Radiative Transfer Problem Research Articles

Generalized lattice Boltzmann method for radiative transfer problem in slab and irregular graded-index media.

The lattice Boltzmann method (LBM) has been developed as a powerful solution method in computational fluid dynamics and heat transfer. However, the development of the LBM for solving radiative transfer problems has been far from perfect. This paper proposes a generalized form of the lattice Boltzmann model for the multidimensional radiative transfer equation (RTE) in irregular geometry with a graded index based on body-fitted coordinates. The macroscopic RTE is recovered from Chapman-Enskog analysis, which provides two possible procedures to formulate the Boltzmann equation in graded-index media and irregular geometries. These proposed models have been tested by considering one-and two-dimensional problems of the RTE, and the benchmark solutions reported in the literature were used for comparisons. Afterwards, the LBM is used to analyze the radiation transport in graded-index media for various forms of scattering law, refractive index, boundary reflection, laser and optical properties, and temperatures. The graded-index function and the geometry type have a significant effect on radiative transport in cases in which the refractive index matches or mismatches the boundary. It is also apparent that the developed LBM is an efficient, powerful, robust, and accurate solver for radiative transport in inhomogeneous media with a graded-index function and irregular geometries.

Global variance reduction method for Monte Carlo simulation of thermal radiation transport

The implicit Monte Carlo (IMC) method is an important numerical approximation method of simulating the thermal radiative transfer problems under high temperature condition. However, one problem plaguing the IMC method is that the calculation error distributions of the radiation specific intensities are highly asymmetric in space and time. By theoretical analysis and numerical simulations, we find that the error is affected by the records of track in the tallying mesh. Accordingly, a global variance reduction method for implicit Monte Carlo simulation is developed and the corresponding formulas are derived. This method includes three key techniques: 1) the automated dynamic distribution method for the Monte Carlo simulation source particles; 2) the dynamic weight-window technique and the none-bias weight revise algorithm that is suited to the particle distribution method; 3) the analytical estimation variance reduction method of the radiation specific intensity. In view of the above, a three-dimensional simulation code, named IMC3D, is developed to simulate the thermal radiative transfer phenomena. The typical thermal radiative transport problem, known as Marshak wave, is simulated. The simulation results indicate that the global variance reduction method for implicit Monte Carlo makes the statistical errors much more symmetric in space and time and the maximum of error is controllable, thereby increasing the calculation speed approximately 10 times. The new IMC method and code are used for simulating the radiative transportation in hohlraum of ICF successfully.

Reduced order models for thermal radiative transfer problems based on moment equations and data-driven approximations of the Eddington tensor

A new group of structure and asymptotic preserving reduced-order models (ROMs) for multidimensional nonlinear thermal radiative transfer (TRT) problems is presented. They are formulated by means of the nonlinear projective approach and data compression techniques. The nonlinear projection is applied to the Boltzmann transport equation (BTE) to derive a hierarchy of low-order moment equations. Approximation of the Eddington tensor that provides exact closure for the system of moment equations is found with projection-based data-driven methodologies. These include the (i) proper orthogonal decomposition (POD), (ii) dynamic mode decomposition (DMD) and (iii) a variant of the DMD. A parameterization is derived for this ROM for the temperature of radiation incoming to the problem domain (the radiation drive temperature). This parameterization is informed from results of a dimensionless study of the TRT problem. Analysis of the ROMs is performed on the classical Fleck-Cummings TRT multigroup test problem in 2D geometry with a radiation-driven Marshak wave. Numerical results are presented to demonstrate the performance of these ROMs for the simulation of evolving radiation and heat waves. Results show these models to be sufficiently accurate for practical computations with rather low-rank representations of the Eddington tensor. As the rank of the approximation is increased, the errors of solutions generated by the ROMs gradually decreases.

Radiative transfer as a Bayesian linear regression problem

ABSTRACT Electromagnetic radiation plays a crucial role in various physical and chemical processes. Hence, almost all astrophysical simulations require some form of radiative transfer model. Despite many innovations in radiative transfer algorithms and their implementation, realistic radiative transfer models remain very computationally expensive, such that one often has to resort to approximate descriptions. The complexity of these models makes it difficult to assess the validity of any approximation and to quantify uncertainties on the model results. This impedes scientific rigour, in particular, when comparing models to observations, or when using their results as input for other models. We present a probabilistic numerical approach to address these issues by treating radiative transfer as a Bayesian linear regression problem. This allows us to model uncertainties on the input and output of the model with the variances of the associated probability distributions. Furthermore, this approach naturally allows us to create reduced-order radiative transfer models with a quantifiable accuracy. These are approximate solutions to exact radiative transfer models, in contrast to the exact solutions to approximate models that are often used. As a first demonstration, we derive a probabilistic version of the method of characteristics, a commonly-used technique to solve radiative transfer problems.

Method and new tabulations for flux-weighted line opacity and radiation line force in supersonic media

Context. In accelerating and supersonic media, understanding the interaction of photons with spectral lines can be of utmost importance, especially in an accelerating flow. However, fully accounting for such line forces is computationally expensive and challenging, as it involves complicated solutions of the radiative transfer problem for millions of contributing lines. This currently can only be done by specialised codes in 1D steady-state flows. More general cases and higher dimensions require alternative approaches. Aims. We present a comprehensive and fast method for computing the radiation line force using tables of spectral-line-strength distribution parameters, which can be applied in arbitrary (multi-D, time-dependent) simulations, including those that account for the line-deshadowing instability, to compute the appropriate opacities. Methods. We assume local thermodynamic equilibrium to compute a flux-weighted line opacity from ~4 million spectral lines. We fit the opacity computed from the line list with an analytic result derived for an assumed distribution of the spectral line strength and found the corresponding line-distribution parameters, which we tabulate here for a range of assumed input densities ρ ∈ [10−20, 10−10] g cm−3 and temperatures T ∊ [104, 1047] K. Results. We find that the variation in the line-distribution parameters plays an essential role in setting the wind dynamics in our models. In our benchmark study, we also find a good overall agreement between the O-star mass-loss rates of our models and those derived from steady-state studies that use a more detailed radiative transfer. Conclusions. Our models reinforce the idea that self-consistent variation in the line-distribution parameters is important for the dynamics of line-driven flows. Within a well-calibrated O-star regime, our results support the proposed methodology. In practice, utilising the provided tables, yielded a factor >100 speed-up in computational time compared to specialised 1D model-atmosphere codes of line-driven winds, which constitutes an important step towards efficient multi-dimensional simulations. We conclude that our method and tables are ready to be exploited in various radiation-hydrodynamic simulations where the line force is important.

Monte Carlo radiative transfer with explicit absorption to simulate absorption, scattering, and stimulated emission

Context. The Monte Carlo method is probably the most widely used approach to solve the radiative transfer problem, especially in a general 3D geometry. The physical processes of emission, absorption, and scattering are easily incorporated in the Monte Carlo framework. Net stimulated emission, or absorption with a negative cross section, does not fit this method, however.Aims. We explore alterations to the standard photon packet life cycle in Monte Carlo radiative transfer that allow the treatment of net stimulated emission without loss of generality or efficiency.Methods. We present the explicit absorption technique that allows net stimulated emission to be handled efficiently. It uses the scattering rather than the extinction optical depth along a photon packet’s path to randomly select the next interaction location, and offers a separate, deterministic treatment of absorption. We implemented the technique in a special-purpose Monte Carlo code for a two-stream 1D radiative transfer problem and in the fully featured 3D code SKIRT, and we studied its overall performance using quantitative statistical tests.Results. Our special-purpose code is capable of recovering the analytical solutions to the two-stream problem in all regimes, including the one of strong net stimulated emission. The implementation in SKIRT is straightforward, as the explicit absorption technique easily combines with the variance reduction and acceleration techniques already incorporated. In general, explicit absorption tends to improve the efficiency of the Monte Carlo routine in the regime of net absorption.Conclusions. Explicit absorption allows the treatment of net stimulated emission in Monte Carlo radiative transfer, it interfaces smoothly with other variance reduction and acceleration techniques, and it tends to improve the efficiency of the simulations in the net absorption regime. We recommend to always include this new technique in Monte Carlo radiative transfer.

The polarization signals of the solar K I D lines and their magnetic sensitivity

Aims. This work aims to identify the relevant physical processes in shaping the intensity and polarization patterns of the solar K I D lines through spectral syntheses, placing particular emphasis on the D2 line. Methods. The theoretical Stokes profiles were obtained by numerically solving the radiative transfer problem for polarized radiation considering one-dimensional semi-empirical models of the solar atmosphere. The calculations account for scattering polarization, partial frequency redistribution (PRD) effects, hyperfine structure (HFS), J- and F-state interference, multiple isotopes, and magnetic fields of arbitrary strength and orientation. Results. The intensity and circular polarization profiles of both D lines can be suitably modeled while neglecting both J-state interference and HFS. The magnetograph formula can be applied to both lines, without including HFS, to estimate weak longitudinal magnetic fields in the lower chromosphere. By contrast, modeling the scattering polarization signal of the D lines requires the inclusion of HFS. The amplitude of the D2 scattering polarization signal is strongly depolarized by HFS, but it remains measurable. A small yet appreciable error is incurred in the scattering polarization profile if PRD effects are not taken into account. Collisions during scattering processes have a clear depolarizing effect, although a quantitative analysis is left for a forthcoming publication. Finally, the D2 scattering polarization signal is particularly sensitive to magnetic fields with strengths around 10 G and it strongly depends on their orientation. Despite this, its center-to-limb variation relative to the amplitude at the limb is largely insensitive to the field strength and orientation. Conclusions. These findings highlight the value of the K I D2 line polarization for diagnostics of the solar magnetism, and show that the linear and circular polarization signals of this line are primarily sensitive to magnetic fields in the lower chromosphere and upper photosphere, respectively.

Heat Transfer Study in Cylindrical Cavity with Heat Absorption or Generation

Lattice Boltzmann mesoscopic (LBM) is applied to solve energy equation of a transient conduction radiation heat transfer problem in a two-dimensional cylindrical participating (absorbing, emitting and scattering) medium in the presence of heat generation/absorption coefficient. Control volume finite element method (CVFEM) formulation is used to obtain the radiative information. To study the effectiveness of the LBM-CVFEM combination on unsteady conduction-radiation problems in cylindrical media, the energy equation of the problem is also solved using the finite difference method (FDM) in which the CVFEM is used to compute radiative information. The effects of heat generation/absorption coefficient on temperature distributions in the medium are studied. Results of the present work are benchmarked against those available in the literature. The hybrid numerical model’s results are also compared with those obtained by the FDM-CVFEM combination. All the results presented in this work show that the present method is accurate and valuable for the analysis of cylindrically axisymmetric radiative heat transfer problems .

Quantitative Evaluation of the Phase Function Effects on Light Scattering and Radiative Transfer in Dispersed Systems

The light scattering properties of particles play important roles in radiative transfer in many dispersed systems, such as turbid atmosphere, ocean water, nanofluids, composite coatings and so on. As one of the scattering property parameters, the scattering phase functions of particles are strongly dependent on the particle size, size distribution, and morphology, as well as on the complex refractive indices of the particles and surrounding media. For the sake of simplicity, the empirical phase function models are widely used in many practical applications. In this work, we focus on the radiative transfer problem in dispersed systems composed of spherical particles, and give quantitative analyses of the impact of scattering phase functions on the radiative transfer process. We fit the scattering phase functions of four different types of practical dispersed systems using four previously proposed empirical phase function models, including the Henyey–Greenstein (HG) model, Cornette Shanks (CS) model, Reynold and McCormick (RM) model and two-term Reynolds–McCormick (TTRM) model. By comparing the radiative transfer characteristics (i.e., hemispherical reflectance, hemispherical transmittance and total absorptance) of dispersed layers calculated using the Monte Carlo method, the relative errors caused by using the empirical phase functions are systematically investigated. The results demonstrate that the HG, CS and RM models cause obvious errors in the calculation of hemispherical reflectance in many cases. Meanwhile, the induced errors show no obvious regularity, but are related to the particle size and layer optical thickness. Due to the good fitting effect in both forward and backward directions, the TTRM model provides significantly higher performances in fitting the phase functions of all considered cases than the widely used single-term parametrizations. Moreover, for different particle sizes and layer optical thicknesses, the induced errors of the TTRM model in radiative transfer characteristics are very small, especially for the case of polydisperse particles. Our results can be used to guide the design, analysis and optimization of dispersed systems in practical optics and photonics applications.

On the efficiency of using correlative randomized algorithms for solving problems of gamma radiation transfer in stochastic medium

Abstract To solve problems of radiation balance, optical sounding, and tomography, it may be necessary to take into account multiple scattering of radiation in a stochastically inhomogeneous medium. In real radiation models, for this purpose, the numerical-statistical ‘majorant cross-section method’ (MCM, delta-Woodcock tracking) is used based on the alignment of the optical density field by adding an artificial ‘delta scattering’ event. However, the computation cost of the corresponding unbiased estimate of the averaged problem solution infinitely increases as the correlation scale (correlation radius L) of standard mosaic models for a random medium density decreases. Previously, we constructed the MCM randomization providing asymptotically (for L → 0) unbiased estimates of the required functionals, in which the value of the physical attenuation coefficient is randomly chosen at the end of the particle free path l under condition l > L. Otherwise the value of the physical attenuation coefficient is the same as at the starting point of the particle (CR algorithm). In a more accurate functional correlative randomized algorithm (FCR algorithm), the coefficient remains the same with a probability determined by the correlation function. These correlative randomized algorithms were implemented for a mixture of homogeneous substance (water) and a Poisson ensemble of ‘empty’ balls. In the present paper, we construct correlative randomized algorithms for problems related to transfer through a ‘thick’ layer containing a water and a Poisson ensemble of ‘empty’ layers. A detailed comparative analysis of the results obtained by exact direct simulation (MCM) and approximate algorithms (CR, FCR) for the problems of gamma radiation transfer through a ‘thick’ water layer containing a Poisson ensemble of ‘empty’ layers or balls is presented.

Construction and optimization of numerically-statistical projection algorithms for solving integral equations

Abstract The problem of minimizing the root-mean-square error of the numerical-statistical projection estimation of the solution to an integral equation is solved. It is shown that the optimal estimator in this sense can be obtained by equalizing deterministic and stochastic components of the error in the case when the norm of the remainder of the utilized decomposition decreases inversely proportional to its length. As a test, the Milne problem of radiation transfer in a semi-infinite layer of matter is solved using Laguerre polynomials. To solve such a problem in the case of a finite layer, a special regularized projection algorithm is used.

Numerical solutions to linear transfer problems of polarized radiation

Context. The polarization signals produced by the scattering of anistropic radiation in strong resonance lines encode important information about the elusive magnetic fields in the outer layers of the solar atmosphere. An accurate modeling of these signals is a very challenging problem from the computational point of view, in particular when partial frequency redistribution (PRD) effects in scattering processes are accounted for with a general angle-dependent treatment. Aims. We aim at solving the radiative transfer problem for polarized radiation in nonlocal thermodynamic equilibrium conditions, taking angle-dependent PRD effects into account. The problem is formulated for a two-level atomic model in the presence of arbitrary magnetic and bulk velocity fields. The polarization produced by scattering processes and the Zeeman effect is considered. Methods. The proposed solution strategy is based on an algebraic formulation of the problem and relies on a convenient physical assumption, which allows its linearization. We applied a nested matrix-free GMRES iterative method. Effective preconditioning is obtained in a multifidelity framework by considering the light-weight description of scattering processes in the limit of complete frequency redistribution (CRD). Results. Numerical experiments for a one-dimensional (1D) atmospheric model show near optimal strong and weak scaling of the proposed CRD-preconditioned GMRES method, which converges in few iterations, independently of the discretization parameters. A suitable parallelization strategy and high-performance computing tools lead to competitive run times, providing accurate solutions in a few minutes. Conclusions. The proposed solution strategy allows the fast systematic modeling of the scattering polarization signals of strong resonance lines, taking angle-dependent PRD effects into account together with the impact of arbitrary magnetic and bulk velocity fields. Almost optimal strong and weak scaling results suggest that this strategy is applicable to realistic 3D models. Moreover, the proposed strategy is general, and applications to more complex atomic models are possible.

Thermal emission in the successive orders of scattering (SOS) radiative transfer approach

The Successive Orders of Scattering (SOS) approach [1] is one of the well known methods for solving the Radiative Transfer (RT) problem. Its efficiency in terms of speed and accuracy of computation was already demonstrated for scattering and absorbing atmospheres in Solar spectrum. Although there are no principle limitations to account for the emission processes, the application of the SOS method for atmospheres with thermal emission is not widely used yet. In this paper we present a SOS-based RT approach accounting for the full source function, which enables its application from the UV (UltraViolet) to the TIR (Thermal InfraRed) parts of the electromagnetic spectrum. The atmospheric vertical discretization in this extended SOS scheme is a key point in order to properly retain the scattering and emission processes. An analysis of different methodologies to perform this vertical discretization is presented. The numerical implementation has been included in GRASP (Generalized retrieval of Atmosphere and Surface Properties) RT code [2]. In comparison with the widely used code DISORT (DIScrete-ORdinatemethod for Radiative Transfer) [3], the developed SOS scheme achieves a mean accuracy of radiance calculation of −0.005 K (−0.003%) expressed in terms of brightness temperature. Under the same configuration and vertically inhomogeneous atmospheric conditions, GRASP SOS RT is approximately eight times faster than DISORT. The analysis of the sensitivity of GRASP TIR SOS scheme to the number of layers and the effect of polarization are also investigated in the paper.

Optical Tomography in Variable Index Media Using the Transient Discrete Transfer Method

This paper presents the optical tomography based on the reconstruction of optical properties in one- and two-dimensional variable index media. A transient discrete transfer method is used to solve the forward problem of transient radiative transfer. A multistart conjugate gradient method is applied to solve the inverse problem, and a straightforward approach is used to obtain the sensitivity coefficients. Different numerical examples are presented to show the efficiency of this method. The results show that the proposed algorithm can effectively estimate the distribution of absorption and scattering coefficients, even with noisy measured data.

On the application of the ADO method to the solution of two-dimensional radiative transfer problems in anisotropic scattering media

In this work, the Analytical Discrete Ordinates (ADO) method is applied to the solution of the discrete ordinates approximation of the two-dimensional radiative transfer equation in Cartesian geometry. The phase function is expanded in terms of Legendre polynomials up to arbitrary order L to model higher-order anisotropy. The discrete ordinates equations are transversally integrated over regions of the domain, yielding one-dimensional equations for average angular intensities in x and y-directions. The ADO method is then applied to the one-dimensional equations, with approximations for the unknown intensities on the contours of the regions, to derive explicit solutions for the spatial variables. This approach, based on the use of the ADO method along with the nodal technique, is usually referred to as ADO-Nodal. The scheme does not use sweeping. Linear systems have to be solved to fully establish the general solutions. Here, using a recent rearrangement of those linear systems, numerical results are provided for more refined domain partitions, into regions, than previously. Higher-order quadrature schemes are also explored. Comparisons with few results available in the literature reinforce the good performance of the method. A detailed analysis is carried out to contribute in the direction of obtaining reference solutions for the class of problems investigated. Furthermore, numerical results for different optical thicknesses and section average intensities are included, that according to our knowledge it has not been presented before in the literature.

Modeling the signatures of interaction in Type II supernovae: UV emission, high-velocity features, broad-boxy profiles

Because mass loss is a fundamental phenomenon in massive stars, an interaction with circumstellar material (CSM) should be universal in core-collapse supernovae (SNe). Leaving aside the extreme CSM density, extent, or mass typically encountered in Type IIn SNe, we investigate the diverse long-term radiative signatures of an interaction between a Type II SN ejecta and CSM corresponding to mass-loss rates up to 10−3 M⊙ yr−1. Because these CSM are relatively tenuous and optically thin to electron scattering beyond a few stellar radii, radiation hydrodynamics is not essential and one may treat the interaction directly as an additional power source in the non-local thermodynamic equilibrium radiative transfer problem. The CSM accumulated since shock breakout forms a dense shell in the outer ejecta and leads to high-velocity absorption features in spectral lines, even for a negligible shock power. In addition to Balmer lines, such features may appear in Na I D and He I lines, among others. A stronger interaction strengthens the continuum flux (preferentially in the UV), quenches the absorption of P-Cygni profiles, boosts the Mg IIλλ 2795, 2802 doublet, and fosters the production of a broad-boxy Hα emission component. The rise in ionization in the outer ejecta may quench some lines (e.g., the Ca II near-infrared triplet). The interaction power emerges preferentially in the UV, in particular at later times, shifting the optical color to the blue, but increasing the optical luminosity modestly. Strong thermalization and clumping seem to be required to make an interaction superluminous in the optical. The UV range contains essential signatures that provide critical constraints to infer the mass-loss history and inner workings of core-collapse SN progenitors at death.

An equation-solving method based on radiation distribution factor for radiative transfer in participating media with diffuse boundaries

An efficient and accurate method to solve the directional radiation intensity in the participating medium has always been a pursuit in radiative transfer fields, especially in radiation detection and inverse radiative transfer problems. As one of the few algorithms with both high accuracy and strong applicability, the reverse Monte Carlo (RMC) method suffers from the disadvantages of unavoidable random statistical errors and time-consuming ray tracing. Recently, a so-called equation-solving integral equation method based on the radiation distribution factor (ES-RDFIEM) can effectively overcome these shortcomings of RMC, which has comparable accuracy as RMC while taking only a few hundredths or thousandths of the calculation time of RMC. In this paper, ES-RDFIEM was extended to a radiation system with diffuse surfaces by constructing the radiative transfer equation (RTE) about the radiation distribution factor (RDF) of the wall and internal medium, respectively. The mathematical principle and formula were introduced in detail, and the computational performance was examined by several samples with different radiative parameters. The results showed that the RDF and radiation intensity solved by ES-RDFIEM are in good agreement with RMC, and ES-RDFIEM is more stable and reliable. More importantly, the computation time of ES-RDFIEM is significantly shorter than that of RMC, and it is not affected by wall emissivity, internal medium optical thickness and scattering albedo.

Conservation Laws in Time-Dependent Radiative Transfer Problems

Conservation Laws in Time-Dependent Radiative Transfer Problems

Analysis of Corner Balance Nonlinear Diffusion Acceleration and Moment-Based, Spatially Linear Transport Discretizations

The recently developed High-Order, Low-Order scheme for the solution of thermal radiative transfer problems is applied as an acceleration method to the neutral particle transport equation. The resulting Corner Balance Nonlinear Diffusion Acceleration (CB-NDA) is derived, and a stability analysis is performed in conjunction with moment-based, spatially linear discretizations. These spatial discretizations correspond to the lumped Linear Discontinuous (LD) and Linear Characteristic (LC) schemes, which possess the thick diffusion limit. The lumped LD and LC schemes satisfy corner balance equations, which in turn are used to derive the CB-NDA. Two variants of the CB-NDA include the net current and partial current formulations. Numerical results are presented that verify the theoretical predictions and implementation. Theoretical spectral radius from the analysis is verified by comparison to values from the numerical solution of a one-dimensional transport problem. Results indicate similar stability between the CB-NDA–accelerated lumped LD and LC schemes. The net current–based CB-NDA is found to be unstable whereas the partial current formulation remains stable over the range of scattering ratios and optical thicknesses.

Numerical results for radiative heat transfer in finite cylindrical medium with isotropic scattering

In recent years, the interest in radiative heat transfer in cylindrical media has increased primarily because of its several applications related to combustion chambers, furnaces, and high temperature heat exchangers. Although there are many papers dealing with the radiative transfer in such geometry, there is lack of high precision benchmarks. Such scarcity is related to the fact that some researches present inaccurate results and most published works are limited to presenting results only in graphics. The shortage is more evident in the case of multidimensional problems, like those in which incident radiation and radiative heat fluxes vary in the axial direction. These problems can be approached in many ways; in the integral formulation, the angular variables are completely eliminated by integration over the solid angle, which allows the elimination of the ray effect and the consequent obtainment of more accurate numerical results. A strategy that can be used simultaneously with the integral formulation is the use of the singularity-subtraction technique with the Nyström method, which provides high quality numerical results with relatively low computational times if combined with mathematical refinements. Following this methodology, in the present research, radiative heat transfer problems in finite cylindrical media with isotropic scattering are solved. High precision numerical results for incident radiation and radiative heat fluxes in the radial and axial directions are provided and validated by comparing with results available in the literature.