Steady-state Radiative Transfer Equation Research Articles

Integrating angular and domain decomposition with space-angle discontinuous Galerkin methods in 2D radiative transfer

A space-angle discontinuous Galerkin (saDG) method is used to solve the steady-state radiative transfer equation (RTE) for 2D problems involving absorption, emission, and scattering for a semitransparent medium. This approach discretizes both spatial and angular domains. Parallel computing is based on angular decomposition (AD), and domain decomposition (DD) techniques. The DD technique directly solves the entire domain using the MUMPS library, whereas the AD technique results in an iterative approach for scattering media. This study proposes a novel hybrid AD-DD method, combining the best aspects of both techniques. Numerical results investigate the scalability, performance, and efficiency of AD and DD techniques. It is shown that a hybrid AD-DD technique is superior to these individual techniques by taking advantage of their strengths. Numerical methods demonstrate the applicability of the method of the best combination of hybrid AD-DD to 2D scattering gray media with complex geometries or enclosures with circular and square obstacles.

Numerical Study of Near-Infrared Light Propagation in Aqueous Alumina Suspensions Using the Steady-State Radiative Transfer Equation and Dependent Scattering Theory

Understanding light propagation in liquid phantoms, such as colloidal suspensions, involves fundamental research of near-infrared optical imaging and spectroscopy for biological tissues. Our objective is to numerically investigate light propagation in the alumina colloidal suspensions with the mean alumina particle diameter of 55 nm at the volume fraction range 1–20%. We calculated the light scattering properties using the dependent scattering theory (DST) on a length scale comparable to the optical wavelength. We calculated the steady-state radiative transfer and photon diffusion equations (RTE and PDE) using the DST results based on the finite difference method in a length scale of the mean free path. The DST calculations showed that the scattering and reduced scattering coefficients become more prominent at a higher volume fraction. The anisotropy factor is almost zero at all the volume fractions, meaning the scattering is isotropic. The comparative study of the RTE with the PDE showed that the diffusion approximation holds at the internal region with all the volume fractions and the boundary region with the volume fraction higher than 1%. Our findings suggest the usefulness of the PDE as a light propagation model for the alumina suspensions rather than the RTE, which provides accurate but complicated computation.

Fast algorithms for integral formulations of steady-state radiative transfer equation

We investigate integral formulations and fast algorithms for the steady-state radiative transfer equation with isotropic and anisotropic scattering. When the scattering term is a smooth convolution on the unit sphere, a model reduction step in the angular domain using the Fourier transformation in 2D and the spherical harmonic transformation in 3D significantly reduces the number of degrees of freedoms. The resulting Fourier coefficients or spherical harmonic coefficients satisfy a Fredholm integral equation of the second kind. We study the uniqueness of the equation and proved an a priori estimate. For a homogeneous medium, the integral equation can be solved efficiently using the FFT and iterative methods. For an inhomogeneous medium, the recursive skeletonization factorization method is applied instead. Numerical simulations demonstrate the efficiency of the proposed algorithms in both homogeneous and inhomogeneous cases and for both transport and diffusion regimes.

Light propagation in biological tissue

We developed a new parallel computational method for solving fluorescence and elastically scattered light propagation through a biological tissue illuminated by a collimated incident beam. The three-dimensional steady state radiative transfer equation was solved using a Modified Finite Volume Method with a cell-vertex formulation. An Exponential spatial differencing scheme was used to efficiently lessen the false scattering. Several test problems are presented to assess the performance and accuracy of the numerical Method. We show that it is possible to obtain a very good level of accuracy. Relative differences less than 1.5% were obtained in comparison with the Monte Carlo reference solution for the selected problems. This study shows the potential of the proposed computational method to be used as an accurate deterministic forward solver in Optical Tomography.

High performance computation of radiative transfer equation using the finite element method

This article deals with an efficient strategy for numerically simulating radiative transfer phenomena using distributed computing. The finite element method alongside the discrete ordinate method is used for spatio-angular discretization of the monochromatic steady-state radiative transfer equation in an anisotropically scattering media. Two very different methods of parallelization, angular and spatial decomposition methods, are presented. To do so, the finite element method is used in a vectorial way. A detailed comparison of scalability, performance, and efficiency on thousands of processors is established for two- and three-dimensional heterogeneous test cases. Timings show that both algorithms scale well when using proper preconditioners. It is also observed that our angular decomposition scheme outperforms our domain decomposition method. Overall, we perform numerical simulations at scales that were previously unattainable by standard radiative transfer equation solvers.

Solution of the 2-D steady-state radiative transfer equation in participating media with specular reflections using SUPG and DG finite elements

The 2D radiative transfer equation coupled with specular reflection boundary conditions is solved using finite element schemes. Both Discontinuous Galerkin and Streamline-Upwind Petrov–Galerkin variational formulations are fully developed. These two schemes are validated step-by-step for all involved operators (transport, scattering, reflection) using analytical formulations. Numerical comparisons of the two schemes, in terms of convergence rate, reveal that the quadratic SUPG scheme proves efficient for solving such problems. This comparison constitutes the main issue of the paper. Moreover, the solution process is accelerated using block SOR-type iterative methods, for which the determination of the optimal parameter is found in a very cheap way.

On the Use of a Direct Radiative Transfer Equation Solver for Path Loss Calculation in Underwater Optical Wireless Channels

In this letter, we propose a fast numerical solution for the steady state radiative transfer equation in order to calculate the optical path loss of light propagation suffering from attenuation due to the absorption and scattering in various water types. We apply an optimal non-uniform method to discretize the angular space and an upwind type finite-difference method to discretize the spatial space. A Gauss-Seidel iterative method is then applied to solve the fully discretized system of linear equations. Finally, we extend the resulting radiance in two-dimensional to three-dimensional by the azimuthal symmetric assumption to compute the received optical power under the given receiver aperture and field of view. The accuracy and efficiency of the proposed scheme are validated by uniform RTE solver and Monte Carlo simulations.

Analytical solution of the radiative transfer equation for infinite-space fluence

This Brief Report presents the derivation of analytical expressions for the fluence of the steady state radiative transfer equation in an infinitely extended and anisotropically scattering medium in arbitrary dimensions for different source types. The fluence, which is composed of an infinite sum of diffusion-like Green's functions, was compared to the Monte Carlo method. Within the stochastic nature of the Monte Carlo simulations, an exact agreement was found in the steady state and time domains. It is shown that the use of low-order approximations is sufficient for many relevant cases.

A new attempt for radiation hydrodynamics simulation with anisotropic and non-equilibrium distribution

We present a computational method for solving time-dependent radiation moment equations with an Eddington tensor which represents an anisotropic field of radiation. For estimating the Eddington tensor, we propose a conical-ray method covering the whole solid angle with which the steady-state radiative transfer equation is solved. Computed radiation flux shows an excellent agreement with an analytical solution using the proposed conical-ray method even if a light source is located at a distant place from the computed point.

Radiative heat transfer in enhanced hydrogen outgassing of glass

This paper explores the physical mechanisms responsible for experimental observations that led to the definition of photo-induced hydrogen outgassing of glass . Doped borosilicate glass samples were placed inside an evacuated silica tube and heated in a furnace or by an incandescent lamp. It was observed that hydrogen release from the glass sample was faster and stronger when heated by an incandescent lamp than within a furnace. Here, sample and silica tube were modeled as plane-parallel slabs exposed to furnace or to lamp thermal radiation. Combined conduction, radiation, and mass transfer were accounted for by solving the one-dimensional transient mass and energy conservation equations along with the steady-state radiative transfer equation. All properties were found in the literature. The experimental observations can be qualitatively explained based on conventional thermally activated gas diffusion and by carefully accounting for the participation of the silica tube to radiation transfer along with the spectral properties of the silica tube and the glass samples. In brief, the radiation emitted by the incandescent lamp is concentrated between 0.5 and 3.0 μm and reaches directly the sample since the silica tube is nearly transparent for wavelengths up to 3.5 μm. On the contrary, for furnace heating at 400°C, the silica tube absorbs a large fraction of the incident radiation which reduces the heating rate and the H2 release rate. However, between 0.8 and 3.2 μm undoped borosilicate does not absorb significantly. Coincidentally, Fe3O4 doping increases the absorption coefficient and also reacts with H2 to form ferrous ions which increase the absorption coefficient of the sample by two orders of magnitude. Thus, doped and reacted samples heat up much faster when exposed to the heating lamp resulting in the observed faster response time and larger H2 release rate.

Measurement of the spectral absorption coefficient in the ocean with an isotropic source

Closed-form equations that describe the vector irradiance from an isotropic source embedded in the oceanare rigorously derived from the steady-state radiative transfer equation. The equations are exact for a homogeneous medium and are believed to be an excellent approximation along the vertical axis for a plane-parallel ocean. The equations are solved for the absorption coefficient as a function of distance from the source. For clear ocean water, it is shown that vector irradiance measurements alone provide sufficient information for an accurate calculation of the absorption coefficient. Measurements in Pacificwater of the vector irradiance from an isotropic source are presented, and the absorption coefficient is computed. The estimated value of the absorption coefficient from a linear least-squares fit to the data has a standard error of ~ 1%.

Radiation transport through a plasma boundary layer between armatures and material surfaces

The ablation of rail and insulator materials by plasma armatures limits the velocity and the component lifetime of electromagnetic launchers. The photon radiation from the armature is the dominant form of energy transfer causing the ablation. However, not all of the escaping radiation actually is transmitted to the ablating surface, and hence vapor shielding is said to occur in the plasma boundary layer. A one-dimensional time-dependent self-consistent hydrodynamics model with a multigroup flux-limited diffusion approximation for radiative transfer is used to estimate the energy transmission and the amount of ablated material. It has been found that the vapor shield is optically thin for photon energies where most of the incident source radiation is distributed, thus the simple diffusion model for radiative transfer would overestimate the radiation flux at the material surface. The multigroup flux limited diffusion model is used in order to overcome the problem of flux overestimation, and a formal solution of the steady-state radiative transfer equation has been derived in order to compare the results. The flux limiter of (c/2)U/sub g/, which represents a weak streaming behavior of photons and corresponds to the variable Eddington factor of 1/2, is found to give good agreement for surface heat fluxes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>