Radiation Transport Problems Research Articles

2D numerical comparison between [formula omitted] and [formula omitted] radiation transport methods

In this article we study the accuracy of the M1 method to solve some relevant radiation transport problems in 2D. We compare two radiation models (Sn and M1) with analytical and numerical tests to highlight the strengths and limitations of each method. These methods give comparable results except when sharp geometry effects are present. We have used these methods in a test that mimics, without fluid motion or electron heat conduction, the cone-target interaction relevant to inertial confinement fusion physics. In this case, we show that Sn and M1 models agree with a quite good accuracy but shows differences in the temperature profiles and heating times inside the target. These results point out that M1 is a possible alternative candidate for 3D simulations, where full energy transport methods are extremely computer time consuming.

KP1 acceleration scheme for inner iterations consistent with the weighted diamond differencing scheme for the transport equation in three-dimensional geometry

For the transport equation in three-dimensional (r, ϑ, z) geometry, a KP1 acceleration scheme for inner iterations that is consistent with the weighted diamond differencing (WDD) scheme is constructed. The P 1 system for accelerating corrections is solved by an algorithm based on the cyclic splitting method (SM) combined with Gaussian elimination as applied to auxiliary systems of two-point equations. No constraints are imposed on the choice of the weights in the WDD scheme, and the algorithm can be used, for example, in combination with an adaptive WDD scheme. For problems with periodic boundary conditions, the two-point systems of equations are solved by the cyclic through-computations method elimination. The influence exerted by the cycle step choice and the convergence criterion for SM iterations on the efficiency of the algorithm is analyzed. The algorithm is modified to threedimensional (x, y, z) geometry. Numerical examples are presented featuring the KP1 scheme as applied to typical radiation transport problems in three-dimensional geometry, including those with an important role of scattering anisotropy. A reduction in the efficiency of the consistent KP1 scheme in highly heterogeneous problems with dominant scattering in non-one-dimensional geometry is discussed. An approach is proposed for coping with this difficulty. It is based on improving the monotonicity of the difference scheme used to approximate the transport equation.

Monte Carlo Modeling of Light Scattering in Paper

The theory of radiation transmission describes the interaction of radiation with the medium which scatters and absorbs the light. The solutions of radiation transport, which were obtained during the previous century, have been applied on the broad fan of problems from neutron diffusion, optical tomography, spreading of infra-red and visible light in atmosphere to paper and prints. At the beginning the majority of problems of radiation transport were unbeatable, but with the improvement of mathematical tools and approximations, and the computers become quicker and powerful the solutions become more specialized and effective. The first approximate solution was given by Schuster (1905) who presumed in the solution the radiation in only forward and backward direction. Under this influence Kubelka and Munk (1931) have developed their well known model. The original Kubelka-Munk theory was developed for spreading the light in parallel colored layers boundlessly extended in xy direction (Kubelka , Emmel ) (in order to avoid the problems connected with the boundary conditions). The basic assumption of Kubelka-Munk theory is that the layer is uniform and the light distribution within the layer is diffuse. From these assumptions the transport of light in the layer was simplified

The problem of radiation transport in the channels of a polysectional thermodynamic system

The problem of radiation energy transfer in ballistic radiative-gas compression facilities is considered. For constructing mathematical models of these facilities, an algorithm for splitting the thermodynamic system into sections (with consideration of the interconnections of the polysectional system) is used. The conditions of applicability of the thermodynamic approach to describing the processes in a ballistic plasmotron are established. It is shown that, in most practical problems, the losses in the intersectional channels are insignificant as compared with the losses in the sections and can be neglected.

Efficient, automated Monte Carlo methods for radiation transport

Monte Carlo simulations provide an indispensible model for solving radiative transport problems, but their slow convergence inhibits their use as an everyday computational tool. In this paper, we present two new ideas for accelerating the convergence of Monte Carlo algorithms based upon an efficient algorithm that couples simulations of forward and adjoint transport equations. Forward random walks are first processed in stages, each using a fixed sample size, and information from stage k is used to alter the sampling and weighting procedure in stage k + 1 . This produces rapid geometric convergence and accounts for dramatic gains in the efficiency of the forward computation. In case still greater accuracy is required in the forward solution, information from an adjoint simulation can be added to extend the geometric learning of the forward solution. The resulting new approach should find widespread use when fast, accurate simulations of the transport equation are needed.

Shielding Calculations for a Spent Fuel Storage Cask: A Comparison of Discrete Ordinates, Monte Carlo, and Hybrid Methods

In this study, discrete ordinates and Monte Carlo methods were applied to solve the radiation transport problem for a simplified spent fuel storage cask considering fixed neutron and gamma-ray sources. The results were compared, and the causes for their differences were investigated. In addition, a hybrid method based on the Consistent Adjoint Driven Importance Sampling (CADIS) methodology has been adopted to accelerate the Monte Carlo simulations. CADIS utilizes a deterministic adjoint function for variance reduction through source biasing and consistent transport biasing. The problem encountered and its possible solution for applying the source biasing in such a large volume source are described. Compared with the unbiased case, the computational efficiency is improved by a factor of several tens for neutron transport, and the efficiency is increased tremendously by about five orders of magnitude for gamma-ray transport. It has been demonstrated that the biasing scheme applied here is very effective in the shielding calculations for a spent fuel storage cask using the Monte Carlo method.

Diffuse conditions for efficient solution of multislab radiation transport problems

In this article, we make use of recently developed spectral nodal methods for anisotropically scattering media and we derive mathematical conditions for the diffuse reflection and transmission of radiation in the discrete ordinates formulation of particle transport theory for plane-parallel applications. The conditions arise from a suitable reformulation of spatially discretized equations defined on the boundary layers of a multislab domain. As a result, the boundary layers can be removed from the radiation transport calculations and replaced with exact and numerically stable equivalent conditions. In order to illustrate the applicability and computational merit of our discrete ordinates conditions for diffuse reflection and transmission in radiation transport calculations, we perform numerical experiments with atmospheric radiative transfer and nuclear reactor core models.

Optimal discontinuous finite element methods for the Boltzmann transport equation with arbitrary discretisation in angle

This paper describes the development of two optimal discontinuous finite element (FE) Riemann methods and their application to the one-speed Boltzmann transport equation in the steady-state. The proposed methods optimise the amount of dissipation applied in the streamline direction. This dissipation is applied within an element using a novel Riemann FE method, which is based on an analogy between control volume discretisation methods and finite element methods when integration by parts is applied to the transport terms. In one-dimension the optimal finite element solutions match the analytical solution exactly at each outlet node. Both schemes couple elements in space via a Riemann approach. The first of the two schemes is a Petrov–Galerkin (PG) method which introduces dissipation via the equation residual. The second scheme uses a streamline diffusion stabilisation term in the discretisation. These two methods provide a discontinuous Petrov–Galerkin (DPG) scheme that can stabilise an element across the full range of radiation regimes, obtaining robust solutions with suppressed oscillation. Three basis functions in angle of particle travel have been implemented in an optimal DPG Riemann solver, which include the PN (spherical harmonic), SN (discrete ordinate) and LWN (linear octahedral wavelet) angular expansions. These methods are applied to a series of demanding two-dimensional radiation transport problems.

An approximate method for solving radiation transport problems in temporally and spatially stochastic media

A new approach has been developed to deal with stochastic transport problems in three-dimensional media. It is assumed that the medium consists of randomly distributed lumps of material embedded in a background matrix and in each lump the properties may vary randomly with time. The coefficients for scattering and absorption are represented mathematically by members of a random characteristic set function, which depend on space and time. Different physical situations can be described by different forms and combinations of these set functions. In order to effect a solution of the resulting stochastic transport equation, which may be for photons or neutrons, we make the, a priori, assumption that the functional form for the solution of the transport equation, i.e. the stochastic flux, can be represented by the same mathematical form as the scattering and absorption coefficients (or cross sections), i.e. we introduce a stochastic ansatz. This procedure leads to a set of deterministic equations from which the mean and variance of the flux in space and time can be obtained. For the case of a two-phase medium, either two or four coupled integro-differential equations are obtained for the deterministic functions that arise (depending on the problem) and expressions are given for the mean and variance of the angular flux. There is a close relationship between these equations and those from the Levermore–Pomraning (LP) theory, but the new equations offer an opportunity to deal with more general forms of stochastic processes and combine simultaneously time and space fluctuations. The stochastic characteristics of the medium are defined by the correlation functions which appear in the equations and, by making plausible assumptions about the functional form of these autocorrelation functions, different physical situations can be simulated, according to the structure of the medium. The main contribution of the present work is to include space and time fluctuations simultaneously as a pseudo-dichotomic Markov process.

View factor calculation in the 2-D geometries using the collapsed dimension method

Usage of the collapsed dimension method (CDM) applied so far in solving radiative transport problems with participating media has been extended for the determination of view factors of the 2-D geometries. Unlike its application in the participating media, in view factor calculations, the CDM does not require tracing of rays, and thus its expressions are exact. To test the applicability of the CDM for the calculation of view factors, two geometries were considered. The CDM results were found to match precisely with the exact results. If in the CDM, the view factor expressions are not considered exact, the ray tracing becomes mandatory and, in this method too like the discrete ordinate method, the discrete transfer method and the finite volume method, ray effect manifests. For this situation, for a given number of control surfaces, effects of number rays were studied and trends were found similar to those found in other methods.

Collision probabilities in spatially stochastic media II

An improved model for calculating collision probabilities in spatially stochastic media is described based upon a method developed by Cassell and Williams [Cassell, J.S., Williams, M.M.R., in press. An approximate method for solving radiation and neutron transport problems in spatially stochastic media. Annals of Nuclear Energy] and is applicable to three-dimensional problems. We shall show how to evaluate the collision probability in an arbitrarily shaped non-re-entrant lump, consisting of a random dispersal of two phases, for any form of autocorrelation function. Specific examples, with numerical values, are given for a sphere and a slab. In the case of the slab we allow the material to have different stochastic properties in the x, y and z directions.

An approximate method for solving radiation and neutron transport problems in spatially stochastic media

A new approach has been developed to deal with stochastic transport problems in three-dimensional media. This is done by assuming, a priori, a functional form for the stochastic flux in terms of the members of a random set function. For the case of a two-phase medium, two coupled integro-differential equations are obtained for the deterministic functions that arise and expressions are given for the mean and variance of the angular flux. There is a close relationship between these equations and those of the Levermore–Pomraning (LP) theory, but they offer an opportunity to deal with more general forms of stochastic processes. It is also shown that the coupling coefficient between the phase equations is directly proportional to the gradient of the autocorrelation function evaluated at the origin; a feature which has been noted in other fields in which random media occur. By making plausible assumptions about the functional form of the autocorrelation function, different forms of the transport equations can be obtained, according to the structure of the medium. For the one-dimensional case, we may show an exact correspondence with the LP equations. Discussions are given regarding the application of the method to three-dimensional problems for which we expect it to be a good approximation for the mean. We also note that the equations are applicable to realistic problems, such as grains embedded in a background matrix, and not restricted to slabs. Investigations into the variance have also been made and a simple approximation scheme developed which gives reasonable agreement with the simulation results of Adams et al. [Adams, M.L., Larsen, E.W., Pomraning, G.C., 1989. Benchmark results for particle transport in a binary Markov statistical medium. Journal of Quantitative Spectroscopy & Radiative Transfer, 42, 253].

Inversion of time‐dependent nuclear well‐logging data using neural networks

ABSTRACTThe purpose of this work was to investigate a new and fast inversion methodology for the prediction of subsurface formation properties such as porosity, salinity and oil saturation, using time‐dependent nuclear well logging data. Although the ultimate aim is to apply the technique to real‐field data, an initial investigation as described in this paper, was first required; this has been carried out using simulation results from the time‐dependent radiation transport problem within a borehole. Simulated neutron and γ‐ray fluxes at two sodium iodide (NaI) detectors, one near and one far from a pulsed neutron source emitting at ∼14 MeV, were used for the investigation. A total of 67 energy groups from the BUGLE96 cross section library together with 567 property combinations were employed for the original flux response generation, achieved by solving numerically the time‐dependent Boltzmann radiation transport equation in its even parity form. Material property combinations (scenarios) and their correspondent teaching outputs (flux response at detectors) are used to train the Artificial Neural Networks (ANNs) and test data is used to assess the accuracy of the ANNs. The trained networks are then used to produce a surrogate model of the expensive, in terms of computational time and resources, forward model with which a simple inversion method is applied to calculate material properties from the time evolution of flux responses at the two detectors.The inversion technique uses a fast surrogate model comprising 8026 artificial neural networks, which consist of an input layer with three input units (neurons) for porosity, salinity and oil saturation; and two hidden layers and one output neuron representing the scalar photon or neutron flux prediction at the detector. This is the first time this technique has been applied to invert pulsed neutron logging tool information and the results produced are very promising. The next step in the procedure is to apply the methodology to real data.

Optical coherence computed tomography

A time-resolved optical tomography, optical coherence computed tomography, is proposed to bridge the gap between diffuse optical tomography and optical coherence tomography. Both ballistic and multiple-scattered photons are measured at multiple source-detection positions by low-coherence interferometry providing a temporal resolution smaller than 100fs. A light-tissue interaction model was established using the time-resolved Monte Carlo method. The optical properties were then reconstructed by solving the inverse transient radiative transport problem under the first Born approximation. Absorbing inclusions of 100μm diameter were imaged through a 2.6-mm-thick (∼30 scattering mean-free-paths) scattering medium.

Type Ia supernovae

At its heart, a Type Ia supernova is a problem in turbulent nuclear combustion. There are four sub-problems, each of which has eluded solution for decades, but which can be addressed by large scale simulation. First is the ignition – where and how often the burning is ignited in the convective core of an exploding white dwarf star. The outcome is sensitive to the initial conditions near the star's center, which may be chaotic. Second is the propagation of the flame. Until near the end, the flame is an unresolvably narrow sheet moved around by instabilities and the turbulence that its own motion produces, yet how fast it moves determines the strength and brightness of the explosion. Third is whether and how the subsonic burning makes a spontaneous transition to a detonation. Observations favor this outcome, but the physics of the transition is obscure. Fourth, is the radiation transport problem. Why does the supernova look the way it does and can its light curve be relied upon to do precision cosmology? Our Consortium has made genuine progress in each of these areas, as well as in planning, with observers, future observational strategies for SNAP/JDEM and LSST. A new generation of codes is being optimized for the four tasks.

An Inverse Phonon Radiative Transport Problem in Estimating the Boundary Temperatures for A Double-Layer Nanoscale Thin Film

An iterative regularization method (or conjugate gradient method, CGM) is utilized in the present inverse phonon radiative transport problem in estimating the unknown boundary temperature distributions, based on the phonon intensity measurements, for a double-layer thin-film structure. The CGM in dealing with the present integrodifferential governing equations is not as straightforward as for the normal differential equations; special treatment is needed to overcome the difficulties. Results obtained in this inverse analysis are justified based on numerical experiments in which three different unknown temperature (or phonon intensity) distributions are to be determined. Finally, it is shown that accurate boundary temperatures can always be obtained with the CGM.

Type Ia Supernova Light Curves

The diversity of Type Ia supernova (SN Ia) photometry is explored using a grid of 130 one-dimensional models. It is shown that the observable properties of SNe Ia resulting from Chandrasekhar-mass explosions are chiefly determined by their final composition and some measure of ``mixing'' in the explosion. A grid of final compositions is explored including essentially all combinations of 56Ni, stable ``iron'', and intermediate mass elements that result in an unbound white dwarf. Light curves (and in some cases spectra) are calculated for each model using two different approaches to the radiation transport problem. Within the resulting templates are models that provide good photometric matches to essentially the entire range of observed SNe Ia. On the whole, the grid of models spans a wide range in B-band peak magnitudes and decline rates, and does not obey a Phillips relation. In particular, models with the same mass of 56Ni show large variations in their light curve decline rates. We identify the physical parameters responsible for this dispersion, and consider physically motivated ``cuts'' of the models that agree better with the Phillips relation. For example, models that produce a constant total mass of burned material of 1.1 +/- Msun do give a crude Phillips relation, albeit with much scatter. The scatter is further reduced if one restricts that set to models that make 0.1 to 0.3 Msun of stable iron and nickel isotopes, and then mix the ejecta strongly between the center and 0.8 Msun. We conclude that the supernovae that occur most frequently in nature are highly constrained by the Phillips relation and that a large part of the currently observed scatter in the relation is likely a consequence of the intrinsic diversity of these objects.

Uncertainties beyond statistics in Monte Carlo simulations

The Monte Carlo method has become an essential tool for the simulation of radiation and particle transport problems. The combination of its ease of use and the power of the method can create a temptation to use Monte Carlo as a 'black box' tool. In this paper, we shall mention a number of important issues that any user of a Monte Carlo computer code should keep firmly in mind when attempting a transport simulation, and we shall present a recent practical example to show the potential significance of such issues.

Optimum shielding in Jovian radiation environment

At Jupiter, the most dominant particle constituent is the high-energy electrons with E>1 MeV (and can be as high as >100 MeV). Huge mass is required to shield these electrons to reduce the radiation doses to the acceptable level for spacecraft electronics. An adjoint Monte Carlo code, NOVICE, which was developed specifically for space radiation transport problems, was used as a primary tool to study an optimum shielding design for a Jupiter mission. Also, a traditional forward Monte Carlo code, MCNPX, was used to verify the NOVICE results. Here we studied two representative spacecraft-shielding materials: aluminum representing low/medium-Z material and tungsten representing high-Z material. A series of NOVICE and MCNPX runs were performed to compute the energy deposition in spherical shell geometry with a small silicon detector at the center of the sphere. Calculation results indicate that, for the radiation attenuation required for typical electronics used in a Jupiter mission, the low-Z material and the low/high-Z combination are a less-efficient shield per the same areal mass than the high-Z material in the Jovian radiation environment studied here. When massive shielding >10 g/cm 2 is required to protect very radiation-sensitive electronics, then the low- /high-Z combination is a better shield per the same areal mass.

On the uniqueness of a bounded solution of the radiation transport problem

On the uniqueness of a bounded solution of the radiation transport problem