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FAMILIES OF NONLINEAR TRANSFORMATIONS FOR ACCURATE EVALUATION OF WEAKLY SINGULAR INTEGRALS

  • BEONG IN YUN (DEPARTMENT OF MATHEMATICS, KUNSAN NATIONAL UNIVERSITY)
  • Received : 2023.06.20
  • Accepted : 2023.09.23
  • Published : 2023.09.25

Abstract

We present families of nonlinear transformations useful for numerical evaluation of weakly singular integrals. First, for end-point singular integrals, we define a prototype function with some appropriate features and then suggest a family of transformations. In addition, for interior-point singular integrals, we develop a family of nonlinear transformations based on the aforementioned prototype function. We take some examples to explore the efficiency of the proposed nonlinear transformations in using the Gauss-Legendre quadrature rule. From the numerical results, we can find the superiority of the proposed transformations compared to some existing transformations, especially for the integrals with high singularity strength.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) [No. 2021R1F1A1047343].

References

  1. M. Doblare, Computational aspects of the boundary element methods,Topics in Boundary Element Research 3, Springer, Berlin, 1987.
  2. P.K. Kythe, An introduction to boundary element method, CRC Press, Boca Raton, 1995.
  3. M.P. Savruk, A. Kazberuk, Method of Singular Integral Equations in Application to Problems of the Theory of Elasticity, Stress Concentration at Notches, Springer, Cham, 2017.
  4. M. Doblare, L. Gracia, On non-linear transformations for integration of weakly-singular and Cauchy principal value integrals, Int J Numer Methods Eng, 40 (1997), 3325-3358. https://doi.org/10.1002/(SICI)1097-0207(19970930)40:18<3325::AID-NME215>3.0.CO;2-Q
  5. J.M. Sanz Serna, M. Doblare, 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
  6. M. Sato, S. Yoshiyoka, K. Tsukui, R. Yuuki, Accurate numerical integration of singular kernels in the two-dimensional boundary element method; Boundary Elements X Vol.1, C.A. Brebbia edt., Springer, Berlin, 1988.
  7. J.C.F. Telles, A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Int J Numer Methods Eng, 24 (1987), 959-973. https://doi.org/10.1002/nme.1620240509
  8. 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
  9. D. Elliott, The Euler Maclaurin formula revised, J Austral Math Soc Ser B, 40(E) (1998), E27-E76. https://doi.org/10.21914/anziamj.v40i0.454
  10. P.R. Johnston, Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals, Int J Numer Methods Eng, 47 (2000), 1709-1730. https://doi.org/10.1002/(SICI)1097-0207(20000410)47:10<1709::AID-NME852>3.0.CO;2-V
  11. P.R. Johnston, D. Elliott, Error estimation of quadrature rules for evaluating singular integrals in boundary element, Int J Numer Methods Eng, 48 (2000), 949-962. https://doi.org/10.1002/(SICI)1097-0207(20000710)48:7<949::AID-NME905>3.0.CO;2-Q
  12. P.R. Johnston, D. Elliott, A generalization of Telles' method for evaluating weakly singular boundary element integrals, J Comput Appl Math, 131 (2001), 223-241. https://doi.org/10.1016/S0377-0427(00)00273-9
  13. G. Monegato, I.H. Sloan, Numerical solutions of the generalized airfoil equation for an airfoil with a flap, SIAM J Numer Anal, 34 (1997), 2288-2305. https://doi.org/10.1137/S0036142995295054
  14. B.I. Yun, An extended sigmoidal transformation technique for evaluating weakly singular integrals without splitting the integration interval, SIAM J Sci Comput, 25 (2003), 284-301. https://doi.org/10.1137/S1064827502414606
  15. B.I. Yun, Rational transformations for evaluating singular integrals by the Gauss quadrature rule, Mathematics, 8, (2020), 677.