CALCULATION OF CAUCHY-TYPE INTEGRALS NEAR CONTOURS IN DIRECT AND INVERSE ELASTIC PROBLEMS
Free (open access)
45 - 55
ALEXANDER N. GALYBIN
This study presents an approach for the calculation of Cauchy-type integrals at points located near contours. It is evident that the kernel of a Cauchy integral becomes close to singular as soon as one intends to calculate the value of the integral close to the contour. As a result, more nodes in a quadrature formula are needed, in order to reach acceptable accuracy in the calculations. This problem is faced in standard formulations when analysing stress–strain states after obtaining numerical solutions of certain singular integral equations; as well as in non-classical formulations, where the data close to the contour are used as input. On the other hand, one can employ, for the contour points, the Plemelj–Sokhotski formulas, assuming calculation of the singular integral is followed by addition of a known non-integral term. In this study, we use expansions into power series to calculate stress characteristics at points near the contours, suggest an algorithm, and numerically analyse two cases that are relevant to direct and inverse formulations in plane elasticity.
plane elasticity, Cauchy integrals, singular integral equations, inverse problem, mathematical optimisation