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Optimality of the Boundary Knot Method for Numerical Solutions of 2D Helmholtz-Type Equations

  • Mathematics
  • Published:
Wuhan University Journal of Natural Sciences

Abstract

The boundary knot method (BKM) is a boundary-type meshfree method. Only non-singular general solutions are used during the whole solution procedures. The effective condition number (ECN), which depends on the right-hand side vector of a linear system, is considered as an alternative criterion to the traditional condition number. In this paper, the effective condition number is used to help determine the position and distribution of the collocation points as well as the quasi-optimal collocation point numbers. During the solution process, we propose an NMN-search algorithm. Numerical examples show that the ECN is reliable to measure the feasibility of the BKM.

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References

  1. Chen J T, Lee Y T, Chen I L, et al. Mathematical analysis and treatment for the true and spurious eigenequations of circular plates by the meshless method using radial basis function [J]. Journal of the Chinese Institute of Engineers, 2004, 27(4): 547–561.

    Article  Google Scholar 

  2. Bui T Q, Nguyen M N, Zhang C Z. An efficient meshfree method for vibration analysis of laminated composite plates [J]. Computational Mechanics, 2011, 48(2): 175–193.

    Article  Google Scholar 

  3. Liu Y J, Mukherjee S, Nishimura N, et al. Recent advances and emerging applications of the boundary element method [J]. ASME Applied Mechanics Reviews, 2011, 64(5): 1–38.

    Google Scholar 

  4. Gu Y, Chen W, Qu W Z, et al. Two general algorithms for nearly singular integrals in two dimensional anisotropic boundary element method [J]. Computational Mechanics, 2014, 53(6): 1223–1234.

    Article  Google Scholar 

  5. Li J, Huang Y, Yang W. Mathematical analysis and finite element simulation of a magnetized ferrite model [J]. Journal of Computational and Applied Mathematics, 2016, 292(1): 279–291.

    Article  Google Scholar 

  6. Khan W, Islam S, Ullah B. Analysis of meshless weak and strong formulations for boundary value problems [J]. Engineering Analysis with Boundary Elements, 2017, 80: 1–17.

    Article  Google Scholar 

  7. Cheng Y M, Peng M J. Boundary element-free method for elastodynamics [J]. Science in China Series G, 2005, 48(6): 641–657.

    Article  Google Scholar 

  8. Liu C S. A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation [J]. Engineering Analysis with Boundary Elements, 2008, 32: 778–785.

    Article  Google Scholar 

  9. Gu Y, Gao H W, Chen W, et al. Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems [J]. International Journal of Heat and Mass Transfer, 2015, 90: 291–301.

    Article  Google Scholar 

  10. Lin J, Zhang C Z, Sun L L, et al. Simulation of seismic wave scattering by embedded cavities in an elastic half-plane using the novel singular boundary method [J]. Advances in Applied Mathematics and Mechanics, 2018, 10(2): 322–342.

    Article  Google Scholar 

  11. Lin J, Reutskiy S Y, Lu J. A novel meshless method for fully nonlinear advection-diffusion-reaction problems to model transfer in anisotropic media [J]. Applied Mathematics and Computation, 2018, 339: 459–476.

    Article  Google Scholar 

  12. Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundary value problems [J]. Advances in Computational Mathematics, 1998, 9(1–2): 69–95.

    Article  Google Scholar 

  13. Chen C S, Karageorghis A, Smyrlis Y S. The Method of Fundamental Solutions—A Meshless Method[M]. Atlanta: Dynamic Publishers, 2008.

    Google Scholar 

  14. Wang F Z, Chen W, Ling L. Combinations of the method of fundamental solutions for general inverse source identification problems [J]. Applied Mathematics and Computation, 2012, 219(3): 1173–1182.

    Article  Google Scholar 

  15. Kang S W, Lee J M, Kang Y J. Vibration analysis of arbitrarily shaped membranes using non-dimensional dynamic influence function [J]. Journal of Sound and Vibration, 1996, 221(1): 117–132.

    Article  Google Scholar 

  16. Chen J T, Kuo S R, Chen K H, et al. Comments on “Vibration analysis of arbitrary shaped membranes using nondimensional dynamic influence function” [J]. Journal of Sound and Vibration, 2000, 235(1): 156–171.

    Article  Google Scholar 

  17. Chen J T, Chang M H, Chen K H, et al. The boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function [J]. Journal of Sound and Vibration, 2002, 257(4): 667–711.

    Article  Google Scholar 

  18. Chen J T, Chang M H, Chung I L, et al. Comment on “Eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point matching” [J. Acoust. Soc. Am. 107, 1153 (2000)][J]. Journal of the Acoustical Society of America, 2002, 11(1): 33–36.

    Article  Google Scholar 

  19. Wang F Z, Zhang J. Numerical simulation of acoustic problems with high wavenumbers [J]. Applied Mathematics & Information Sciences, 2015, 9(2): 1011–1014.

    Google Scholar 

  20. Wang F Z, Zheng K H. Analysis of the boundary knot method for 3D Helmholtz-type problems [J]. Mathematical Problems in Engineering, 2014, 2014: 1–9.

    Google Scholar 

  21. Zheng K H, Ma H W. Combinations of the boundary knot method with analogy equation method for nonlinear problems [J]. CMES: Computer Modeling in Engineering and Sciences, 2012, 87(3): 225–238.

    Google Scholar 

  22. Jin B T, Zheng Y. Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation [J]. Engineering Analysis with Boundary Elements, 2005, 29(10): 925–935.

    Article  Google Scholar 

  23. Zhang J Y, Wang F Z. Boundary knot method: An overview and some novel approaches [J]. CMES: Computer Modeling in Engineering and Sciences, 2012, 88(2): 141–154.

    Google Scholar 

  24. Wang F Z, Ling L, Chen W. Effective condition number for boundary knot method [J]. CMC: Computers, Materials, & Continua, 2009, 12(1): 57–70.

    CAS  Google Scholar 

  25. Ling L, Schaback R. Stable and convergent unsymmetric meshless collocation methods [J]. SIAM Journal on Numerical Analysis, 2008, 46(3): 1097–1115.

    Article  Google Scholar 

  26. Ling L, Schaback R. An improved subspace selection algorithm for meshless collocation methods [J]. International Journal for Numerical Methods in Engineering, 2009, 80(13): 1623–1639.

    Article  Google Scholar 

  27. Fornberg B, Larsson E, Wright G. A new class of oscillatory radial basis functions [J]. Computers & Mathematics with Applications, 2006, 51(8): 1209–1222.

    Article  Google Scholar 

  28. Drombosky T W, Meyer A L, Ling L. Applicability of the method of fundamental solutions [J]. Engineering Analysis with Boundary Elements, 2009, 33: 637–643.

    Article  Google Scholar 

  29. Li Z C, Huang H T, Chen J T, et al. Effective condition number and its applications [J]. Computing, 2010, 89(1–2): 87–112.

    Article  Google Scholar 

  30. Li Z C, Huang G T, Wei Y M, et al. Effect Condition Number for Numerical Partial Differential Equations [M]. Second Edition. Beijing: Science Press, 2015(Ch).

    Google Scholar 

  31. Carlos J S A. On the choice of source points in the method of fundamental solutions [J]. Engineering Analysis with Boundary Elements, 2009, 33(12): 1348–1361.

    Article  Google Scholar 

  32. Wong K Y, Ling L. Optimality of the method of fundamental solutions [J]. Engineering Analysis with Boundary Elements, 2011, 35(1): 42–46.

    Article  Google Scholar 

  33. Wang F Z, Chen W, Jiang X R. Investigation of regularized techniques for boundary knot method [J]. International Journal for Numerical Methods in Biomedical Engineering, 2010, 26(12): 1868–1877.

    Article  Google Scholar 

  34. Ma H W, Wang F Z. New investigations into the boundary knot method for inverse problems of Helmholtz equation [J]. Journal of the Chinese Institute of Engineers, 2016, 39(4): 455–460.

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Huaibei Normal University, Huaibei, Anhui, 235000, China

    Fuzhang Wang & Juan Zhang

  2. College of Water Conservancy and Ecological Engineering, Nanchang Institute of Technology, Nanchang, Jiangxi, 330099, China

    Kehong Zheng

  3. Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China

    Congcong Li

Authors
  1. Fuzhang Wang
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  2. Kehong Zheng
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  3. Congcong Li
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  4. Juan Zhang
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Corresponding author

Correspondence to Congcong Li.

Additional information

Foundation item: Supported by the Natural Science Foundation of Anhui Province (1908085QA09) and Higher Education Department of the Ministry of Education (201802358008)

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Wang, F., Zheng, K., Li, C. et al. Optimality of the Boundary Knot Method for Numerical Solutions of 2D Helmholtz-Type Equations. Wuhan Univ. J. Nat. Sci. 24, 314–320 (2019). https://doi.org/10.1007/s11859-019-1402-x

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  • DOI: https://doi.org/10.1007/s11859-019-1402-x

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