Integral equation solutions using radial basis functions for radiative heat transfer in higher-dimensional refractive media

A collocation method based on radial basis functions (RBFs) is applied to solve the integral equations of intensity moments for radiative heat transfer in scattering media with spatially varying refractive index (VRI). Since the method does not require predefined meshes, it can be readily applied to the problem with irregular geometry. Efficient codes of the collocation method with discrete ray tracing are developed. The codes are applied to analyze radiative equilibrium in a semicircular medium with an inner circular boundary. Since rigorous solutions on radiative equilibrium in three-dimensional refractive media are seldom reported, we also apply the present method to analyze radiative heat transfer in cubic media with VRI. The results obtained by using multiquadric RBFs for cases with various optical sizes and boundary conditions are presented. Comparisons of the present results and those obtained by Monte Carlo discrete ray tracing simulation show a good agreement; the discrepancy between the results of the two methods decreases with the increase of the distinct data points used. The present results also show that the temperatures of the cases with a diffusely reflecting semicircular surface are larger than those of the cases with a black semicircular surface and a larger variation of temperature may be observed in the cubic medium with a larger optical thickness.