SINGULAR INTEGRALS IN AXISYMMETRIC TASKS OF POTENTIAL THEORY

Abstract

Axisymmetric structures are widely used in the chemical and aerospace industries, logistics, power engineering and other engineering sectors. Usually these structures and their components work at high loads, interact with moving fluids and are exposed to high temperatures. Therefore, the main tasks are assessment of critical loads, identification and separation of hazardous resonance frequencies, and prediction of a reliable safety forecast for equipment operation. Experimental research in this area is financially costly, complex and sometimes impossible for a number of reasons. That is why the methods of mathematical and computer modeling are the most effective engineering tools for assessing the strength characteristics of existing and projected objects. The calculation of elements of a matrix of a system of linear algebraic equations does not cause difficulties, since the integral functions are continuous and for them it is possible to use standard Gaussian quadratures. However, in the case where the collocation point coincides with the boundary element on which the integration occurs, the argument of the complete elliptic integral becomes at the point of collocation equal to one, so the calculated integral becomes an improper integral of the second kind. It is necessary to investigate its convergence and develop quadrature formulas for its calculation. In the above study, the existing approaches to solving the axisymmetric problem of the theory of potential are analyzed and their own algorithms and schemes are presented and proved. In the piecewise linear approximation, singular integrals are analyzed and quadrature formulas are obtained for their program realization that arise when calculating the matrix coefficients of a system of linear algebraic equations. Integrals with a logarithmic singularity were compared with analytic expressions for some functions, and the general scheme for the singular integral was tested using the surface Gaussian integral with a fixed point on the surface.