Application of the modified differential approximation for radiative transfer to arbitrary geometry

The first-order spherical harmonics method (or P1 approximation) has found prolific usage for approximate solution of the radiative transfer equation (RTE) in participating media. However, the accuracy of the P1 approximation deteriorates as the optical thickness of the medium is decreased. The modified differential approximation (MDA) was originally proposed to remove the shortcomings of the P1 approximation in optically thin situations. This article presents algorithms to apply the MDA to arbitrary geometry-in particular, geometry with obstructions, and inhomogeneous media. The wall-emitted component of the intensity was computed using a combined view-factor and ray-tracing approach. The Helmholtz equation, arising out of the medium-emitted component, was solved using an unstructured finite-volume procedure. The general procedure was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results. The accuracy of MDA was found to be superior to the P1 approximation for all optical thicknesses. Its accuracy, when compared with the discrete ordinates method (both S6 and S8), was found to be clearly superior in optically thin situations, but problem dependent in optically intermediate and thick situations. For 3D geometries, calculation and storage of the view-factor matrix was found to be a major shortcoming of the MDA. In addition, for inhomogeneous media, calculation of optical distances requires a ray-tracing procedure, which was found to be a bottleneck from a computational efficiency standpoint. Several strategies to reduce both memory and computational time are discussed and demonstrated.