Solution Of Radiative Transfer Problems Research Articles

Quasi-Random Discrete Ordinates Method to Radiative Transfer Equation with Linear Anisotropic Scattering

The modeling of energy transport via radiative transfer is important to many practical high temperature engineering applications. Furthermore, the computation of solutions to the radiative transfer equation (RTE) plays a fundamental part in it. The quasi-random discrete ordinates method was developed as an alternative to mitigate the ray effect found in the classical discrete ordinates method solutions. The former method was originally developed for transport problems with isotropic scattering and it is here extended and tested to problems with linear anisotropic scattering. Its main idea is to approximate the integral term of the RTE by a quasi-Monte Carlo integration. The discrete system of differential equations arising from it can be solved by a variety of classical discretization methods, here computed with a SUPG finite element scheme. The novel developments are tested for selected manufactured solutions. The achieved good results indicate the potential of the novel method to be applied to the solution of radiative transfer problems with anisotropic scattering.

Multiparameter Spectrophotometry Platform for Turbid Sample Measurement by Robust Solutions of Radiative Transfer Problems

Multiparameter spectrophotometer instruments based on the radiative transfer (RT) theory differ fundamentally from conventional ones in their ability to determine absorption and scattering coefficients and anisotropy factor. Ease of use and robustness of inverse solutions are of necessity for general-purpose applications. We have developed a new instrument platform design incorporating a compact three-photodiode detection configuration without integrating spheres, four-channel signal-processing unit, and Monte Carlo (MC) simulation-based software for accurate and rapid inverse solutions. To verify the platform design, two sphere suspension samples of different size distributions were measured and RT parameters were determined by a sample model with the Henyey and Greenstein (HG) phase function between 460 and 1000 nm. System validation is achieved by comparison of the RT parameters and spectral dependence against the results by Mie theory for single-sized spheres of nominal diameter values. Five wavelengths from 560 to 960 nm were selected for quantitative evaluation of the robustness of the HG sample model and two Mie theory-based sample models assuming light scattering by independent and single spheres. We found that only the HG-based sample model yields high accuracy and strong robustness for inverse determination of RT parameters even for the cases of sphere suspensions in which the assumptions of sample uniformity and scattering by independent and single spheres are not satisfied. These results demonstrate that the new platform with MC simulation-based inverse algorithm combined with the HG sample model is capable of robust and rapid measurement of RT parameters for turbid samples with optical thickness up to 30.

Line by line benchmark solutions for radiative heat transfer in 2D irregular enclosures with non-gray media

Thermal radiation heat transfer in two-dimensional complex (irregular) enclosures containing non-gray gases, namely CO2/H2O/N2 mixtures, is calculated. The meshless Natural Element Method (NEM) is used for spatial discretization and the Discrete Ordinates Method (DOM) is employed for angular discretization of the radiative transfer equation. The Line-by-Line (LBL) and the correlated-k model (CK) models are used to model the spectral radiative properties of the participating gases. To obtain the most accurate results, the updated HITEMP spectroscopic database i.e. HITEMP2010 is employed in LBL calculations and the updated parameters (based on the HITEMP2010) is used in the CK calculation. The objectives of this study are threefold: (1) to combine a meshless natural elements method with LBL and the CK spectral models for the solution of radiative transfer problems in irregular (complex) enclosures, (2) to evaluate the accuracy of the CK model by comparing its results against the LBL results, and (3) to provide benchmark solutions for radiative heat transfer in irregular enclosures with the HITEMP2010 as the most accurate spectral database at high temperatures. Calculations are carried out for isothermal/non-isothermal and homogeneous/inhomogeneous mixtures of CO2/H2O/N2 gases. The present LBL results can serve as benchmark results for validating other approximate models in gas radiation heat transfer calculations in 2D complex enclosures. Results show that the CK results are in good agreement with those of the LBL model.

Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method

The contribution of this paper relies in the development of numerical algorithms for the mathematical treatment of specular reflection on borders when dealing with the numerical solution of radiative transfer problems. The radiative transfer equation being integro-differential, the discrete ordinates method allows to write down a set of semi-discrete equations in which weights are to be calculated. The calculation of these weights is well known to be based on either a quadrature or on angular discretization, making the use of such method straightforward for the state equation. Also, the diffuse contribution of reflection on borders is usually well taken into account. However, the calculation of accurate partition ratio coefficients is much more tricky for the specular condition applied on arbitrary geometrical borders. This paper presents algorithms that calculate analytically partition ratio coefficients needed in numerical treatments. The developed algorithms, combined with a decentered finite element scheme, are validated with the help of comparisons with analytical solutions before being applied on complex geometries.

A 3D radiative transfer framework

Context. Time-dependent, 3D radiation transfer calculations are important for the modeling of a variety of objects, from supernovae and novae to simulations of stellar variability and activity. Furthermore, time-dependent calculations can be used to obtain a 3D radiative equilibrium model structure via relaxation in time. Aims. We extend our 3D radiative transfer framework to include direct time dependence of the radiation field; i.e., the $\partial I/ \partial t$ terms are fully considered in the solution of radiative transfer problems. Methods. We build on the framework that we have described in previous papers in this series and develop a subvoxel method for the $\partial I/\partial t$ terms. Results. We test the implementation by comparing the 3D results to our well tested 1D time dependent radiative transfer code in spherical symmetry. A simple 3D test model is also presented. Conclusions. The 3D time dependent radiative transfer method is now included in our 3D RT framework and in PHOENIX/3D.

Modified finite-volume method based on a cell vertex scheme for the solution of radiative transfer problems in complex 3D geometries

A modified finite-volume method based on a cell vertex scheme was applied to solve radiative transfer problems within a participating medium of complex three-dimensional shaped domain. The computational spatial domain of interest was divided into four-node tetrahedron elements with unstructured meshes while the adopted formulation was combined with a closure relation based on an exponential scheme. The studied medium was assumed to be grey, non-scattering and was bounded by black surfaces. Our results were then compared with those found in other articles on the subject. The approach shows a very good level of performance for wall heat transfer evaluation. Accurate results were obtained on coarse computational meshes and solution errors were found to decrease with grid refinement.

Group-theoretical description of radiative transfer in one-dimensional media

Group theory is used to describe a procedure for adding inhomogeneous absorbing and scattering atmospheres in a one-dimensional approximation. The inhomogeneity originates in the variation of the scattering coefficient with depth. Group representations are derived for the composition of media in three different cases: inhomogeneous atmospheres in which the scattering coefficient varies continuously with depth, composite or multicomponent atmospheres, and the special case of homogeneous atmospheres. We extend an earlier proposal to solve problems in radiative transfer theory by first finding global characteristics of a medium (reflection and transmission coefficients) and then determining the internal radiation field for an entire family of media without solving any new equations. Semi-infinite atmospheres are examined separately. For some special depth dependences of the scattering coefficients it is possible to obtain simple analytic solutions expressed in terms of elementary functions. An algorithm for numerical solution of radiative transfer problems in inhomogeneous atmospheres is described.

Spectral Element Method with Adaptive Artificial Diffusion for Solving the Radiative Transfer Equation

An adaptive isotropically artificial diffusion (AISO) scheme is developed for the spectral element method to mitigate the ray effects encountered in the solution of radiative transfer problems. By considering the coupled manner of ray effects and false scattering, the artificial diffusion coefficient is determined heuristically from both local angular discretization scale and local spatial discretization scale. The scheme is easily and efficiently implemented under the spectral or finite-element method framework. The performance of various artificial diffusion schemes is studied and compared. The streamwise artificial diffusion schemes are only responsible for stabilization of strong convection-induced instability, while the isotropically artificial diffusion scheme shows very good performance in mitigating the “wiggles” in both low- and high-order spatial approximation. Numerical experiments show that the spectral element method with the AISO scheme is stable, high-order-accurate, and effective for solving radiative transfer in simple and complex geometries, and also robust for mitigating ray effects of various origins.

A New Kinetic Equation for Compton Scattering

A kinetic equation for Compton scattering is given that differs from the Kompaneets equation in several significant ways. By using an inverse differential operator, this equation allows treatment of problems for which the radiation field varies rapidly on the scale of the width of the Compton kernel. This inverse operator method describes, among other effects, the thermal Doppler broadening of spectral lines and continuum edges and automatically incorporates the process of Compton heating/cooling. It is well adapted for inclusion into a numerical iterative solution of radiative transfer problems. The equivalent kernel of the new method is shown to be a positive function and with reasonable accuracy near the initial frequency, unlike the Kompaneets kernel, which is singular and not wholly positive. It is shown that iterations of the inverse operator kernel can be easily calculated numerically, and a simple summation formula over these iterations is derived that can be efficiently used to compute Comptonized spectra. It is shown that the new method can be used for initial-value and other problems with no more numerical effort than the Kompaneets equation and that it more correctly describes the solution over times comparable to the mean scattering time.

Application of adaptive multi-frequency grids to three-dimensional astrophysical radiative transfer

We present a method to solve the three-dimensional (3D) radiative transfer equation for astrophysical applications using adaptive photon transport grids. Contrary to earlier treatments, they are calculated for each frequency separately. Generated minimizing the first-order discretization error in the scattered radiation intensity, they provide global error control for solutions of radiative transfer problems on the grid. We discuss minimization of the grid point number in regions where the optical depth becomes large and show that the method allows for treating applications with optical depth of any value using the concept of penetration depth. The proposed grid generation algorithm is easy to implement, allows pre-calculation of the grids and storage in integer arrays, making a fast solution of the 3D radiative transfer equation possible. The grid generation algorithm is suitable for optimization in cases where simple radiation source distributions are given. Besides discussing application to simple density distribution commonly occurring in astrophysical objects, we illustrate the capabilities of the method by generating grids for an accretion disk around a young star.

Numerical Solution of Radiative Transfer Problems in a Slab Medium by Galerkin-type Approximation Techniques

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using a Galerkin-type method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.

Stochastic radiative transfer in finite plane for Rayleigh scattering phase function

Formulation and solution of radiative transfer problem in stochastic finite plane medium composed of two immiscible mixed materials is considered. The mixing statistics is described as two-state homogeneous Markov process. Pomraning–Eddington technique is used to obtain an explicit solution of the problem in deterministic case. A weight function, related to Chandrasekhar function, with its special forms is used to get this solution. A developed formalism, to treat radiation transfer in stochastic medium, is used to get the ensemble-averaged solution of the problem under consideration. Explicit analytical expressions and numerical results for averaged reflectivity, transmitivity and scalar flux of the medium in case of Rayleigh scattering of radiation are given.

Modified Collapsed Dimension Method for Radiative Heat Transfer Problems

Introduction C OLLAPSED dimensionmethod (CDM) is one of the ray tracing methods used for the solution of radiative transfer problems. Development of this method is partly based on the works of Shih and Chen1 and Shih and Ren2 on the discretized intensity method. Some light on the method was thrown by Blank. Subsequent developmentof the method was made by Blank andMishra.4 Detailed description of this method is available in Ref. 5. In CDM, three-dimensionalradiative information is mapped into a two-dimensionalsolutionplane in terms of effective intensity (EI) andoptical thicknesscoefŽ cient (OTC). Thus, unlikeothermethods, analysis and computations in this method are performed in a twodimensional solution plane instead of three-dimensionalspace. CDMhas beenused for the solutionof radiativetransferproblems with highaccuracy.4;5 Thismethodhasbeenfoundtowork for awide rangeof optical thickness (very low to high optical thickness). Furthermore, CDM has also been found to work well for the conjugate mode heat transfer problems. In CDM, for determinationof heat  ux and temperature information, at each point of interest EIs have to be integrated over planar angle in the solutionplane.Theseangularintegrationsareperformed by dividing the planar angle into intervals of equal sizes. In each subinterval, EI is assumed isotropic. Hence, for higher accuracy, CDM requiresmore EIs and, thus, higher computational time. In the presentwork,with the objectiveofmakingCDMmore economical, the method is modiŽ ed. In the modiŽ ed CDM (MCDM), angular integrations of the EIs are performed differently. The discretizationand integrationsperformed in thismethod originatefrom the concept of the discrete ordinate method (DOM). Here, the planar angle is divided into a Ž nite number of subintervals according to the number of Gaussian quadraturepoints considered.Hence, in MCDM, angular subintervals are unequal and instead of considering average values of interval, the weighted mean corresponding to the Gauss points is considered. With this, the method becomes more realistic, and it derives computational efŽ ciency with a more realistic representationof the radiative transfer process. In the present work, improvements in MCDM over CDM are tested by solving radiative transfer problems in oneand twodimensional Cartesian enclosures with gray and homogeneous, absorbing, emitting, and anisotropically scattering medium. Both radiative and nonradiative equilibrium situations are considered. MCDM and CDM results are compared with results from the exact method and DOM.

Tools for atmospheric radiative transfer: Streamer and FluxNet

Two tools for the solution of radiative transfer problems are presented. Streamer is a flexible medium spectral resolution radiative transfer model based on the plane-parallel theory of radiative transfer. Capable of computing either fluxes or radiances, it is suitable for studying radiative processes at the surface or within the atmosphere and for the development of remote-sensing algorithms. FluxNet is a fast neural network-based implementation of Streamer for computing surface fluxes. It allows for a sophisticated treatment of radiative processes in the analysis of large data sets and potential integration into geophysical models where computational efficiency is an issue. Documentation and tools for the development of alternative versions of FluxNet are available. Collectively, Streamer and FluxNet solve a wide variety of problems related to radiative transfer: Streamer provides the detail and sophistication needed to perform basic research on most aspects of complex radiative processes, whereas the efficiency and simplicity of FluxNet make it ideal for operational use.

Radiative heat transfer in participating media — A review

This paper presents an overview of various exact analytic and approximate numerical methods for the solution of radiative heat transfer problems in participating media. Review of each method is followed by its strengths and limitations. Importance of radiative heat transfer analysis and difficulties in the solution of radiative transfer problems have been emphasized.

Parallel radiative transport using a unified matrix approach

This paper describes the use of parallel computational methods applied to the solution of radiative transfer problems involving participating media. A unified matrix approach, involving a range of operations (ray tracing, numerical integration, and solution of large, dense linear systems), is employed. Several issues associated with massively parallel architectures which affect performance are examined and applied to problems in two and three dimensions.

Auxiliary functions for radiative transfer problems in plane-parallel geometry—I. The infinite medium

The auxiliary functions occurring in the solution of radiative transfer problems in plane-parallel, homogeneous media with isotropic light scattering are investigated. In an infinite medium, they can be deduced from the classical resolvent function φ ∞ and its generalization η ∞ to non-vanishing values of a generalized angular variable. The analytical properties of the η ∞-function are studied and exact expressions are derived. An integral term is involved in these expressions, which is the only difficulty raised when proceeding to their numerical evaluation. Ten-digit values of the φ ∞- and η ∞-functions are finally tabulated.

A Newton-Raphson approach for nonlinear diffusion equations in radiation hydrodynamics

Numerical solutions of radiative transfer problems via a flux-limited diffusion equation coupled with the material energy balance equation are in general characterized by a standard explicit treatment of all the nonlinearities. An increase in accuracy has been recently obtained by analyzing the rate of convergence of a fully implicit treatment. In this paper a complete Newton-Raphson linearization algorithm at each time step is investigated. Numerical results are presented for radiation penetration into a cold slab driven by a constant incident surface intensity.

Reflection operators and contemporary applications to radiative transfer

The paper presents some results related to the theory of reflection operators and contemporary numerical algorithms for solution of radiative transfer problems. They include estimates of reflection operator norms, construction of domain decomposition methods, and estimation of their convergence rate.

Convergence properties of the accelerated Λ-iteration method for the solution of radiative transfer problems

We investigate the convergence properties and the computational speed of the accelerated Λ-iteration method for the solution of a wide variety of radiative transfer problems. The formal solution of the transfer equation is done using the short characteristics method. As the approximate Λ-operator we use bands of the discretized Λ-operator and derive the formulae for computing operators with arbitrary bandwidth. We define the optimum bandwidth of the approximate Λ-operator as the bandwidth for which the CPU time used for the solution of any particular radiative transfer problems is minimum. The radiative transfer equation is solved for a number of continuum and line transfer problems in spherical symmetry and for static to very rapidly expanding media. We find the optimum bandwidth is around 5–15 for workstation class computers whereas it is around 1–2 for supercomputers.