On the solution of exterior plane problems by the boundary element method: A physical point of view

The paper is devoted to the solution of Laplace equation by the boundary element method. The coupling between a finite element solution inside a bounded domain and a boundary integral formulation for an exterior infinite domain can be performed by producing a “stiffness” or “impedance matrix”. It is shown in a first step that the use of classical Green's functions for plane domains can lead to impedance matrices which are not satisfying, being singular or not positive-definite. Avoiding the degenerate scale problem is classically overcome by adding to Green's function a constant which is large compared to the size of the domain. However, it is shown that this constant affects the solution of exterior problems in the case of non-null resultant of the normal gradient at the boundary. It becomes therefore important to define this constant related to a characteristic length introduced into Green's function. Using a “'slender body theory” allows to show that for long cylindrical domains with a given cross section, the characteristic length is shown as being asymptotically equal to the length of the cylindrical domain. Comparing numerical or analytical 3D and 2D solutions on circular cylindrical domains confirms this result for circular cylinders.