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Meshfree Particle Methods in the Framework of Boundary Element Methods for the Helmholtz Equation

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Abstract

In this paper, we study electromagnetic wave scattering from periodic structures and eigenvalue analysis of the Helmholtz equation. Boundary element method (BEM) is an effective tool to deal with Helmholtz problems on bounded as well as unbounded domains. Recently, Oh et al. (Comput. Mech. 48:27–45, 2011) developed reproducing polynomial boundary particle methods (RPBPM) that can handle effectively boundary integral equations in the framework of the collocation BEM. The reproducing polynomial particle (RPP) shape functions used in RPBPM have compact support and are not periodic. Thus it is not ideal to use these RPP shape functions as approximation functions along the boundary of a circular domain. In order to get periodic approximation functions, we consider the limit of the RPP shape function as its support is getting infinitely large. We show that the basic approximation function obtained by the limit of the RPP shape function yields accurate solutions of Helmholtz problems on circular, or annular domains as well as on the infinite domains.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1970)

    Google Scholar 

  2. Atkinson, K.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  3. Atluri, S., Shen, S.: The Meshless Method. Tech Science Press, Dulath (2002)

    MATH  Google Scholar 

  4. Babuška, I., Banerjee, U., Osborn, J.E.: Survey of meshless and generalized finite element methods: a unified approach. Acta Numer. 12, 1–125 (2003). Cambridge Press

    Article  MathSciNet  MATH  Google Scholar 

  5. Barnes, J., Hut, P.: A hierarchical O(NlogN) force-calculation algorithm. Nature 324, 446 (1986)

    Article  Google Scholar 

  6. Betcke, T., Trefethen, N.: Reviewing the method of particular solutions. SIAM Rev. 47, 469–491 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  8. Cai, W., Deng, S.: An upwinding embedded boundary method for Maxwell’s equations in media with material interfaces: 2D case. J. Comput. Phys. 190, 159 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cho, M.H., Cai, W.: A wideband fast multipole method for the two-dimensional complex Helmholtz equation. Comput. Phys. Commun. 181, 2086 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cho, M.H., Lee, Y.P., Cai, W.: Modeling of 2D photonic crystals with a boundary integral equation. J. Korean Phys. Soc. 51, 1507–1512 (2007)

    Article  Google Scholar 

  11. Cho, M.H., Cai, W., Her, T.-S.: A boundary integral equation method for photonic crystal fibers. J. Sci. Comput. 28, 263–278 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, J.T., Lin, J.H., Kuo, S.R., Chyuan, S.W.: Boundary element analysis for the Helmholtz eigenvalue problem. Proc. R. Soc. A 457, 2521–2546 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, J.T., Liu, L.W., Hong, H.K.: Spurious and true eigensolution of Helmholtz BIEs and BEMs for a multiply connected problem. Proc. R. Soc. A 459, 1897–1924 (2003)

    MathSciNet  Google Scholar 

  14. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  15. Conway, J.B.: Functions of One Complex Variable. Springer, Berlin (1978)

    Book  Google Scholar 

  16. Davis, C.: Meshless boundary particle methods for boundary integral equations and meshfree particle methods for plates. Ph.D. thesis, University of North Carolina at Charlotte (2011)

  17. Duarte, C.A., Oden, J.T.: An hp adaptive method using clouds. Comput. Methods Appl. Mech. Eng. 139, 237–262 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Duffy, M.G.: Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J. Numer. Anal. 19, 1260–1262 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Greengard, L.: The Rapid Evaluation of Potential Fields in Particle Systems. MIT Press, Cambridge (1988)

    MATH  Google Scholar 

  20. Griebel, M., Schweitzer, M.A. (eds.): Meshfree Methods for Partial Differential Equations I. Lect. Notes in Comput. Science and Engr., vol. 57. Springer, Berlin (2007)

    Google Scholar 

  21. Han, W., Meng, X.: Error analysis of reproducing kernel particle method. Comput. Methods Appl. Mech. Eng. 190, 6157–6181 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hollig, K.: Finite Element Methods with B-spline. SIAM, Philadelphia (2003)

    Book  Google Scholar 

  23. Hunter, P., Pullan, A.: FEM and BEM notes. Dept. of Eng Science, Univ. of Auckland (2001)

  24. Joannopoulos, J.D., Johnson, S.G., Winn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light, 2 edn. Princeton University Press, Princeton (2008)

    MATH  Google Scholar 

  25. Knight, J.C.: Photonic crystal fibers. Nature 424, 847–851 (2003)

    Article  Google Scholar 

  26. Kress, R.: Linear Integral Equations. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  27. Li, S., Liu, W.K.: Meshfree Particle Methods. Springer, Berlin (2004)

    MATH  Google Scholar 

  28. Liu, W.K., Jun, S., Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Methods Fluids 20, 1081–1106 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, W.K., Jun, S., Li, S., Adee, J., Belytschko, T.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Methods Eng. 38, 1655–1679 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liu, W.K., Li, S., Belytschko, T.: Moving least square reproducing kernel method part I: methodology and convergence. Comput. Methods Appl. Mech. Eng. 143, 422–453 (1997)

    Article  MathSciNet  Google Scholar 

  31. Marin, L., Lesnic, D., Mantic, V.: Treatment of singularities in Helmholtz-type equations using the boundary element method. J. Sound Vib. 278, 39–62 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Melenk, J.M., Babuška, I.: The partition of unity finite element method: theory and application. Comput. Methods Appl. Mech. Eng. 139, 239–314 (1996)

    Article  Google Scholar 

  33. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)

    Article  MATH  Google Scholar 

  34. Oh, H.-S., Davis, C., Kim, J.G., Kwon, Y.H.: Reproducing polynomial particle methods for boundary integral equations. Comput. Mech. 48, 27–45 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Oh, H.-S., Davis, C., Jeong, J.W.: Meshfree particle methods for thin plates. Comput. Methods Appl. Mech. Eng. 209–212, 156-171 (2012)

    Article  MathSciNet  Google Scholar 

  36. Oh, H.-S., Jeong, J.W., Hong, W.T.: The generalized product partition of unity for the meshless methods. J. Comput. Phys. 229, 1600–1620 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Oh, H.-S., Jeong, J.W.: Reproducing polynomial (singularity) particle methods and adaptive meshless methods for two-dimensional elliptic boundary value problems. Comput. Methods Appl. Mech. Eng. 198, 933–946 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Oh, H.-S., Kim, J.G., Hong, W.T.: The piecewise polynomial partition of unity shape functions for the generalized finite element methods. Comput. Methods Appl. Mech. Eng. 197, 3702–3711 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Oh, H.-S., Kim, J.G., Jeong, J.W.: The closed form reproducing polynomial particle shape functions for meshfree particle methods. Comput. Methods Appl. Mech. Eng. 196, 3435–3461 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Oh, H.-S., Kim, J.G., Jeong, J.W.: The smooth piecewise polynomial particle shape functions corresponding to patch-wise non-uniformly spaced particles for meshfree particles methods. Comput. Mech. 40, 569–594 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Oh, H.-S., Jeong, J.W., Kim, J.G.: The reproducing singularity particle shape function for problems containing singularities. Comput. Mech. 41, 135–157 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rokhlin, V.: Rapid solution of integral equations of scattering theory in two dimensions. J. Comput. Phys. 86, 441 (1990)

    Article  MathSciNet  Google Scholar 

  43. Saad, Y.: Iterative methods for sparse linear systems. PWS-Kent, Boston (1996)

    MATH  Google Scholar 

  44. Sommerfeld, A.: Partial Differential Equations in Physics. Academic Press, San Diego (1961)

    Google Scholar 

  45. Stroubolis, T., Copps, K., Babuska, I.: Generalized finite element method. Comput. Methods Appl. Mech. Eng. 190, 4081–4193 (2001)

    Article  Google Scholar 

  46. Stroubolis, T., Zhang, L., Babuska, I.: Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids. Comput. Methods Appl. Mech. Eng. 192, 3109–3161 (2003)

    Article  Google Scholar 

  47. Szabo, B., Babuska, I.: Finite Element Analysis. Wiley, New York (1991)

    MATH  Google Scholar 

  48. Tsai, C.C., Young, D.L., Chen, C.W., Fan, C.M.: The method of fundamental solutions for eigenproblems in domains with and without interior holes. Proc. R. Soc. A 462, 1443–1466 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  49. Vahala, K.: Optical Microcavities. World Scientific, Singapore (2005)

    Google Scholar 

  50. Yablonovitch, E.: Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059–2062 (1987)

    Article  Google Scholar 

  51. Yu, D.: Natural Boundary Integral Method and Its Applications. Kluwer Academic, Norwell (2010)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    Christopher Davis, Hae-Soo Oh & Min Hyung Cho

  2. Department of Mathematics, Kangwon National University, Chunchon, 200-701, Korea

    June G. Kim

Authors
  1. Christopher Davis
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  2. June G. Kim
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  3. Hae-Soo Oh
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  4. Min Hyung Cho
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Corresponding author

Correspondence to Hae-Soo Oh.

Additional information

H.S. Oh was supported by NSF grant DMS-10-16060. M.H. Cho was supported by US Army Office of Research W911NF-11-1-0364.

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Davis, C., Kim, J.G., Oh, HS. et al. Meshfree Particle Methods in the Framework of Boundary Element Methods for the Helmholtz Equation. J Sci Comput 55, 200–230 (2013). https://doi.org/10.1007/s10915-012-9645-0

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