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Two trigonometric quadrature formulae for evaluating hypersingular integrals
Author(s) -
Kim Philsu,
Choi U. Jin
Publication year - 2002
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.582
Subject(s) - mathematics , trigonometric substitution , trigonometric integral , quadrature (astronomy) , gauss–jacobi quadrature , trigonometry , mathematical analysis , cauchy principal value , gauss–kronrod quadrature formula , trigonometric functions , clenshaw–curtis quadrature , numerical integration , principal value , integration using euler's formula , tanh sinh quadrature , trigonometric interpolation , boundary value problem , nyström method , polynomial , polynomial interpolation , gaussian quadrature , geometry , linear interpolation , physics , optics , bicubic interpolation , cauchy boundary condition , free boundary problem
Two trigonometric quadrature formulae, one of non‐interpolatory type and one of interpolatory type for computing the hypersingular integral ${\int\hskip-0.33cm=}_{-1}^{1} w(\tau)g(\tau)/(\tau-t)^{2} \,{\rm d}\tau$ are developed on the basis of trigonometric quadrature formulae for Cauchy principal value integrals. The formulae use the cosine change of variables and trigonometric polynomial interpolation at the practical abscissae. Fast three‐term recurrence relations for evaluating the quadrature weights are derived. Numerical tests are carried out using the current formula. As applications, two simple crack problems are considered. One is a semi‐infinite plane containing an internal crack perpendicular to its boundary and the other is a centre cracked panel subjected to both normal and shear tractions. It is found that the present method generally gives superior results. Copyright © 2002 John Wiley & Sons, Ltd.