Radiative transfer equation accounting for rotational Raman scattering and its solution by the discrete-ordinates method

2014 ◽  
Vol 133 ◽  
pp. 603-618 ◽  
Author(s):  
Vladimir V. Rozanov ◽  
Marco Vountas
Keyword(s):  
Radiative Transfer ◽  
Raman Scattering ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Discrete Ordinates ◽  
Discrete Ordinates Method

Hybrid SN-PN Solution of the Radiative Transfer Equation in Multi-Dimensional Media

2011 ◽  
Author(s):  
Maathangi Sankar ◽  
Sandip Mazumder
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Complex Geometry ◽  
Radiative Transfer Equation ◽  
Discrete Ordinates ◽  
Differential Approximation ◽  
Discrete Ordinates Method ◽  
Surface Exchange ◽  
New Formulation ◽  
Ray Effects

The Modified Differential Approximation (MDA) was originally proposed for solution of the radiative transfer equation (RTE) in order to remove the shortcomings of the P1 approximation in scenarios where the radiation intensity is strongly directionally dependent. In the original MDA approach, the wall-emitted component of the intensity is determined using a surface-to-surface exchange formulation that makes use of geometric viewfactors. Such an approach is computationally very expensive for complex geometry and/or inhomogeneous media. This article presents a new formulation in which the wall-emitted component is solved using the Discrete Ordinates Method (SN approximation), while the medium-emitted component is solved using the P1 approximation, resulting in a hybrid SN-PN RTE solver. Results show that the hybrid Discrete Ordinates-P1 method (DOM-P1) is computationally very efficient, but its accuracy is poor in optically thin situations where ray effects, inherent in the Discrete Ordinates Method, are pronounced. To circumvent this problem, the control-angle Discrete Ordinates Method (CADOM) is finally employed, and the accuracy of the hybrid CADOM-P1 method is found to be far superior to the hybrid DOM-P1 method.

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.

Solution of the Radiative Transfer Equation in Three-Dimensional Participating Media Using a Hybrid Discrete Ordinates: Spherical Harmonics Method

2012 ◽  
Vol 134 (11) ◽  
Author(s):  
Maathangi Sankar ◽  
Sandip Mazumder
Keyword(s):  
Monte Carlo ◽  
Radiative Transfer ◽  
Spherical Harmonics ◽  
Hybrid Method ◽  
Transfer Equation ◽  
Large Range ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Medium Component

In this article, a new hybrid solution to the radiative transfer equation (RTE) is proposed. Following the modified differential approximation (MDA), the radiation intensity is first split into two components: a “wall” component, and a “medium” component. Traditionally, the wall component is determined using a viewfactor-based surface-to-surface exchange formulation, while the medium component is determined by invoking the first-order spherical harmonics (P1) approximation. Recent studies have shown that although the MDA approach is accurate over a large range of optical thicknesses, it is prohibitive for complex three-dimensional geometry with obstructions, both from a computational efficiency as well as memory standpoint. The inefficiency stems from the use of the viewfactor-based approach for determination of the wall-emitted component. In this work, instead, the wall component is determined directly using the control angle discrete ordinates method (CADOM). The new hybrid method was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results for gray media in which the optical thickness was varied over a large range. In all cases, the accuracy of the hybrid method was found to be within a few percent of Monte Carlo results, and comparable to the solutions of the RTE obtained directly using CADOM. Finally, the new hybrid method was explored for 3D nongray media in the presence of reflecting walls and various scattering albedos. As a noteworthy advantage, irrespective of the conditions used, it was always found to be computationally more efficient than standalone CADOM and up to 15 times more efficient than standalone CADOM for optically thick media with strong scattering.

An Efficient Sparse Finite Element Solver for the Radiative Transfer Equation

2009 ◽  
Vol 132 (2) ◽  
Author(s):  
Gisela Widmer
Keyword(s):  
Finite Elements ◽  
Linear System ◽  
Radiative Transfer ◽  
Transfer Equation ◽  
Degrees Of Freedom ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Five Dimensions ◽  
Dimensional Domain

The stationary monochromatic radiative transfer equation is posed in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. For nonscattering radiative transfer, sparse finite elements [2007, “Sparse Finite Elements for Non-Scattering Radiative Transfer in Diffuse Regimes,” ICHMT Fifth International Symposium of Radiative Transfer, Bodrum, Turkey; 2008, “Sparse Adaptive Finite Elements for Radiative Transfer,” J. Comput. Phys., 227(12), pp. 6071–6105] have been shown to be an efficient discretization strategy if the intensity function is sufficiently smooth. Compared with 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 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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