Finite-element analysis of radiative transport in gray participating media

Abstract

A finite-element solution scheme for radiative transport problems in gray participating media is devised, and its validity is substantiated through application to representative problems involving plane-parallel geometry with azimuthal symmetry. The governing boundary value problem of radiative transport in gray participating media is first posed, and its, simplification in the case of plane-parallel geometry with azimuthal symmetry is considered in great detail with emphasis on the physics assumed in the simplification process. To provide the necessary basis for the application of the finite-element approximation technique, the governing boundary value problem is formu.lated in the weak sense, and subsequently the Galerkin approximation of the resulting weak formulation is stated. In the weak formulation. and therefore in its corresponding Galerkin approximation boundary conditions are incorporated as natural rather than essential conditions.The advantages of such an approach are clear and are discusied briefly. After a short discussion of the relevant concepts of the finite-element approximatio technique, the finite-element model of the Galerkin approximation of the weak formulation of the governing boundary value problem is developed. The resulting equations describing this model are simple, well-conditioned algebraic equations. With the general underlying theory thus established, a specific finite-element model applicable to any radiative transport problem in plane-parallel azimuthally symmetric gray participating media is, derived, vlith particular emphasis laid on accounting for the angular discontinuities in the intensity distribution. For the sake of simplicity, linear rectangular finite elements are incorporated in this specific model. For this particular choice of elements, specific expressions for the (elevant element matrices and the element vectors are derived and presehted with the intention of facilitating the tailoring of a finite-element solution scheme to problems in the application range of the aforementioned specific model. Finally; the application of this specific model to selected homogeneous as viell as nonhomogeneous problems, which are well-documented, is considered. Numerical results obtained for these problems are tabulated and compared with the corresponding exact results reported in the literature. The agreement between the finite-element and the corresponding exact results is seen to be highly satisfactory. On the basis of the aforementioned theoretical and numerical results, it is found that the finite-element approximation technique provides an efficient and reliable solution scheme for radiative transport problems in gray participating media.