P1 Approximation Research Articles

Steady-state reflectance spectroscopy in the P_3 approximation

The Green’s function of the steady-state radiative-transport equation is derived in the P3 approximation for isotropic sources. It is demonstrated that the P3 approximation models the radiance in highly absorbing media or close to sources more accurately than does diffusion theory. Boundary conditions consistent with the P3 approximation are also developed for semi-infinite media bounded by a nonscattering medium. Expressions for the reflectance remitted from media interrogated by a normally incident pencil beam are derived and fitted to simulated and experimental reflectance data from media having optical properties typical of biological tissue. The reconstructed optical properties are accurate to within ±10% in absorption and scattering for source–detector separations as small as 0.43 mm and albedos as low as 0.59. Methods for simplifying these expressions given some a priori knowledge of the scattering phase function are discussed.

Overlapping multigroup approach to a linear extended Boltzmann equation

We introduce a linear Boltzmann equation governing the evolution of charged structureless particles scattering inelastically with heavy field particles. External electric and magnetic fields are accounted for. Conservation of mass as well as balance equations for momentum and energy transfer are discussed in the Lorentz gas limit. A multigroup approximation based on the method of weighted residuals is presented. This approach incorporates external fields in a natural manner. A P1 approximation of the multigroup equations is applied to simple test cases.

Landauer equation in three-dimensional amorphous materials

We use the Boltzmann transport equation to calculate the three-dimensional (3D) version of the Landauer equation for incoherent electrons diffusing in a slab of amorphous material. We use the P1 approximation to calculate the multiple-scattering transmission coefficient as a function of the diffusion coefficient. The 3D transmission coefficient is, apart from numerical coefficients, similar to that for 1D. The result is valid only for positions far away from the boundaries and for slabs having a thickness much greater than one mean free path.

Penetration of beta rays through gold absorbers

The linear transport equation for electrons has been solved adopting the P1 approximation and assuming two main types of interactions with matter: elastic scattering and soft inelastic collisions. The differential cross-section for elastic scattering is calculated using the screened Mott formula. For inelastic processes the differential cross-section is obtained from a simple oscillator strength model. Comparisons are made between the attenuation coefficients for beta rays obtained by solution of the linear transport equation and those obtained experimentally, obtaining a fairly good result.

Combined Conductive and Radiative Heat Transfer through Open-Cellular Porous Plates.

The ability of the cubic lattice model for the thermal and radiative properties of open-cellular porous materials in predicting the combined effect of conduction and radiation in plane-parallel, open-cellular porous plates was investigated in comparison with available experimental data. The validities of the P1 approximation to radiative transfer and the simple adding method for the total heat flux were also addressed. It is concluded that the cubic lattice model can be utilized accurately to evaluate the temperature porfiles and heat transfer characteristics in open-cellular porous plates and that the simple adding method gives the fairly accurate total heat flux across the media. In addition, it is found that there exists only a little discrepancy between the results for the temperature profiles and mean total effective thermal conductivities obtained by the P1 approximation and those obtained by the exact numerical method.

The Simplified P3 Approximation

The simplified P3 (SP3) approximation to the multigroup neutron transport equation in arbitrary geometries is derived using a variational analysis. This derivation yields the SP3 equations along with material interface and Marshak-like boundary conditions. The multigroup SP3 approximation is reformulated as a system of within-group problems that can be solved iteratively. An “explicit” iterative algorithm for solving the within-group problem is described, Fourier analyzed, and shown to be more efficient than the traditional FLIP implicit algorithm. Numerical results compare diffusion (P1), simplified P2 (SP2), and simplified P3 calculations of a mixed-oxide (MOX) fuel benchmark problem to a reference transport calculation. The SP3 approximation can eliminate much of the inaccuracy in the diffusion and SP2 calculations of MOX fuel problems.

Nonuniqueness in optical tomography:?relevance of the P1 approximation

The question of whether a unique distribution of the absorption coefficient, the scattering coefficient, and the refractive index of a turbid medium can be reconstructed by optical tomography is considered. A recent publication [Opt. Lett. 23, 882 (1998)] established that such reconstruction is not possible when photon transport is well modeled by the diffusion equation. A simple proof is offered that, when measurements at high modulation frequencies are included, the more exact P1 model of light transport suggests that this reconstruction may be possible.

More on boundary conditions for differential approximations

The spherical harmonic PN approximation for a half space, non-equilibrium Marshak wave problem with an isotropic incident radiation flux impinging on the surface is shown to yield exact radiation energy density and material temperature at the boundary for all times when the Mark boundary condition is used. Numerical results of the standard P1 and diffusion approximations for the problem are compared amongst different boundary conditions.

A residual based error estimator for mortar finite element discretizations

A residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed. In particular, we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using P1 approximations in each subdomain. Additionally, a residual based error estimator for Crouzeix-Raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation. It is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the Lagrange multiplier is taken into account.

The Coupling of Radiation and Hydrodynamics

The coupling of radiation transport and hydrodynamics is discussed for the Eulerian frame. The discussion is aimed at developing a suitable set of equations for nonrelativistic radiation hydrodynamics (RHD) that can be numerically integrated using high-resolution methods for conservation laws. We outline how numerical methods based on a wave decomposition may be developed, along with the importance of conservation, particularly in the equilibrium regime. The properties of the RHD equations are examined through asymptotic and dispersion analyses. The conditions required to obtain the classical equilibrium limit are rigorously studied. The results show that a simple coupling term developed recently by Morel, which retains a minimum of relativistic corrections, may be sufficient for nonrelativistic flows. We also give two constraints on the relativistic corrections that result in retaining terms on the order of the truncation. In addition, the dispersion results for the P1 approximation are studied in detail and are compared with both the exact-transport results and a full relativistic treatment. We also examine some nonintuitive behavior in the dispersion results.

A Boundary Functional for the Least-Squares Finite- Element Solution of Neutron Transport Problems

The least-squares finite-element framework for the neutron transport equation is based on the minimization of a least-squares functional applied to the properly scaled neutron transport equation. This approach is extended by incorporating the boundary conditions into the least-squares functional. The proof of the V-ellipticity and continuity of the new functional leads to bounds of the discretization error for different regimes. For a P1 approximation of the angular dependence the resulting system of partial differential equations for the moments is explicitly derived. In the diffusion limit this system is essentially a Poisson equation for the zeroth moment and has a divergence structure for the set of moments of order 1. One of the key features of the least-squares approach is that it produces a posteriori error bounds. The use of these bounds is demonstrated in numerical examples for a spatial discretization using trilinear finite elements on a uniform tessellation into cubes.

Radiance modelling using the P3 approximation

Light dosimetry is an essential component of effective photodynamic therapy (PDT) of tumours. Present PDT light dosimetry techniques rely on fluence-based models and measurements. However, in a previous paper by Barajas et al, radiance-based light dosimetry was explored as an alternative approach. Although successful in demonstrating the use of Monte Carlo (MC) simulations of radiance in tissue optical characterization, the MC proved time consuming and impractical for clinical applications. It was proposed that an analytical solution to the transport equation for radiance would be desirable as this would facilitate and increase the speed of tissue characterization. It has been found that the P3 approximation is one such potential solution. Radiance and fluence expressions based on the P3 approximation were used to optically characterize an Intralipid-based tissue phantom of varying concentration of scatterer (Intralipid) and absorber (methylene blue) using a plane wave illuminated, semi-infinite medium geometry. The results obtained compare favourably with the Grosjean approximation of fluence (a modified diffusion theory) using the same optical parameters . The results illustrate that radiance-based light dosimetry is a viable alternative approach to tissue characterization and dosimetry. It is potentially useful for clinical applications because of the limited number of invasive measurements needed and the speed at which the tissue can be characterized.

A radiative transport model for large-eddy fire simulations

Three-dimensional simulations of fires cannot be performed on present-day computers without devising simplifications to the governing equations. One such method is the large-eddy simulation (LES) approach for fires developed at NIST. This method results in computationally efficient fire simulations in which the buoyancy-generated motion of hot gases and smoke is driven by Lagrangian particles that carry the heat released by combustion. Complex geometries are represented by blocking cells interior to a rectangular domain. A P1 approximation to the radiation transport equation was developed to be consistent with the exact transport equation for scenarios based upon this model. An isolated fire plume above a semi-infinite solid with a constant absorption coefficient in each half-space was studied as an example. A direct elliptic solver required only a fraction of the total LES computational cost. Radiative fluxes and intensities from the numerical and exact solutions to the P1 approximation were in excellent agreement.

Electromagnetic response of the mantle to long-period geomagnetic variations over the globe

Long-period electromagnetic response must be estimated to derive information on the electrical conductivity distribution within the deep mantle of the Earth. For this, we estimated the electromagnetic response function for long-period geomagnetic variations, on the basis of the P1 approximation. The data used for analyses are geomagnetic daily mean values for 10 years (1965–1974) from 59 stations distributed over the globe and for about 20 years (1960–1980) from 9 stations. It turned out that the P1 approximation generally holds except for the periods corresponding to annual and semi-annual variations. We also found that accurate estimation is difficult for periods longer than a few years, probably because of contamination due to secular variations of core origin.

Simulation of natural Gamma-ray spectrometry logs in random media

Neste trabalho, é desenvolvido um modelo teórico para simular o perfil de espectrometria de raios gama naturais em meios aleatórios. O transporte da radiação gama é simulado através do método de elementos finitos (FEM) que calcula o fluxo escalar proveniente da aproximação P1 da equação de transporte de Boltzmann para meios radialmente simétricos. A resposta espectral baseada na distribuição randômica de potássio-40, urânio-238 e tório-232 permite determinar as concentrações em peso destes nuclídeos conforme sugerido pelo método Monte Carlo. Os parâmetros físicos das formações, isto é, o coeficiente de difusão e as seções de choque total e de espalhamento, são determinados através da distribuição de Klein-Nishina e de regressão linear múltipla de dados experimentais. O método é então aplicado na geração de perfis sintéticos de poço e mostra a utilidade da técnica de elementos finitos na predição de informações importantes contidas nos perfis nucleares corridos em ambientes litológicos complexos.

Evaluation of the Neutron and Gamma-Ray Heating in the Radiation Shielding and Magnet Coils of the VISTA Spacecraft

Basic nuclear data for a design concept with inertial fusion energy propulsion for manned or heavy cargo deep space missions beyond earth orbit have been evaluated. Fusion power deposited in the inertial confined fuel pellet debris delivers the rocket propulsion with the help of a magnetic nozzle.The superconducting magnets of the magnetic nozzle are protected against neutron and gamma-ray radiation by a massive shielding. Throughout the shielding, the nuclear heating, caused by neutrons and gamma rays has been calculated. As a critical issue for this design concept, special attention is paid to the nuclear heating in the superconducting magnet coils. The neutron and gamma-ray penetration into the coils is calculated using the Sn methods with a high angular resolution in r-z geometry in S16 P3 approximation by dividing the solid space angle in 160 sectors.Total peak nuclear heat generation density in the coils is calculated as 64.5 μW/cm3 by a fusion power of 17 500 MW. Peak neutron heating density is 30.8 μW/cm3, and peak gamma-ray heating density is 40.6 μW/cm3 (on a different point). However, volume-averaged heat generation in the coils is much lower, namely, 2.17, 8.49, and 10.66 μW/cm3 for neutron, gamma-ray, and total nuclear heating, respectively.A conically shaped frozen hydrogen expellant reduces the neutron streaming toward the spacecraft by a factor of ~12.5 via neutron scattering on hydrogen and deflection into vacuum, in addition to the geometric neutron flux attenuation in space by 1/r2. The results of these calculations can help to increase the credibility of the vehicle for interplanetary space transport applications design concept.

A Bubble Element for the Compressible Euler Equations

In this paper, we present a new Galerkin finite element method with bubble function for the compressible Euler equations. This method is derived from the scaled bubble element for the advection-diffusion problems developed by Simo and his colleagues, which is based on the equivalence between the Galerkin method employing piecewise linear interpolation with bubble functions and the Streamline-Upwind/Petrov Galerkin (SUPG) finite element method using P1 approximation in the steady advection-diffusion problem. Simo and this author have applied this approach to transient advection-diffusion problems by using a special scaled bubble function called P-scaled bubble, which is designed to work in the transient advection-diffusion problems for any Peclet number from 0 to ∞. The method presented in this paper is an application of this p-scaled bubble element to a pure hyperbolic system.

The diffusion and P1 approximations for modeling buoyant flow of an optically thick fluid

Buoyant flow of an optically thick fluid representative of molten glass or an oxide crystal melt is solved via a finite element method using four different radiation models—the Rosseland diffusion approximation with and without radiation slip, the P 1, approximation, and a rigorous numerical treatment. The results indicate that both diffusion approximations fail to accurately predict thermal and flow fields in this problem due to their inability to represent thermal boundary layers. However, the P 1 approximation matches well with the rigorous numerical solution. This accuracy, coupled with implementation ease, favors the P 1 approximation for solution of problems of this type.

New approach to simulation of radiative-conductive heat transfer in semi-transparent crystals being pulled from a melt

Abstract A method is suggested for the solution of multi-dimensional radiative heat transfer problems arising in growing crystals. The basic idea of the approach lies in the construction of a special division of the unit sphere into a set of solid angles (cells) and in the approximation of the radiation intensity in each solid angle by the P1 approximation. The radiant transport equation is satisfied in the mean over each elementary cell and the system of partial differential equations of the second order relative to the local zeroth-moments of radiation intensity are obtained. It is shown that the solid angle subdivision can be carried out in different ways with respect to specific features of the heat transfer problem under consideration. As a result even a very rough partition permits satisfactory accuracy of the numerical solutions. One of the main advantages of the method consists in using solid angle subdivisions which can be varied from point to point of the spatial domain. The latter gave possibility to simulate the radiative heat transfer in a circular cylinder of finite length with specular side surface. On this basis the calculation of the temperature field in a cylindrical sapphire crystal being pulled from the melt has been carried out without any restriction on the size of the crystal

New method for solving radiation transfer problems in emitting, absorbing, and scattering media

The proposed method is based on a novel technique for approximating the angular dependence of the radiated intensity. The entire range of solid angles is divided into N cells, which are symmetric relative to the center of the sphere. In each of the cells the radiation is assigned in the form of the P1 approximation, and a system of differential equations is obtained to determine the set of local zeroth and first moments. In some special cases the proposed approach can be regarded as a generalization of the discrete-ordinates method, which makes it possible to solve the problem of selecting the weights in the quadrature formulas in a natural manner. The effectiveness of the method is demonstrated in two one-dimensional test cases. It is shown that in these cases fairly high accuracy is achieved in the solution of the problem already for N=2.