On the convergence of the irradiation coefficient

This work is devoted to the study of a singular integral that is the irradiation coefficient (angular coefficient) between surfaces of finite sizes. The exact calculation of the irradiation coefficients between arbitrary surfaces is very difficult. Formulas for calculation are available only for a limited class of standard surfaces located in space in a certain way. The irradiation coefficients for a wider class of surfaces can be analytically calculated using the properties of these coefficients, such as distributivity, reciprocity, and closeness; this method is also laborious. Therefore, it is necessary to develop methods that make it possible to calculate the irradiation coefficient between arbitrary surfaces with a predefined high accuracy. These methods include the use of cubature formulas based on the definition of these irradiation coefficients, which are surface integrals. However, this approximate method is effective in the case of distant surfaces of finite sizes. The theoretical question of the convergence of singular integrals expressing the irradiation coefficient for adjoining or intersecting surfaces has not been studied. As was shown by numerical experiments, the accuracy and convergence rate of the cubature formulas are extremely low in the singular case.
We prove the convergence of the singular integral which is the irradiation coefficient (angular coefficient) between two adjoining flat surfaces of finite sizes. A cubature formula with non-uniform mesh is proposed for approximate calculation with high accuracy. The results of numerical experiments are presented.