichlet boundary condition

Zengtao Chen ¹,
Fajie Wang ^{1,2
,
,}

1.
College of Mechanical and Electrical Engineering, National Engineering Research Center for Intelligent Electrical Vehicle Power System, Qingdao University, Qingdao 266071, China
2.
Institute of Mechanics for Multifunctional Materials and Structures, Qingdao University, Qingdao 266071, China

Received: 23 March 2021 Accepted: 21 April 2021 Published: 27 April 2021
MSC : 35J05, 65N35, 68W99

This paper proposes a simple, accurate and effective empirical formula to determine the number of supporting nodes in a newly-developed method, the localized method of fundamental solutions (LMFS). The LMFS has the merits of meshless, high-accuracy and easy-to-simulation in large-scale problems, but the number of supporting nodes has a certain impact on the accuracy and stability of the scheme. By using the curve fitting technique, this study established a simple formula between the number of supporting nodes and the node spacing. Based on the developed formula, the reasonable number of supporting nodes can be determined according to the node spacing. Numerical experiments confirmed the validity of the proposed methodology. This paper perfected the theory of the LMFS, and provided a quantitative selection strategy of method parameters.
- localized method of fundamental solutions,
- supporting nodes,
- empirical formula,
- potential problems,
- meshless method
Citation: Zengtao Chen, Fajie Wang. On the supporting nodes in the localized method of fundamental solutions for 2D potential problems with Dirichlet boundary condition[J]. AIMS Mathematics, 2021, 6(7): 7056-7069. doi: 10.3934/math.2021414

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Abstract

This paper proposes a simple, accurate and effective empirical formula to determine the number of supporting nodes in a newly-developed method, the localized method of fundamental solutions (LMFS). The LMFS has the merits of meshless, high-accuracy and easy-to-simulation in large-scale problems, but the number of supporting nodes has a certain impact on the accuracy and stability of the scheme. By using the curve fitting technique, this study established a simple formula between the number of supporting nodes and the node spacing. Based on the developed formula, the reasonable number of supporting nodes can be determined according to the node spacing. Numerical experiments confirmed the validity of the proposed methodology. This paper perfected the theory of the LMFS, and provided a quantitative selection strategy of method parameters.

References

[1]	G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), 69-95. doi: 10.1023/A:1018981221740
[2]	M. A. Golberg, C. S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, In: Boundary Integral Methods-Numerical and Mathematical Aspects (Ed. M. A. Golberg), (1999), 103-176.
[3]	J. T. Chen, C. S. Wu, Y. T. Lee, K. H. Chen, On the equivalence of the Trefftz method and method of fundamental solutions for Laplace and biharmonic equations, Comput. Math. Appl., 53 (2007), 851-879. doi: 10.1016/j.camwa.2005.02.021
[4]	J. T. Chen, Y. T. Lee, S. R. Yu, S. C. Shieh, Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept, Eng. Anal. Bound. Elem., 33 (2009), 678-688. doi: 10.1016/j.enganabound.2008.10.003
[5]	J. A. Kołodziej, J. K. Grabski, Many names of the Trefftz method, Eng. Anal. Bound. Elem., 96 (2018), 169-178. doi: 10.1016/j.enganabound.2018.08.013
[6]	G. Fairweather, A. Karageorghis, P. A. Martin, The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem., 27 (2003), 759-769. doi: 10.1016/S0955-7997(03)00017-1
[7]	Y. C. Hon, T. Wei, A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem., 28 (2004), 489-495. doi: 10.1016/S0955-7997(03)00102-4
[8]	C. S. Chen, The method of fundamental solutions for non-linear thermal explosions, Commun. Numer. Methods Eng., 11 (1995), 675-681. doi: 10.1002/cnm.1640110806
[9]	F. J. Wang, W. Cai, B. Zheng, C. Wang, Derivation and numerical validation of the fundamental solutions for constant and variable-order structural derivative advection-dispersion models, Z. Angew. Math. Phys., 71 (2020), 135. doi: 10.1007/s00033-020-01360-2
[10]	W. Cai, F. J. Wang, Numerical investigation of three-dimensional hausdorff derivative anomalous diffusion model, Fractals, 28 (2020), 2050020. doi: 10.1142/S0218348X20500206
[11]	A. H. D. Cheng, Y. Hong, An overview of the method of fundamental solutions-Solvability, uniqueness, convergence, and stability, Eng. Anal. Bound. Elem., 120 (2020), 118-152. doi: 10.1016/j.enganabound.2020.08.013
[12]	F. F. Dou, L. P. Zhang, Z. C. Li, C. S. Chen, Source nodes on elliptic pseudo-boundaries in the method of fundamental solutions for Laplace's equation; selection of pseudo-boundaries, J. Comput. Appl. Math., 377 (2020), 112861. doi: 10.1016/j.cam.2020.112861
[13]	M. R. Hematiyan, M. Mohammadi, C. C. Tsai, The method of fundamental solutions for anisotropic thermoelastic problems, Appl. Math. Model., 95 (2021), 200-218. doi: 10.1016/j.apm.2021.02.001
[14]	C. J. S. Alves, On the choice of source points in the method of fundamental solutions, Eng. Anal. Bound. Elem., 33 (2009), 1348-1361. doi: 10.1016/j.enganabound.2009.05.007
[15]	F. J. Wang, C. S. Liu, W. Z. Qu, Optimal sources in the MFS by minimizing a new merit function: Energy gap functional, Appl. Math. Lett., 86 (2018), 229-235. doi: 10.1016/j.aml.2018.07.002
[16]	J. K. Grabski, On the sources placement in the method of fundamental solutions for time-dependent heat conduction problems, Comput. Math. Appl., 88 (2021), 33-51. doi: 10.1016/j.camwa.2019.04.023
[17]	W. Chen, F. J. Wang, A method of fundamental solutions without fictitious boundary, Eng. Anal. Bound. Elem., 34 (2010), 530-532. doi: 10.1016/j.enganabound.2009.12.002
[18]	L. Qiu, F. J. Wang, J. Lin, Y. Zhang, A meshless singular boundary method for transient heat conduction problems in layered materials, Comput. Math. Appl., 78 (2019), 3544-3562. doi: 10.1016/j.camwa.2019.05.027
[19]	D. L. Young, K. H. Chen, J. T. Chen, J. H. Kao, A modified method of fundamental solutions with source on the boundary for solving Laplace equations with circular and arbitrary domains, Cmes-Comp. Model. Eng. Sci., 19 (2007), 197-221.
[20]	Q. G. Liu, B. Šarler, Non-singular method of fundamental solutions for anisotropic elasticity, Eng. Anal. Bound. Elem., 45 (2014), 68-78. doi: 10.1016/j.enganabound.2014.01.020
[21]	C. M. Fan, Y. K. Huang, C. S. Chen, S. R. Kuo, Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations, Eng. Anal. Bound. Elem., 101 (2019), 188-197. doi: 10.1016/j.enganabound.2018.11.008
[22]	Y. Gu, C. M. Fan, R. P. Xu, Localized method of fundamental solutions for large-scale modelling of two-dimensional elasticity problems, Appl. Math. Lett., 93 (2019), 8-14. doi: 10.1016/j.aml.2019.01.035
[23]	Y. Gu, C. M. Fan, W. Z. Qu, F. J. Wang, Localized method of fundamental solutions for large-scale modelling of three-dimensional anisotropic heat conduction problems-Theory and MATLAB code, Comput. Struct., 220 (2019), 144-155. doi: 10.1016/j.compstruc.2019.04.010
[24]	Y. Gu, C. M. Fan, W. Z. Qu, Localized method of fundamental solutions for three-dimensional inhomogeneous elliptic problems: Theory and MATLAB code, Comput. Mech., 64 (2019), 1567-1588. doi: 10.1007/s00466-019-01735-x
[25]	Q. G. Liu, C. M. Fan, B. Šarler, Localized method of fundamental solutions for two-dimensional anisotropic elasticity problems, Eng. Anal. Bound. Elem., 125 (2021), 59-65. doi: 10.1016/j.enganabound.2021.01.008
[26]	W. Z. Qu, C. M. Fan, Y. Gu, F. J. Wang, Analysis of three-dimensional interior acoustic fields by using the localized method of fundamental solutions, Appl. Math. Model., 76 (2019), 122-132. doi: 10.1016/j.apm.2019.06.014
[27]	W. Z. Qu, L. L. Sun, P. W. Li, Bending analysis of simply supported and clamped thin elastic plates by using a modified version of the LMFS, Math. Comput. Simulat., 185 (2021), 347-357. doi: 10.1016/j.matcom.2020.12.031
[28]	W. Z. Qu, C. M. Fan, Y. Gu, Localized method of fundamental solutions for interior Helmholtz problems with high wave number, Eng. Anal. Bound. Elem., 107 (2019), 25-32. doi: 10.1016/j.enganabound.2019.06.018
[29]	F. J. Wang, C. M. Fan, Q. S. Hua, Y. Gu, Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations, Appl. Math. Comput., 364 (2020), 124658.
[30]	F. J. Wang, C. M. Fan, C. Z. Zhang, A localized space-time method of fundamental solutions for diffusion and convection-diffusion problems, Adv. Appl. Math. Mech., 12 (2020), 940-958. doi: 10.4208/aamm.OA-2019-0269
[31]	L. Qiu, J. Lin, Q. H. Qin, W. Chen, Localized space-time method of fundamental solutions for three-dimensional transient diffusion problem, Acta Mech. Sinica-PRC, 36 (2020), 1051-1057. doi: 10.1007/s10409-020-00979-8
[32]	L. Qiu, F. J. Wang, J. Lin, Q. H. Qin, Q. H. Zhao, A novel combined space-time algorithm for transient heat conduction problems with heat sources in complex geometry, Comput. Struct., 247 (2021), 106495. doi: 10.1016/j.compstruc.2021.106495
[33]	W. W. Li, Localized method of fundamental solutions for 2D harmonic elastic wave problems, Appl. Math. Lett., 112 (2021), 106759. doi: 10.1016/j.aml.2020.106759
[34]	S. N. Liu, P. W. Li, C. M. Fan, Y. Gu, Localized method of fundamental solutions for two-and three-dimensional transient convection-diffusion-reaction equations, Eng. Anal. Bound. Elem., 124 (2021), 237-244. doi: 10.1016/j.enganabound.2020.12.023
[35]	X. L. Li, S. L. Li, On the augmented moving least squares approximation and the localized method of fundamental solutions for anisotropic heat conduction problems, Eng. Anal. Bound. Elem., 119 (2020), 74-82. doi: 10.1016/j.enganabound.2020.07.007
[36]	W. Z. Qu, C. M. Fan, X. L. Li, Analysis of an augmented moving least squares approximation and the associated localized method of fundamental solutions, Comput. Math. Appl., 80 (2020), 13-30. doi: 10.1016/j.camwa.2020.02.015
[37]	F. J. Wang, W. Z. Qu, X. L. Li, Augmented moving least squares approximation using fundamental solutions, Eng. Anal. Bound. Elem., 115 (2020), 10-20. doi: 10.1016/j.enganabound.2020.03.003
[38]	X. L. Li, S. L. Li, A linearized element-free Galerkin method for the complex Ginzburg-Landau equation, Comput. Math. Appl., 90 (2021), 135-147. doi: 10.1016/j.camwa.2021.03.027
[39]	T. Zhang, X. L. Li, Analysis of the element-free Galerkin method with penalty for general second-order elliptic problems, Appl. Math. Comput., 380 (2020), 125306.
[40]	F. J. Wang, Y. Gu, W. Z. Qu, C. Z. Zhang, Localized boundary knot method and its application to large-scale acoustic problems, Comput. Methods Appl. Mech. Eng., 361 (2020), 112729. doi: 10.1016/j.cma.2019.112729
[41]	F. J. Wang, C. Wang, Z. T. Chen, Local knot method for 2D and 3D convection-diffusion-reaction equations in arbitrary domains, Appl. Math. Lett., 105 (2020), 106308. doi: 10.1016/j.aml.2020.106308
[42]	F. J. Wang, Q. H. Zhao, Z. T. Chen, C. M. Fan, Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains, Appl. Math. Comput., 397 (2021), 125903.
[43]	X. X. Yue, F. J. Wang, P. W. Li, C. M. Fan, Local non-singular knot method for large-scale computation of acoustic problems in complicated geometries, Comput. Math. Appl., 84 (2021), 128-143. doi: 10.1016/j.camwa.2020.12.014
[44]	X. X. Yue, F. J. Wang, C. Z. Zhang, H. X. Zhang, Localized boundary knot method for 3D inhomogeneous acoustic problems with complicated geometry, Appl. Math. Model., 92 (2021), 410-421. doi: 10.1016/j.apm.2020.11.022
[45]	P. W. Li, Space-time generalized finite difference nonlinear model for solving unsteady Burgers' equations, Appl. Math. Lett., 114 (2021), 106896. doi: 10.1016/j.aml.2020.106896
[46]	W. Z. Qu, H. He, A spatial-temporal GFDM with an additional condition for transient heat conduction analysis of FGMs, Appl. Math. Lett., 110 (2020), 106579. doi: 10.1016/j.aml.2020.106579
[47]	H. Xia, Y. Gu, Generalized finite difference method for electroelastic analysis of three-dimensional piezoelectric structures, Appl. Math. Lett., 117 (2021), 107084. doi: 10.1016/j.aml.2021.107084
[48]	Z. J. Fu, Z. Y. Xie, S. Y. Ji, C. C. Tsai, A. L. Li, Meshless generalized finite difference method for water wave interactions with multiple-bottom-seated-cylinder-array structures, Ocean Eng., 195 (2020), 106736. doi: 10.1016/j.oceaneng.2019.106736
[49]	T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless methods: An overview and recent developments, Comput. Methods Appl. Mech. Eng., 139 (1996), 3-47. doi: 10.1016/S0045-7825(96)01078-X

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