Sparse Tensor Product Approximation for Radiative Transfer

2007 ◽  
Author(s):  
Gisela Widmer ◽  
Ralf Hiptmair
Keyword(s):  
Radiative Transfer ◽  
Tensor Product ◽  
Degrees Of Freedom ◽  
Convergence Result ◽  
Superior Performance ◽  
Discrete Ordinates ◽  
Strong Regularity ◽  
Scattering Media ◽  
Translation Invariant ◽  
Tensor Product Approximation

The stationary monochromatic radiative transfer equation is stated in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. In order to overcome the high dimensionality of the problem, we propose and analyse a new multiscale Galerkin Finite Element discretizaton that, under strong regularity assumptions on the solution, reduces the complexity of the problem to the number of degrees of freedom in space only (up to logarithmic terms). The sparse tensor product approximation adapts the idea of so-called ‘Sparse Grids’ for the product space of functions on the physical domain and the unit sphere. We present some details of the sparse tensor product construction including a convergence result that shows that, assuming strong regularity of the solution, the method converges with essentially optimal asymptotic rates while its complexity grows essentially only as that for a linear transport problem. Numerical experiments in a translation invariant setting in non-scattering media agree with predictions of theory and demonstrate the superior performance of the sparse tensor product method compared to the discrete ordinates 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.

Transient Radiative Transfer in Anisotropically Scattering Media Using Monotonicity-Preserving Schemes

2000 ◽  
Author(s):  
M. Sakami ◽  
K. Mitra ◽  
P.-F. Hsu
Keyword(s):  
Radiative Transfer ◽  
Research Work ◽  
Discrete Ordinates ◽  
Scattering Media ◽  
Discrete Ordinates Method ◽  
Piecewise Parabolic ◽  
Phase Functions ◽  
Monotonicity Preserving ◽  
Made In ◽  
Thick Medium

Abstract This research work deals with the analysis of transient radiative transfer in one-dimensional scattering medium. The time-dependant discrete ordinates method was used with an upwind monotonic scheme: the piecewise parabolic scheme. This scheme was chosen over a total variation diminishing version of the Lax-Wendroff scheme. These schemes were originally developed to solve Eulerian advection problem in hydrodynamics. The capability of these schemes to handle sharp discontinuity in a propagating electromagnetic wave front was compared. The accuracy and the efficiency of the discrete ordinates method associated with the piecewise parabolic advection scheme were studied. Comparisons with Monte Carlo and integral formulation methods show the accuracy and the efficiency of this proposed method. Parametric study for optically thin and thick medium, different albedos and phase functions is then made in the unsteady state zone.

Least-Squares Finite Element Analysis for Transient Radiative Transfer in Absorbing and Scattering Media

2005 ◽  
Vol 128 (5) ◽  
pp. 499-503 ◽  
Author(s):  
W. An ◽  
L. M. Ruan ◽  
H. P. Tan ◽  
H. Qi
Keyword(s):  
Finite Element ◽  
Radiative Transfer ◽  
Least Squares ◽  
Present Model ◽  
Element Model ◽  
Discrete Ordinates ◽  
Scattering Media ◽  
Element Analysis ◽  
Volume Method ◽  
Transfer Problems

In some radiative transfer processes, the time scales are usually on the order of 10−9-10−15s, so the transient effect of radiation should be considered. In present research, a finite element model, which is based on the discrete ordinates method and least-squares variational principle, is developed to simulate the transient radiative transfer in absorbing and scattering media. The numerical formulations and detailed steps are given. Moreover, two transient radiative transfer problems are investigated and the results are compared with those by integral method and finite volume method. It indicates that the present model can simulate the transient radiative transfer effectively and accurately.

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.

scholarly journals Sparse Discrete Ordinates Method in Radiative Transfer

2011 ◽  
Vol 11 (3) ◽  
pp. 305-326 ◽  
Author(s):  
Konstantin Grella ◽  
Christoph Schwab
Keyword(s):  
Radiative Transfer ◽  
Degrees Of Freedom ◽  
Convergence Rates ◽  
Cartesian Product ◽  
Solution Space ◽  
Physical Space ◽  
Discrete Ordinates ◽  
Discrete Ordinates Method ◽  
Combination Technique ◽  
Product Approach

AbstractThe stationary monochromatic radiative transfer equation (RTE) is a partial differential transport equation stated on a five-dimensional phase space, the Cartesian product of physical and angular domain. We solve the RTE with a Galerkin FEM in physical space and collocation in angle, corresponding to a discrete ordinates method (DOM). To reduce the complexity of the problem and to avoid the "curse of dimension", we adapt the sparse grid combination technique to the solution space of the RTE and show that we obtain a sparse DOM which uses essentially only as many degrees of freedom as required for a purely spatial transport problem. For smooth solutions, the convergence rates deteriorate only by a logarithmic factor. We compare the sparse DOM to the standard full DOM and a sparse tensor product approach developed earlier with Galerkin FEM in physical space and a spectral method in angle. Numerical experiments confirm our findings.

scholarly journals Variational Iteration Method for Infrared Radiative Transfer in a Scattering Medium

2017 ◽  
Vol 74 (2) ◽  
pp. 419-430 ◽  
Author(s):  
Feng Zhang ◽  
Yi-Ning Shi ◽  
Jiangnan Li ◽  
Kun Wu ◽  
Hironobu Iwabuchi
Keyword(s):  
Radiative Transfer ◽  
Iteration Method ◽  
Climate Models ◽  
Single Layer ◽  
Variational Iteration Method ◽  
Discrete Ordinates ◽  
Scattering Medium ◽  
Scattering Media ◽  
First Order ◽  
Variational Iteration

Abstract A new scheme is proposed for using the variational iteration method (VIM) to solve the problem of infrared radiative transfer in a scattering medium. This scheme allows the zeroth-order solution to be identified as the absorption approximation and the scattering effect is included in the first-order iteration. The upward and downward intensities are calculated separately in VIM, which simplifies the calculation process. By applying VIM to two single-layer scattering media and a full radiation algorithm with gaseous transmission, it is found that VIM is generally more accurate than the discrete-ordinates method (DOM), especially for cirrostratus. Computationally, VIM is slightly faster than DOM in the two-stream case but more than twice as fast in the four-stream case. In view of its high overall accuracy and computational efficiency, VIM is well suited to solving infrared radiative transfer in climate models.

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.

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