Preconditioned conjugate gradient method for three-dimensional non-convex enclosure geometries with diffuse and grey surfaces

Our main concern in this paper is the numerical simulation of the heat radiation exchange in a three-dimensional non-convex enclosure geometry with a diffuse and grey surface. This physical phenomena is governed by a boundary integral equation of the second kind. Due to the non-convexity of the enclosure the presence of the shadow zones must be taken into account in the heat radiation analysis. For that purpose we have developed a geometrical algorithm to provide an efficient detection of these shadow zones that are needed to calculate the visibility function. For the discretization of the boundary integral equation we have used the boundary element method based on the Galerkin-Bubnov scheme. The system of linear equations which subsequently arise has been solved by the conjugate gradient method with preconditioning. To demonstrate the high efficiency of this method a numerical experiment has been constructed for non-convex geometry; the heat radiation in an aperture has been considered.