Numerical solution for degenerate scale problem for exterior multiply connected region

Abstract

Based on some previous publications, this paper investigates the numerical solution for degenerate scale problem for exterior multiply connected region. In the present study, the first step is to formulate a homogenous boundary integral equation (BIE) in the degenerate scale. The coordinate transform with a magnified factor, or a reduced factor h is performed in the next step. Using the property ln( hx)=ln( x)+lg( h), the new obtained BIE equation can be considered as a non-homogenous one defined in the transformed coordinates. The relevant scale in the transformed coordinates is a normal scale. Therefore, the new obtained BIE equation is solvable. Fundamental solutions are introduced. For evaluating the fundamental solutions, the right-hand terms in the non-homogenous equation, or a BIE, generally take the value of unit or zero. By using the obtained fundamental solutions, an equation for evaluating the magnified factor “ h” is obtained. Finally, the degenerate scale is obtainable. Several numerical examples with two ellipses in an infinite plate are presented. Numerical solutions prove that the degenerate scale does not depend on the normal scale used in the process for evaluating the fundamental solutions.