ON THE USE OF EXPLICIT FORMULATIONS IN THE HIGH ORDER DISCRETE ORDINATES SOLUTION OF THE RADIATIVE TRANSFER EQUATION IN ANISOTROPIC SCATTERING MEDIA

2020 ◽  
Author(s):  
Karine Rui ◽  
Liliane Barichello ◽  
Rudnei Dias da Cunha
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Anisotropic Scattering ◽  
High Order ◽  
Discrete Ordinates ◽  
Scattering Media

An Acceleration Technique for the Computation of Participating Radiative Heat Transfer

2009 ◽  
Author(s):  
Sanjay R. Mathur ◽  
Jayathi Y. Murthy
Keyword(s):  
Heat Transfer ◽  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Radiation Heat Transfer ◽  
Solution Procedure ◽  
Discrete Ordinates ◽  
Average Intensity ◽  
Scattering Media ◽  
Acceleration Techniques

It is known that the finite volume and discrete ordinates methods for computing participating radiation are slow to converge when the optical thickness of the medium becomes large. This is a result of the sequential solution procedure usually employed to solve the directional intensities, which couples the ordinate directions and the energy equation loosely. Previously published acceleration techniques have sought to employ a governing equation for the angular-average of the radiation intensity to promote inter-directional coupling. These techniques have not always been successful, and even where successful, have been found to destroy the conservation properties of the radiative transfer equation. In this paper, we develop an algorithm called Multigrid Acceleration using Global Intensity Correction (MAGIC) which employs a multigrid solution of the average intensity and energy equations to significantly accelerate convergence, while ensuring that the conservative property of the radiative transfer equation is preserved. The method is shown to perform well for radiation heat transfer problems in absorbing, emitting and scattering media, both and without radiative equilibrium, and across a range of optical thicknesses.

Completely Spectral Collocation Solution of Radiative Heat Transfer in an Anisotropic Scattering Slab With a Graded Index Medium

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Jing Ma ◽  
Ya-Song Sun ◽  
Ben-Wen Li
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Anisotropic Scattering ◽  
Discrete Ordinates ◽  
Spectral Collocation ◽  
Graded Index ◽  
Wide Range ◽  
Derivative Term ◽  
Scattering Phase Function

A completely spectral collocation method (CSCM) is developed to solve radiative transfer equation in anisotropic scattering medium with graded index. Different from the Chebyshev collocation spectral method based on the discrete ordinates method (SP-DOM), the CSCM is used to discretize both the angular domain and the spatial domain of radiative transfer equation. In this approach, the angular derivative term and the integral term are approximated by the high order spectral collocation scheme instead of the low order finite difference approximations. Compared with those available data in literature, the CSCM has a good accuracy for a wide range of the extinction coefficient, the scattering albedo, the scattering phase function, the gradient of refractive index and the boundary emissivity. The CSCM can provide exponential convergence for the present problem. Meanwhile, the CSCM is much more economical than the SP-DOM. Moreover, for nonlinear anisotropic scattering and graded index medium with space-dependent albedo, the CSCM can provide smoother results and mitigate the ray effect.

Resolved Order of Scattering for the Solution of Radiative Transfer Equation

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
Liangyu Wang ◽  
Robert J. Hall ◽  
Meredith B. Colket
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Anisotropic Scattering ◽  
Scattering Media ◽  
Radiative Intensity ◽  
Reflecting Boundaries ◽  
Strong Scattering

The solution of the radiative transfer equation (RTE) becomes complicated when the participating medium is scattering and/or the boundary walls are reflecting. To reduce the complexity, the resolved order of scattering (ROS) formulation described in this paper separates the radiative intensities being solved by RTE into a series of intensities corresponding to different orders of the scattering and reflection events. The resulting equation of transfer for each order of radiative intensity is not only much simpler to solve but also represents the physical scattering/reflection processes that are hidden in the original full RTE. The ROS formulation provides a mathematically rigorous and elegant means of solving RTE for strong scattering media with or without reflecting boundaries. Sample calculations are presented for a droplet-laden, 3D enclosure with strong anisotropic scattering.

An Efficient Sparse Finite Element Solver for the Radiative Transfer Equation

2009 ◽  
Author(s):  
Gisela Widmer
Keyword(s):  
Linear System ◽  
Radiative Transfer ◽  
Transfer Equation ◽  
Degrees Of Freedom ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Five Dimensions ◽  
Computational Costs ◽  
Dimensional Domain

The stationary monochromatic radiative transfer equation (RTE) is posed in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. For non-scattering radiative transfer, sparse finite elements [1, 2] have been shown to be an efficient discretization strategy if the intensity function is sufficiently smooth. Compared to the discrete ordinates method, they make it possible to significantly reduce the number of degrees of freedom N in the discretization with almost no loss of accuracy. However, using a direct solver to solve the resulting linear system requires O(N3) operations. In this paper, an efficient solver based on the conjugate gradient method (CG) with a subspace correction preconditioner is presented. Numerical experiments show that the linear system can be solved at computational costs that are nearly proportional to the number of degrees of freedom N in the discretization.

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