The numerical evaluation of Hadamard finite-part integrals

Abstract

Summary. A quadrature rule is described for the numerical evaluation ofHadamard finite-part integrals with a double pole singularity within the range of integration. The rule is based upon the observation that such an integral is the derivative of a Cauchy principal value integral.

Distributed dislocation approach for cracks in couple-stress elasticity: shear modes

“…Although a lot of work has been done in the literature for Hadamard type integrals ( 2 a  ) (see e.g. Kutt, 1975;Paget, 1981;Ioakimidis, 1983Ioakimidis, , 1995Kaya and Erdogan, 1987;Monegato, 1987Monegato, , 1994Tsamasphyros and Dimou, 1990;Korsunsky, 1998;Kabir et al, 1998;Hui and Shia, 1999), only a few papers have been published concerning integrals with 2  a . In a recent work by Chan et al (2003), a systematic treatment of hypersingular integrals was presented based on the Kaya / Erdogan approach.…”

“…This approach leads to very good results, with the only caveat that when the kernel cannot be explicitly given in terms of a sum of the hypersingular part and a remainder, the extraction of a strong singularity may lead to a loss of accuracy. Our intention here is to derive a numerical quadrature for the hypersingular integral is to be understood in the Hadamard finite-part sense (Kutt, 1975;Paget, 1981). The basic steps in the development of the quadrature follow the strategy introduced by Korsunsky (1998).…”

Evaluation of finite part integrals using a regularization technique that decreases instability

“…The calculations in case B are performed using formula (10) with M = 1000, whereas case A is worked out using Eq. (11). To compare with a Gauss quadrature formula of pure rational type we refer the reader to [22,[26][27][28] where other methods different from that presented here are used.…”

“…Paget [11] seems to be one of the first in using (3) to remove the singularity of (1). Nevertheless, when using finite precision arithmetic and x ≈ t, then E(x, t) is unstable, even when F (x) exists.…”

Modified Gauss rules for approximate calculation of some strongly singular integrals

“…Here it is examined the theoretical role played by ε in (6). For this purpose, suppose that f can be expressed as a power series expansion at x, which converges uniformly on (α, β) ⊃ [a, b], so that the following equality holds true.…”