Local RBF Meshless Method For Solving Scalar And Vector Radiative Transfer Equation

Radiative transfer theroy has many important applications in diverse fields such as radiative heat transfer, astrophysics, atmospheric and ocean optics, remote sensing, biomedical optics and imaging. Two kind of fundamental equations exist in the radiative transfer theroy, that are scalar radiative transfer equation and polarized radiative transfer equation. Exact analytical solutions are not available in most realistic situations, therefore, numerical methods become important approaches to obtain approximation solutions of radiative transfer problems.Collocation meshless methods have emerged as a potential alternative in the field of the numerical. As compared to the convection-diffusion equation, the discrete ordinates form of the radiative transfer equation can be considered as a special kind of convection-dominated equation. Many developed meshless collocation methods suffer from numerical oscillations without special stabilization techniques. Two kinds of special stabilization techniques were often adopted in collocation meshless methods,(1) taking various numerical stabilization schemes, such as least squares collocation based on auxiliary nodes scheme, upwind technique, and(2) transforming the RTE into a numerical stable equation, for example, a second order form of RTE. These stabilization schemes have been demonstrated to be effective for some cases. However, obviously, the complexity of implementation and the computation time would increase. Besides, the most of developed meshless collocation methods are not easy to be implemented for solving multidimensional radiative transfer problems due to the approximation functions of which are related to spatial coordinates.The meshless methods based on local radial basis function(RBF) interpolation have gained popularity in the engineering and science and emerged as a potential alternative in the field of numerical methods of PDEs. RBFs are expressed in the Euclidean distance variable and radially isotropic. Multiquadric RBF interpolation is spectrally convergent which can be considered as a generalization of the pseudo spectral methods for unstructured grids and complex domains. Shape functions of the local RBF interpolation approximation have Kronecker delta function property which ensures the essential boundary conditions imposed easily and exactly.In this paper, a new implementation of local radial basis function meshless method was proposed to solve the scalar in homogeneous media. For solving the radiative heat transfer in strongly inhomogeneous media, the local radial basis function meshless method based on the upwind scheme was developed. The local radial basis function meshless method was also extended to solve polarized radiative transfer in participating media containing randomly oriented axisymm etric particles.Based on the correction interpolation of outflow-boundary intensity which can be easily implemented, the improved local radial basis function meshless method is of high accuray and excellent stability to solve radiative heat transfer in homogeneous participating media. Based on the correction interpolation of outflow-boundary intensity, the improved direct collocation meshless method is also of high accuray and excellent stability to solve radiative heat transfer in homogeneous participating media, which demonstrates the proposed new scheme is of high stability.An upwind support domain scheme is introduced for the solution to radiative heat transfer in strongly inhomogeneous media with the local radial basis function meshless method. The upwind scheme is implemented by moving the support domain of local radial basis function interpolation approximation to the opposite direction of each streamline, which can fully capture the information from upstream and improve the accuracy and stability of LRBFM. It is demonstrated that the local radial basis function meshless method with upwind support domain scheme provides high accuracy and great stability to solve radiative heat transfer in strongly inhomogeneous media.The local radial basis function meshless method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional par ticipating media. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular nodes distributions, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. Meanwhile, it is extremely simple to implement.The local radial basis function meshless scheme is also extended to solve polarized radiative transfer in participating media containing randomly oriented axisymmetric particles. Performances of the LRBFM are verified with analytical solutions and other numerical results reported earlier in the literature s via five various test cases. It is demonstrated that the LRBFM is also accurate to solve vector radiative transfer in participating media with randomly oriented axisymmetric particles.