Nonlinear Radiative Transfer Research Articles

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. The ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

Quantification of Shortwave Surface Albedo Feedback Using a Neural Network Approach

Radiative transfer is a nonlinear process. Despite this, most current methods to evaluate radiative feedback, such as the kernel method, rely on linear assumptions. Neural network (NN) models can emulate nonlinear radiative transfer due to their structure and activation functions. This study aims to test whether NNs can be used to evaluate shortwave radiative feedbacks and to assess their performance. This study focuses on the shortwave radiative feedback driven by surface albedo. An NN model is first trained using idealized cases, simulating truth values from a radiative transfer model via the partial radiative perturbation method. Two heuristic cases are analyzed: univariate feedback, perturbing the albedo; and bivariate feedback, perturbing the albedo and cloud cover concurrently. These test the NN’s ability to capture nonlinearity in the albedo–flux and albedo–cloud–flux relationships. We identify the minimal NN structure and predictor variables for accurate predictions. Then, an NN model is trained with realistic radiation flux and atmospheric variable data and is tested with respect to its predictions at different order levels: zero-order for the flux itself, first-order for radiative sensitivity (kernels), and second-order for kernel differences. This paper documents the test results and explains the NN’s ability to reproduce the complex nonlinear relationship between radiation flux and different atmospheric variables, such as surface albedo, cloud optical depth, and their coupling effects.

An iterative method for solving one class of non-linear integral equations with Nemytskii operator on the positive semi-axis

A class of non-linear integral equations with monotone Nemytskii operator on the positive semi-axis is considered. This class of integral equations appears in many areas of modern natural science. In particular, such equations, with various restrictions on non-linearity and the kernel, arise in the dynamic theory of $p$-adic strings for the scalar field of tachyons, in the kinetic theory of gases and plasma within the framework of the usual and modified non-linear Bhatnagar-Gross-Crook models for the Boltzmann kinetic equation. Equations of similar nature appear also in the non-linear radiative transfer theory in inhomogeneous media and in the mathematical theory of the spread of epidemic diseases within the framework of the modified Diekmann-Kaper model. A constructive theorem for the existence of a bounded positive and continuous solution is proved. As a result, we get a uniform estimate of the difference between the previous and subsequent iterations, the corresponding successive approximations converge uniformly to a bounded continuous solution to the equation. The asymptotic behaviour of the constructed solution at infinity is studied. In particular, it is proved that the limit of this solution at infinity exists and is positive, and is uniquely determined from a certain characteristic equation. It is also proved that the difference between the limit and the solution is a summable function on the positive semi-axis. Using some geometric estimates for convex and concave functions, and employing the theorem on integral asymptotics obtained here, we prove the uniqueness of the solution in a certain subclass of non-negative non-trivial continuous and bounded functions. The results obtained are also applied to the study of a special class of non-linear Urysohn type integral equations on the positive semi-axis. In particular, the existence of a positive bounded solution to this class of equations is verified, and some qualitative properties of the constructed solution are studied. We also give specific applied examples of the corresponding kernels and non-linearities to illustrate the importance of the results obtained.

Unified gas-kinetic particle method for frequency-dependent radiation transport

This paper proposes a unified gas-kinetic particle (UGKP) method for the frequency-dependent photon transport process. The photon transport is a typical multiscale process governed by the nonlinear radiative transfer equations (RTE). The flow regime of photon transport varies from the ballistic regime to the diffusive regime with respect to optical depth and photon frequency. The UGKP method is an asymptotic preserving (AP) scheme and a regime adaptive scheme. The teleportation error is significantly reduced, and the computational efficiency is remarkably improved in the diffusive regime. Distinguished from the standard multigroup treatment, the proposed UGKP method solves the frequency space in a non-discretized way. Therefore the Rosseland diffusion system can be precisely preserved in the optically thick regime. Taking advantage of the local integral solution of RTE, the distribution of the emitted photon can be constructed from its macroscopic moments. The Monte Carlo particles in the UGKP method need only be tracked before their first collision events, and a re-sampling process is performed to close the photon distribution for each time step. The high computational cost of calculating excessive scattering events can be saved, especially in the optically thick regime. The proposed UGKP method is implicit and removes the light speed constraint on the time step. The particle tracking approach combining the implicit formulation makes the proposed UGKP method an efficient solution algorithm for frequency-dependent radiative transfer problems. We demonstrate with numerical examples the capability of the proposed multi-frequency UGKP method.

An efficient asymptotic preserving Monte Carlo method for radiative transfer equations

In this work, an efficient asymptotic preserving Monte Carlo method is developed for nonlinear thermal radiative transfer equations. We derive a new approximate macroscopic equation for the radiation energy density, from an integral solution of the radiation intensity along characteristics of the microscopic equation. We will solve a coupled macro-micro system with a hybrid numerical method. The macroscopic radiation and material energy densities are updated first by using a deterministic finite volume method. The microscopic equation, with the emission source provided from the macroscopic variables, can then be efficiently solved using a particle-based Monte Carlo method. This new approach is uniformly stable, with large time steps independent of scaling parameters and the speed of light. Besides, it can be shown that the method is asymptotic preserving in the diffusive limit for an optically thick regime. Numerical experiments are performed to demonstrate the high efficiency and good performances of our proposed method.

An Asymptotic-Preserving Hybrid Angular Discretization for the Gray Radiative Transfer Equations

The aim of this paper is to construct a new numerical scheme for the nonlinear gray radiative transfer (GRT) equations, namely, the asymptotic-preserving (AP) H N T -based unified gas kinetic scheme (UGKS). The constructed scheme is obtained by combing the UGKS for spatial discretization with the hybrid H N T method for angular discretization. Since the H N T is a hybrid angular discrete method of both P N and S N methods, the current H N T -based UGKS can not only mitigate the ray effects of the S N method largely, but also suppress the oscillations of the original P N method. Furthermore, we show that the current H N T -based UGKS also inherits the AP property of UGKS. A number of one-dimensional and two-dimensional numerical experiments are presented that validate the performance of the current scheme in both optically thin and thick regimes, as well as in mitigating the ray effects. Moreover, it can capture the initial layer solution without requiring additional treatments.

X-ray radiative transfer in full 3D with SKIRT

Context. Models of active galactic nuclei (AGN) suggest that their circumnuclear media are complex with clumps and filaments, while recent observations hint towards polar extended structures of gas and dust, as opposed to the classical torus paradigm. The X-ray band could form an interesting observational window to study these circumnuclear media in great detail. Aims. We want to extend the radiative transfer code SKIRT with the X-ray processes that govern the broad-band X-ray spectra of obscured AGN, to study the structure of AGN circumnuclear media in full 3D, based on their reflected X-ray emission. Methods. We extended the SKIRT code with Compton scattering on free electrons, photo-absorption and fluorescence by cold atomic gas, scattering on bound electrons, and extinction by dust. This includes a novel treatment of extreme-forward scattering by dust, and a detailed description of anomalous Rayleigh scattering. To verify our X-ray implementation, we performed the first dedicated benchmark of X-ray torus models, comparing five X-ray radiative transfer codes. Results. The resulting radiative transfer code covers the X-ray to millimetre wavelength range self-consistently, has all the features of the established SKIRT framework, is publicly available, and is fully optimised to operate in arbitrary 3D geometries. In the X-ray regime, we find an excellent agreement with the simulation results of the MYTORUS and REFLEX codes, which validates our X-ray implementation. We find some discrepancies with other codes, which illustrates the complexity of X-ray radiative transfer and motivates the need for a robust framework that can handle non-linear 3D radiative transfer effects. We illustrate the 3D nature of the code by producing synthetic X-ray images and spectra of clumpy torus models. Conclusions. SKIRT forms a powerful new tool to model circumnuclear media in full 3D, and make predictions for the X-ray band in addition to the dust-dominated infrared-to-UV wavelength range. The new X-ray functionalities of the SKIRT code allow for uncomplicated access to a broad suite of 3D X-ray models for AGN that can easily be tested and modified. This will be particularly useful with the advent of X-ray microcalorimeter observations in the near future.

Reflectance spectra of mascagnite and salammoniac minerals with varying viewing geometry

We analyse the quantitative effects of viewing geometry variations on the near-infrared reflectance spectra of mascagnite-(NH4)2SO4 and salammoniac-NH4Cl samples. Bi-directional reflectance spectra are collected in the 1–4.2 μm range considering a set of 3 incidence (i) angles (i = 0°; 30°; 60°) and 9 emergence (e) angles between −70° and 70° at room temperature and computed with respect to the normal direction. The NH4+ overtone and combinations bands located at ∼1.09, 1.32, 1.62, 2.04, 2.2 and 3.05 μm are experimentally investigated. The bidirectional reflectance spectra of these samples show significant variations with the observation geometry in terms of reflectance values, spectral slope, and absorption bands parameters. The band positions remain essentially unchanged by varying the incident and emergence angles. On the other hand, bands' area and depth show the highest variability for i ≥ 30° and e greater than ±40°(up to a factor 2.3 in relative mean variation). The area and depth parameters of these bands show a dual behaviour: (i) for the weak-medium spectral features below 2 μm the area and depth decrease as the phase angle increases. (ii) The strong spectral features above 2 μm increase their values only at phase angles above 90°, but also at low phase angles for high incidences, i ≥ 30°. This behaviour is linked both to the non-linear radiative transfer in particulate media and to the way the band depth and area are defined, relative to the local continuum. We observe important dependence (up to ∼60% relative mean variation) of band depth and area on the incidence angle, up to 60°, compared to moderate variation with emergence angles (up to ∼20% relative mean variation). Furthermore, the ∼3 μm features becomes more saturated at ±70° emergence angles. A general trend of spectral bluing with change in observation geometry is observed. The current dataset is a contribution in the framework of present and future space missions focused on understanding the nature and quantification of ammonium-bearing minerals on icy bodies. The NH4+ − bearing minerals identification could provide information on: (i) ocean/brine compositions, (ii) possible explanations of geological phenomena and (iii) implications for biological activity.

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.

Manifold Learning and Nonlinear Homogenization

We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress local solution manifolds. Our framework is applied to a semilinear elliptic equation with oscillatory media and a nonlinear radiative transfer equation; in both cases, significant improvements in efficacy are observed. This new method does not rely on detailed analytical understanding of the multiscale PDEs, such as their asymptotic limits, and thus is more versatile for general multiscale problems.

An Asymptotic-Preserving IMEX Method for Nonlinear Radiative Transfer Equation

We present an asymptotic preserving method for the radiative transfer equations in the framework of \(P_N\) method. An implicit and explicit numerical scheme is proposed to solve the \(P_N\) system based on the order analysis of the expansion coefficients of the specific intensity, where the order of each expansion coefficient is derived by the Chapman-Enskog method. The coefficients at higher-order are treated explicitly while those at lower-order are treated implicitly in each equation of the \(P_N\) system. Energy inequality is proved for this numerical scheme. Several numerical examples validate the efficiency of this scheme in both optically thick and thin regions.

Bayesian Instability of Optical Imaging: Ill Conditioning of Inverse Linear and Nonlinear Radiative Transfer Equation in the Fluid Regime

For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio-)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well- to ill-posed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback–Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is ill-posed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime.

Axisymmetric radiative titanium dioxide magnetic nanofluid flow on a stretching cylinder with homogeneous/heterogeneous reactions in Darcy-Forchheimer porous media: Intelligent nanocoating simulation

Modern nanomaterials coating processes feature high temperature environments and complex chemical reactions required for the precise synthesis of bespoke designs. Such flow processes are extremely complex and feature both heat and mass transfer in addition to viscous behaviour. Intelligent nano-coatings exploit magnetic nanoparticles and can be manipulated by external magnetic fields. Mathematical models provide an inexpensive insight into the inherent characteristics of such coating dynamics processes. Motivated by this, in the current work, a novel mathematical model is developed for dual catalytic reactive species diffusion in axisymmetric coating enrobing forced convection boundary layer flow from a linearly axially stretching horizontal cylinder immersed in a homogenous non-Darcy porous medium saturated withmagnetic nanofluid. Homogeneous and heterogeneous reactions, heat source (e.g. laser source) and non-linear radiative transfer are included. The Tiwari-Das nanoscale model is deployed. A Darcy-Forchheimerdrag force formulation is utilized to simulate both bulk porous drag and second order inertial drag of the porous medium fibres. The magnetic nanofluid is an aqueous electroconductive polymer comprising base fluid water and magnetic TiO2 nanoparticles. The TiO2 nanoparticles are one chemically reacting species (A) and a second species (B) is also present (e.g. oxygen) which also reacts chemically. Viscous heating and Ohmic dissipation are also included to produce a more physically realistic thermal analysis. The non-linear conservation equations proposed here with species diffusion (species A and B) are transformed via an appropriate stream function and scaling variables into a set of non-linear united multi-degree ODEs. The rising nonlinear ordinary differential boundary value problem is solved with four-point Gauss-Lobotto formulae in the MATLAB bvp5c routine. Validation is conducted with an Adams-Moulton predictor–corrector numerical scheme (AM2 coded in Unix). The widespread visualization of velocity, temperature, species A concentration, species B concentration, skin friction, local Nusselt number and species A and B local Sherwood numbers is included. For the higher Schmidt number the momentum diffusion rate exceeds the species diffusion rate and this will produce a depression in concentration values. Further, increasing Darcian parameter retards the local Nusselt number magnitudes since higher permeability corresponds to sparsity in the solid fibres, reduction in thermal conduction and a concomitant cooling of the cylinder surface.

A positive and asymptotic preserving filtered PN method for the gray radiative transfer equations

This paper presents a positive and asymptotic preserving scheme for the nonlinear gray radiative transfer equations. The scheme is constructed by combining the filtered spherical harmonics (FPN) method for the discretization of angular variable and with the framework of the unified gas kinetic scheme (UGKS) for the spatial- and time-discretization. The constructed scheme is almost free of ray effects and can also mitigate oscillations in the spherical harmonics (PN) approximation. Moreover, it can be shown that the current scheme is asymptotic preserving. Consequently, in the optically thick regimes the current scheme can exactly capture the solution of the diffusion limit equation without requiring the cell size being smaller than the photon's mean free path, while the solution in optically thin regimes can also be well resolved in a natural way. In addition, the FPN angular discretization induces a natural macro-micro decomposition, with this help we can obtain the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Then, a linear scaling limiter is given to enforce that sufficient conditions. With the process of such construction, we finally obtain a scheme, called the PPFPN-based UGKS scheme, that is positive and asymptotic preserving. Various numerical experiments are included to validate the robustness, positive- and asymptotic-preserving property as well as the property of almost ray effect free.

Accurate Front Capturing Asymptotic Preserving Scheme for Nonlinear Gray Radiative Transfer Equation

We develop an asymptotic preserving scheme for the gray radiative transfer equation. Two asymptotic regimes are considered: one is a diffusive regime described by a nonlinear diffusion equation for...

Reconstruction of the Emission Coefficient in the Nonlinear Radiative Transfer Equation

In this paper, we investigate an inverse problem for the radiative transfer equation that is coupled with a heat equation in a nonscattering medium in $\mathbb{R}^n, n\geq 2$. The two equations are...

An asymptotic preserving unified gas kinetic particle method for radiative transfer equations

In this work, we propose a unified gas kinetic particle (UGKP) method for solving the nonlinear thermal radiative transfer equations. The UGKP method is a multiscale method in that the macro and microscopic variables are coupled and updated in a consistent way. It employs a finite volume formulation for the macroscopic variable evolution, and a particle-based Monte Carlo solver for tracking the non-equilibrium transport. The stiff coupling between the radiation and material energy is included and resolved efficiently by the coupled macroscopic equations. Compared with the implicit Monte Carlo (IMC) method, it does not employ the effective scattering events. Moreover, we demonstrate that the UGKP method has the asymptotic preserving property in capturing the diffusion limit in optically thick regions. Numerical simulations show that our method is comparable in computational cost to the IMC method for optically thin problems, and becomes much more efficient for optically thick problems. More importantly, the UGKP method doesn't require the cell size being less than the photon's mean free path and it achieves the highest efficiency in the optically thin and thick regions self-adaptively.

Stability analysis of a multilevel quasidiffusion method for thermal radiative transfer problems

In this paper we analyze a multilevel quasidiffusion (QD) method for solving time-dependent multigroup nonlinear radiative transfer problems which describe interaction of photons with matter. The multilevel method is formulated by means of the high-order radiative transfer equation and a set of low-order moment equations. The fully implicit scheme is used to discretize equations in time. The stability analysis is applied to the method in semi-continuous and discretized forms. To perform Fourier analysis, the system of equations of the multilevel method is linearized about an equilibrium solution. The effects of discretization with respect to different independent variables are studied. The multilevel method is shown to be stable and fast converging. We also consider a version of the method in which time evolution in the radiative transfer equation is treated by means of the α-approximation. The Fleck–Cummings test problem is used to demonstrate performance of the multilevel QD method and study its iterative stability.

Analytical solution of the simple nonlinear radiative transfer problem

Exploring the "Principle of invariance" and the method of "Linear images", the simple nonlinear conservative problem of radiative transfer is analyzed. The solutions of nonlinear reection-transmission and internal field problems of one dimensional scattering-absorbing medium of finite optical thickness are obtained, whereas both boundaries of medium illuminated by powerful radiation beams. Using two different approaches - a direct and inverse problem, the analytical solution of the internal field problem is derived.

Nonlinear radiative transfer in high temperature plasmas

An approximation to the nonlinear radiative transfer equation is considered in the context of magnetohydrodynamics. This approximation retains nonlinear terms which are responsible for the Compton plasma heating in addition to the radiation cooling. The effect is studied numerically. In particular, the Kelvin–Helmholtz instability and the Dai-Woodward case are modeled. It is shown that radiation-plasma coupling results in damping of small scale instabilities and alters the shock wave structure.