Finite Element Method For Solving Radiative Transfer Equation In Graded Index Medium

Radiative transfer is a basic way of heat and information transfer, and plays an important role in many scientific and engineering applications. Due to the structural characteristics of material or possible temperature dependency, the refractive index of medium may be not uniform as the hypothesis used in traditional radiative heat transfer analysis. The radiative heat transfer in semitransparent medium with graded index is of great importance in many phenomenon and processes, and has evoked the wide interest of many researchers. In graded index medium, the ray goes along a curved path determined by the Fermat principle. The solution of radiative transfer in graded index medium is more difficult than that in uniform index medium. The traditional methods used in solving radiative transfer equation in uniform index medium can not be directly used in graded index medium.The finite element method has many special merits, such as, considerable flexibility to deal with the complex geometric shape, the higher approximation for the field variables in a volume or surface element and so on. The finite element method developed recently based on discretization of the differential form of radiative transfer equation suffer from stabilization problems, hence its application in solving radiative heat transfer need to be studied further.This dissertation develops finite element methods and studies their characteristics and performance to solve radiative transfer in multidimensional graded index medium under both Cartesian and cylindrical coordinate systems. The scope of present research mainly contains five parts:1. Based on the radiative transfer equation of graded index medium in Cartesian coordinate system, finite element methods are developed for solving radiative transfer in multidimensional semitransparent graded index medium. The performance of the finite element methods based on different stabilization schemes are studied and compared for solving radiative transfer in semitransparent graded index medium. The least square finite element method and streamline upwind based finite element method can successfully eliminate the nonphysical oscillatory of solutions that may appear in Galerkin scheme based finite element method.2. The radiative transfer equations of graded index medium in the traditional cylindrical coordinate system and in a new cylindrical coordinate system are derived. Based on these two equations, finite element methods which are suitable to solve radiative transfer in multidimensional semitransparent cylindrical graded index medium with complex boundary conditions are developed. The performance of the methods is examined for solving radiative transfer in multidimensional semitransparent graded index medium under cylindrical coordinate system. The finite element method based on the radiative transfer equation under the new cylindrical coordinate system can conveniently solve the radiative heat transfer in cylindrical graded index medium with specular and Fresnel boundary.3. Due to the second order diffusion term has good numerical stability, the second order radiative transfer equation in graded index medium is derived, and a finite element method is developed based on the derived second order equation. The performance of the finite element method for solving radiative transfer in multidimensional semitransparent graded index medium is examined. This second order radiative transfer equation can be conveniently applied to solve radiative transfer in absorbing, emitting and anisotropically scattering media. The numerical study verifies that the method has very good numerical property.4. An unstructured hybrid finite volume/finite element method is presented to solve the radiative transfer problem in graded index medium. The performance of the hybrid method for solving radiative transfer in graded index medium is examined. In this method, the spatial discretization is carried out using the finite volume method, while the angular discretization is performed using the finite element method. The hybrid finite volume/finite element method can effectively combine the merits of the two methods.5. Based on the radiative transfer equation in cylindrical coordinate system, the finite element methods which are suitable to solve radiative transfer in multilayer semitransparent graded index cylinder and in optical fiber coated with thin film are developed and studied. An interpolation method based on radial basis functions is presented to interpolate the radiative intensity between two adjacent layers. A finite element analysis model is established to obtain the radiative property of optical fiber coated with thin film.