Spectral Element Method For Solving Radiative Transfer Equation

Many numerical methods have been developed for solving radiative heat transfer in semitransparent participant media. By comprehensive comparison, the methods based on discretization of the differential form of radiative transfer equation possess very good adaptabities. As for solving radiative transfer in uniform index media, discrete ordinates method and finite volume method have many advantages, such as, they are convenient and efficient to deal with multidimensional problem, flexible to deal with problem with complex media and boundary properties, and easy to be adapted to coupled solution with other heat transfer processes, etc., as a result, they are the mostly often used methods until recently. As for solving radiative transfer in graded index media, the finite volume method and the finite element method avoid the time-consuming computation of curved ray trajectories, so they show very good adaptabities. However, these already developed methods are all low order methods, generally with only first or second order of accuracy, furthermore they provide only h-convergence property, as a result, in order to gain wanted accuracy, cumbersome mesh refining or remeshing is necessary.Spectral element method can effectively overcome the drawbacks of low order methods, which possesses both the advantages of low order finite element method and high order spectral method and best combines the p-convergence property of spectral methods, namely, convergence by increasing order of polynomial, and the advantage to be easy to applied to complex enclosures and the h-convergence property of finite element method, namely, convergence by mesh refining. Two convergence strategies make spectral element method are more effective, of which the p-convergence property make spectral element method can achieve convergence by just increasing the order of polynomial.By considering the numerical properties of radiative transfer equation for uniform and graded index media, this paper develops and studies the characteristics and performances of spectral element methods based on different discretization schemes to solve radiative transfer in semitransparent participant media, coupled radiative and conductive heat transfer and transient radiative transfer. The scope of present research contains five parts:1. A spectral element method based on least squares stabilization scheme is developed to solve radiative transfer in multidimensional uniform index media and an efficient implementation algorithm is presented. The proposed method is also extended to solve radiative transfer in graded index media. The stability and p-convergence characteristics of the method are studied and its p-convergence speed is very fast and follows exponential law. The performances of the method for solution of radiative transfer in multidimensional semitransparent uniform and graded index media are examined. Some other stabilization schemes of spectral element method and their imposing techniques are presented and the performances of spectral element method based on these stabilization schemes for solving radiative transfer in semitransparent media are studied and compared.2. A second order radiative transfer equation of primitive variable is derived by using a different way than the even parity formulation. This second order equation possesses major advantages of the even parity formulation of radiative transfer, but overcomes most of its drawbacks, and can be conveniently applied to solve radiative transfer in absorbing, emitting and anisotropically scattering media. Perturbation error analysis are made for both the first order and the second order equation, by comparison, the second order radiative transfer equation shows better numerical properties. A general formulation of Galerkin discretization of the second order radiative transfer equation is presented, and the performances of the second order equation are examined for solving multidimensional radiative transfer problems.3. A spectral element method based on discontinuous Galerkin scheme is developed to solve radiative transfer in multidimensional semitransparent media. The discontinuous Galerkin scheme need not to impose inter-elemental continuity and the resulting discretized equation own local conservativity. This method effectively eliminates the numerical instability caused by the convection property of discrete ordinates equation. The p-convergence characteristics of the proposed method are studied on both structured and unstructured meshes. Influences of different schemes for dealing with elemental boundary numerical flux on numerical properties of the proposed method are compared. Two kinds of ray effect are discussed for methods based on discretization of radiative transfer equation, namely, ray effect induced by boundary loading and ray effect induced by shielding of interior obstacle.4. Based on the second order radiative transfer equation, a spectral element method is developed to solve coupled radiative and conductive transfer in multidimensional semitransparent media. The performances of the proposed method are examined for solving coupled radiative and conductive heat transfer in multidimensional semitransparent media. The h- and p-convergence characteristics of the proposed method are studied. The p-convergence speeds are very fast and follow exponential law under different values of Plank number and are superior to the h-convergence speed. Numerical examinations show that spectral element method possesses very good adaptability to skewed meshes.5. Considering the discontinuous spectral element method has advantages, such as local conservativity, high order of accuracy and minimal artificial diffusion, it is extended to solve transient radiative transfer problems. Special treatment of the collimated irradiation and transient diffusive boundary are studied. The implementation of the discontinuous spectral element method for the discretization of transient radiative transfer equation is presented. The proposed discontinuous spectral element method for transient radiative transfer is examined by several classical examples, and its performances for solving transient radiative transfer are studied.