A generalized non-linear transformation for evaluating singular integrals

International Journal for Numerical Methods in Engineering

Volumn 65, Issue 12, 2006, Pages 1947-1969

  Yun, Beong In ^a  

^a Kunsan National University   (South Korea)

Author keywords

Euler Maclaurin formula; Polynomial transformation; Sigmoidal transformation; Singular integrals

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; INTEGRAL EQUATIONS; NUMERICAL ANALYSIS; POLYNOMIALS;

EULER-MACLAURIN FORMULA; NONLINEAR TRANSFORMATION; POLYNOMIAL TRANSFORMATION; SIGMOIDAL TRANSFORMATION; SINGULAR INTEGRALS;

MATHEMATICAL TRANSFORMATIONS;

CONVERGENCE OF NUMERICAL METHODS; FINITE ELEMENT METHOD; INTEGRAL EQUATIONS; MATHEMATICAL TRANSFORMATIONS; NUMERICAL ANALYSIS; POLYNOMIALS;

EID: 33645309365     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1529     Document Type: Article

References (35)

1. Kythe PK. An Introduction to Boundary Element Method. CRC Press: Boca Raton, FL, 1995.
2. Singh KM, Tanaka M. Analytical integration of weakly singular integrals in boundary element analysis of Helmholtz and advection-diffusion equations. Computer Methods in Applied Mechanics and Engineering 2000; 189:625-640.
3. Doblaré M. Computational Aspects of the Boundary Element Methods, Topics in Boundary Element Research, vol. 3. Springer: Berlin, 1987.
4. Aliabadi MH, Hall WS, Phemister TG. Taylor expansion for singular kernels in the boundary element method. International Journal for Numerical Methods in Engineering 1985; 21:2221-2236.
5. Qiu ZH, Wrobel LC, Power H. Numerical solution of convection-diffusion problems at high Peclét number using boundary elements. International Journal for Numerical Methods in Engineering 1998; 41:899-914.
6. Du QK. Evaluations of certain hypersingular integrals on interval. International Journal for Numerical Methods in Engineering 2001; 51:1195-1210.
7. Hui CY, Shia D. Evaluations of hypersingular integrals using Gaussian quadrature. International Journal for Numerical Methods in Engineering 1999; 44:205-214.
8. Ioakimidis NI. On the numerical evaluation of derivatives of Cauchy principal value integrals. Computing 1981; 27:81-88.
9. Kutt HR. The numerical evaluation of principal value integrals by finite-part integrals. Numerische Mathematik 1975; 24:205-210.
10. Linz P. On the approximate computation of certain strongly singular integrals. Computing 1985; 35:345-353.
11. Monegato G. On the weight of certain quadratures for Cauchy principal value integral. Numerische Mathematik 1987; 50:273-281.
12. Cerrolaza M, Alarcon E. A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary elements. International Journal for Numerical Methods in Engineering 1989; 28:987-999.
13. Doblaré M, Gracia L. On non-linear transformations for the integration of weakly-singular and Cauchy principal value integrals. International Journal for Numerical Methods in Engineering 1997; 40:3325-3358.
14. Elliott D. The cruciform crack problem and sigmoidal transformations. Mathematical Methods in the Applied Sciences 1997; 20:121-132.
15. Elliott D. Sigmoidal transformations and the trapezoidal rule. Journal of the Australian Mathematical Society, Series B 1998; 40(E):E77-E137.
16. Elliott D. The Euler Maclaurin formula revised. Journal of the Australian Mathematical Society, Series B 1998; 40(E):E27-E76.
17. Elliott D, Venturino E. Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals. Numerische Mathematik 1997; 77:453-465.
18. Johnston PR. Application of sigmoidal transformations to weakly singular and near singular boundary element integrals. International Journal for Numerical Methods in Engineering 1999; 45:1333-1348.
19. Johnston PR. Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals. International Journal for Numerical Methods in Engineering 2000; 47:1709-1730.
20. Johnston PR, Elliott D. Error estimation of quadrature rules for evaluating singular integrals in boundary element. International Journal for Numerical Methods in Engineering 2000; 48:949-962.
21. Johnston PR, Elliott D. A generalization of Telles' method for evaluating weakly singular boundary element integrals. Journal of Computational and Applied Mathematics 2001; 131:223-241.
22. Johnston PR, Elliott D. Transformations for evaluating singular boundary element integrals. Journal of Computational and Applied Mathematics 2002; 146:231-251.
23. Korobev NM. Number-Theoretic Method of Approximate Analysis. GIFL: Moscow, 1963.
24. Lyness JN. The Euler-Maclaurin expansion for the Cauchy principal value integral. Numerische Mathematik 1985; 46:611-622.
25. Monegato G, Sloan IH. Numerical solutions of the generalized airfoil equation for an airfoil with a flap. SIAM Journal on Numerical Analysis 1997; 34:2288-2305.
26. Prössdorf S, Rathsfeld A. On an integral equation of the first kind arising from a cruciform crack problem. In Integral Equations and Inverse Problems, Petkov V, Lazarov R (eds). Longman: Coventry, 1991; 210-219.
27. Sanz Serna JM, Doblaré M, Alarcon E. Remarks on methods for the computation of boundary-element integrals by co-ordinate transformation. Communications in Applied Numerical Methods 1990; 6:121-123.
28. Sato M, Yoshiyoka S, Tsukui K, Yuuki R. Accurate numerical integration of singular kernels in the two-dimensional boundary element method. In Boundary Elements X, Brebbia CA (ed.), vol. 1. Springer: Berlin, 1988; 279-296.
29. Sidi A. A new variable transformation for numerical integration. In Numerical Integration IV, Brass H, Hämmerlin G (eds), ISNM vol. 112. Birkhauser: Berlin, 1993; 359-373.
30. Singh KM, Tanaka M. On non-linear transformations for accurate numerical evaluation of weakly singular boundary integrals. International Journal for Numerical Methods in Engineering 2001; 50:2007-2030.
31. Sládek V, Sládek J, Tanaka M. Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity. International Journal for Numerical Methods in Engineering 1993; 36:1609-1628.
32. Telles JCF. A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. International Journal for Numerical Methods in Engineering 1987; 24:959-973.
33. Yun BI. An efficient transformation with Gauss quadrature rule for weakly singular integrals. Communications in Numerical Methods in Engineering 2001; 17:881-891.
34. Yun BI. An extended sigmoidal transformation technique for evaluating weakly singular integrals without splitting the integration interval. SIAM Journal on Scientific Computing 2003; 25:284-301.
35. Yun BI, Kim P. A new sigmoidal transformation for weakly singular integrals in the boundary element method. SIAM Journal on Scientific Computing 2003; 24:1203-1217.