Gauss-Chebyshev quadrature formulae for strongly singular integrals
This paper presents some explicit results concerning an extension of the mechanical quadrature technique, namely, the Gauss-Jacobi numerical integration scheme, to the class of integrals whose kernels exhibit second order of singularity (i.e. one degree more singular than Cauchy). In order to ascribe numerical values to these integrals they must be understood in Hadamard's finite-part sense. The quadrature formulae are derived from those for Cauchy singular integrals. The resulting discretizations are valid at a number of fixed points, determined as the zeros 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 2 n − 1, where n is the number of nodes. These properties make this approach rather attractive for applications to fracture mechanics problems, where often numerical solution of integral equations with strongly singular kernels is the objective. Numerical examples of the application of the Gauss-Chebyshev rule to some plane and axisymmetric crack problems are given.