Numerische Lösungstechniken der Strahlungstransportgleichung mittels verallgemeinerter mittlerer Intensitäten (original) (raw)
Numerical Methods for Multidimensional Radiative Transfer
Reactive Flows, Diffusion and Transport, 2007
This paper presents a continuous finite element method for solving the resonance line transfer problem in moving media. The algorithm is capable of dealing with three spatial dimensions, using hierarchically structured grids which are locally refined by means of duality-based a-posteriori error estimates. Application of the method to coherent isotropic scattering and complete redistribution gives a result of matrix structure which is discussed in the paper. The solution is obtained by way of an iterative procedure, which solves a succession of quasi-monochromatic radiative transfer problems. It is therefore immediately evident that any simulation of the extended frequency-dependent model requires a solution strategy for the elementary monochromatic transfer problem, which is fast as well as accurate. The present implementation is applicable to arbitrary model configurations with optical depths up to 10 3 -10 4 . Additionally, a combination of a discontinuous finite element method with a superior preconditioning method is presented, which is designed to overcome the extremely poor convergence properties of the linear solver for optically thick and highly scattering media. The contents of this article is as follows: * This work has been supported by the German Research Foundation (DFG) through SFB 359 (Project C2).
Journal of The Atmospheric Sciences - J ATMOS SCI, 2006
The multiresolution radiative transfer equations of Part I of this paper are solved numerically for the case of inhomogeneous model clouds using Meyer's basis functions. After analyzing the properties of Meyer's connection coefficients and effective coupling operators (ECOs) for two examples of extinction functions, the present approach is validated by comparisons with Spherical Harmonic Discrete Ordinate Method (SHDOM) and Monte Carlo codes, and a preliminary analysis of the local-scale coupling between the cloud inhomogeneities and the radiance fields is presented. It is demonstrated that the contribution of subpixel-scale cloud inhomogeneities to pixel-scale radiation fields may be very important and that it varies considerably as a function of local cloud inhomogeneities.
On View-Factor Approach for Radiation Transfer Equation
45th AIAA Plasmadynamics and Lasers Conference, 2014
View-factor approach is applied to calculate the radiation heating rates in some important engineering applications. The transfer of radiative energy is considered in the axisymmetric enclosure with the spherical temperature inhomogeneity and for the re-entry of the space vehicle. The numerical results by the view factor approach is then verified against the ray tracing method, diffusion and tangent slab approximations. Spectral radiative transfer on the surface of space vehicle during the orbital re-entry is resolved by means of the view-factor approach. An asymptotic accuracy of radiative flux density in two-dimensional axisymmetric geometry is demonstrated. Semi-analytical expression for the radiation flux density in two-dimensional geometry is derived. General recommendations are formulated for the application in the optically thin and thick media and for the application in three-dimensional geometry. Nomenclature u, v Convective and diffusion velocity J Spectral intensity of radiation J em Spectral emissivity J b Spectral intensity of the black body − → s Vector of spatial coordinates T Attenuation factor Ω Solid angle κ ν Absorption coefficient τ Optical thickness q ν Spectral flux A Surface of element V Volume of element F, E Incomplete elliptic integral of first and second kind Subscript * Graduate Student, Mechanical Engineering Department, student member AIAA. Current address:
Calculation of Jacobians for inverse radiative transfer: An efficient hybrid method
Journal of Quantitative Spectroscopy & Radiative Transfer, 2006
We present an accurate and numerically efficient procedure of calculating Jacobians by finite difference that consists of two components: (1) a method employing the saving of atmospheric layers that accelerates the solution to the equation of radiative transfer for solvers that use the Discrete Space formulation and (2) a method of perturbing the eigenmatrix spectrum associated with a reduced attenuation matrix. The procedure eliminates the need to call the eigenmatrix package, here, LAPACK a second time and provides insights into the fundamental properties of the attenuation matrix, useful for characterizing the accuracy of the derivatives calculated by finite difference methods. The computational complexity of the perturbation method is 8n 3 þ 22n 2 , where n is one half the number of streams in the radiance field as opposed to 16n 3 using LAPACK. The method is not limited to the calculation of base state radiances IðoÞ and those associated with an 'infinitesimal' perturbation Iðo þ doÞ (from which the numerical derivative of Iðo þ doÞ with respect to do may be approximated), but is also useful in the calculation of radiances associated with a 'finite' perturbation Iðo þ DoÞ from which a sensitivity can be calculated. r
Pakal: A Three-Dimensional Model to Solve the Radiative Transfer Equation
The Astrophysical Journal Supplement Series, 2010
We present a new numerical model called "PAKAL" intended to solve the radiative transfer equation in a three dimensional (3D) geometry, using the approximation for a locally plane parallel atmosphere. Pakal uses pre-calculated radial profiles of density and temperature (based on hydrostatic, hydrodynamic or MHD models) to compute the emission from 3D source structures with high spacial resolution. Then, Pakal solves the radiative transfer equation in a set of (3D) ray-paths, going from the source to the observer. Pakal uses a new algorithm to compute the radiative transfer equation by using an Intelligent System consisting of three structures: a cellular automaton; an expert system; and a program coordinator. The code outputs can be either two dimensional maps or one dimensional profiles, which reproduce the observations with high accuracy, giving in this way, detailed physical information about the environment where the radiation was generated and/or transmitted. We present the model applied to a 3D solar radial geometry, assuming a locally plane-parallel atmosphere, and thermal free-free radio emission from a Hydrogen-Helium gas in thermodynamic equilibrium. We also present the convergences test of the code. We computed the synthetic spectrum of the centimetric-millimetric solar emission and found better agreement with observations (up to 10 4 K at 20 GHz) than previous models reported in literature. The stability and convergence test show the high accuracy of the code. Finally, Pakal can improve the integration time by up to an order of magnitude compared against linear integration codes.
A Fourier–Riccati Approach to Radiative Transfer. Part I: Foundations
Journal of the Atmospheric Sciences, 1993
The three-dimensional equation of radiative transfer is formally solved using a Fourier-Riccati approach while calculations are performed on cloudy media embedded in a two-dimensional space. An extension to Stephens' work, this study addresses the coupling between space and angle asserted by the equation of transfer. In particular, the accuracy of the computed radiation field as it is influenced by the angular resolution of the phase function and spatial discretization of the cloudy medium is discussed. The necessity of using a large number of quadrature points to calculate fluxes even when the phase function is isotropic for media exhibiting vertical and horizontal inhomogeneities is demonstrated. Effects of incorrect spatial sampling on both radiance and flux fields are also quantified by example. Radiance and flux comparisons obtained by the Fourier-Riccati model and the independent pixel approximation for inhomogeneous cloudy media illustrate the inadequacy of the latter even for tenuous clouds.
Generalized method for evaluating scattering parameters used in radiative transfer models
Journal of The Optical Society of America A-optics Image Science and Vision, 1997
The effective scattering and absorption coefficients used to describe the optical properties of particulate materials in radiative transfer models are determined by the average path-length parameter of the diffuse radiation, as well as by the fraction of energy that each particle scatters into the forward and backward hemispheres relative to the direction of the impinging radiation. Until now, there were no well-established methods to calculate these parameters. We have devised an approach for evaluating average path-length parameters and forward-scattering ratios for both forward and backward diffuse radiation intensities. Single-scattering processes are described by Lorenz-Mie theory, and multiple-scattering effects have been taken into account by a generalization of Hartel theory. As a consequence of the formalism, the Kubelka-Munk scattering and absorption coefficients are explicitly related to average path-length parameters and forward-scattering ratios. These parameters display an optical depth dependence, characterized by values smoothly increasing or decreasing from the perpendicularly illuminated interface and saturation values at large optical depths.
Limitations of approximate Fourier techniques in solving radiative-transfer problems
Journal of the Optical Society of America, 1979
Limitations on the use of Fourier analysis to solve the radiative-transfer equation in the smallangle-scattering approximation are discussed. The low-order position and angular spatial frequency components of the transformed scattered radiance are shown to be associated with diffusive multiply scattered radiation, and the high-order components with multiply forward scattered radiation. In addition, qualitative and quantitative comparisons of the theory with numerical and experimental results are given.