GAUSS-CHEBYSHEV QUADRATURE-FORMULAS FOR STRONGLY SINGULAR-INTEGRALS

This paper presents some explicit results concerning an extension of t he mechanical quadrature technique, namely, the Gauss-Jacobi numerical integration scheme, to the class of integrals whose kernels exhibit s econd order of singularity (i.e., one degree more singular than Cauchy ). In order to ascribe numerical values to these integrals they must b e understood in Hadamard's finite-part sense. The quadrature formulae are derived from those for Cauchy singular integrals. The resulting di scretizations are valid at a number of fixed points, determined as the zeroes of a certain Jacobi polynomial. As in all Gaussian quadratures , the final quadrature formulae involve fixed nodal points and provide exact results for polynomials of degree 2n - 1, where n is the number of nodes. These properties make this approach rather attractive for a pplications to fracture mechanics problems, where often numerical solu tion of integral equations with strongly singular kernels is the objec tive. Numerical examples of the application of the Gauss-Chebyshev rul e to some plane and axisymmetric crack problems are given.