Strategies using boundary integral equations for solving exterior anisotropic conduction problems outside long cylinders

Abstract

2D approximations can greatly alleviate the computing effort required to solve anisotropic conduction problems outside long 3D cylindrical domain using boundary integral methods. Two strategies can be used to this aim: either transform the anisotropic conduction problem into an isotropic one, or deal with the anisotropic 2D Green's function. In the first case, it is necessary to provide not only the new features of the transformed domain, but also the new expressions of the boundary conditions over the domain. Conversely, the anisotropic 2D Green's function is defined upto a constant which depends on the length of the cylindrical domain, as shown in the isotropic case. In addition, the use of anisotropic Green's function cannot avoid the occurrence in some cases of degenerate scales, which is well known in the isotropic case. The paper addresses these different points: construction of the anisotropic 2D Green's function and its relation with line sources, description of the transformation leading to an equivalent isotropic problem and finally study of the boundary integral solution of the equivalent 2D problem, including the occurence of degenerate scales.