대한수학회지 (Journal of the Korean Mathematical Society)

  • 제41권6호
  • /
  • Pages.957-976
  • /
  • 2004
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

THE TRAPEZOIDAL RULE WITH A NONLINEAR COORDINATE TRANSFORMATION FOR WEAKLY SINGULAR INTEGRALS

  • Yun, Beong-In (Faculty of Mathematics, Informatics and Statistics Kunsan National University)
  • 발행 : 2004.11.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

It is well known that the application of the nonlinear coordinate transformations is useful for efficient numerical evaluation of weakly singular integrals. In this paper, we consider the trapezoidal rule combined with a nonlinear transformation $\Omega$$_{m}$(b;$\chi$), containing a parameter b, proposed first by Yun [14]. It is shown that the trapezoidal rule with the transformation $\Omega$$_{m}$(b;$\chi$), like the case of the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error by using a moderately large b. By several examples, we compare the numerical results of the present method with those of some existing methods. This shows the superiority of the transformation $\Omega$$_{m}$(b;$\chi$).TEX>).

키워드

참고문헌

  1. M. Cerrolaza and E. Alarcon, A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary elements, Internat. J. Numer. Methods Engrg. 28 (1989), 987–999. https://doi.org/10.1002/nme.1620280502
  2. M. Doblare and L. Gracia, On non-linear transformations for the integration of weakly-singular and Cauchy principal value integrals, Internat. J. Numer. Methods Engrg. 40 (1997), 3325–3358. https://doi.org/10.1002/(SICI)1097-0207(19970930)40:18<3325::AID-NME215>3.0.CO;2-Q
  3. D. Elliott, The cruciform crack problem and sigmoidal transformations, Math. Methods Appl. Sci. 20 (1997), 121–132. https://doi.org/10.1002/(SICI)1099-1476(19970125)20:2<121::AID-MMA840>3.0.CO;2-7
  4. D. Elliott, Sigmoidal transformations and the trapezoidal rule, J. Aust. Math. Soc. Ser. B 40(E) (1998), E77–E137.
  5. P. R. Johnston, Application of sigmoidal transformations to weakly singular and neer singular boundary element integrals, Internat. J. Numer. Methods Engrg. 45 (1999), 1333–1348. https://doi.org/10.1002/(SICI)1097-0207(19990810)45:10<1333::AID-NME632>3.0.CO;2-Q
  6. P. R. Johnston, Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals, Internat. J. Numer. Methods Engrg. 47 (2000), 1709–1730. https://doi.org/10.1002/(SICI)1097-0207(20000410)47:10<1709::AID-NME852>3.0.CO;2-V
  7. P. R. Johnston and D. Elliott, Error estimation of quadrature rules for evaluating singular integrals in boundary element, Internat. J. Numer. Methods Engrg. 48 (2000), 949–962. https://doi.org/10.1002/(SICI)1097-0207(20000710)48:7<949::AID-NME905>3.0.CO;2-Q
  8. N. M. Korobev, Number –Theoretic Method of Approximate Analysis, GIFL, Moscow, 1963.
  9. J. M. Sanz Serna, M. Doblare, and E. Alarcon, Remarks on methods for the computation of boundary-element integrals by co-ordinate transformation, Commun. Appl. Numer. Methods 6 (1990), 121–123. https://doi.org/10.1002/cnm.1630060208
  10. M. Sato, S. Yoshiyoka, K. Tsukui and R. Yuuki, Accurate numerical integration of singular kernels in the two-dimensional boundary element method, in: C.A. Brebbia(ed.), Boundary Elements X, 1 (1988), Springer, Berlin, 279–296.
  11. A. Sidi, A new variable transformation for numerical integration, in: H. Brass and G. Hammerlin(ed.), numerical Integration IV, ISNM Vol. 112, Birkhaiser-Verlag, Berlin, 1993, 359–373.
  12. K. M. Singh and M. Tanaka, On non–linear transformations for accurate numerical evaluation of weakly singular boundary integrals, Internat. J. Numer. Methods Engrg. 50 (2001), 2007–2030. https://doi.org/10.1002/nme.117
  13. J. C. F. Telles, A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Internat. J. Numer. Methods Engrg. 24 (1987), 959–973. https://doi.org/10.1002/nme.1620240509
  14. B. I. Yun, An efficient transformation with Gauss quadrature rule for weakly singular integrals, Comm. Numer. Methods Eng. 17 (2001), 881–891. https://doi.org/10.1002/cnm.457
  15. B. I. Yun and P. Kim, A new sigmoidal transformation for weakly singular integrals in the boundary element method, SIAM J. Sci. Comput. 24 (2003), 1203–1217. https://doi.org/10.1137/S1064827501396191