Special Boundary Integral Equations for Approximate Solution of Potential Problems in Three-dimensional Regions with Slender Cavities of Circular Cross-section

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Abstract
A special boundary integral method is developed for solving potential problems in a general three-dimensional region with slender internal cavities of circular cross-section. The solution on the boundary of each cavity is assumed to be axisymmetric so that the surface integrals on a cavity boundary may be reduced to contour integrals along the centre line of the cavity. Special integral equations are introduced to determine the solution along each cavity. Although the contour integrals in these new equations are never singular, they have a nearly singular character which gives them computational advantages similar to traditional boundary integral equations with out the danger of ill-conditioning caused by the strongly contrasting length scales introduced by the slender cavities. For the special case of parallel toroidal cavities, the method gives results with accuracy comparable to previous perturbation methods. The numerical characteristics of the new integral equations are explored by solving the problems of two perpendicular, interlocking tori and two perpendicular, finite cylindrical cavities, both in unbounded regions. The equations exhibit excellent numerical characteristics over a broad range of parameters.

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