A Source Function Expansion in Radiative Transfer

1980 ◽  
Vol 102 (4) ◽  
pp. 715-718 ◽  
Author(s):  
M. N. O¨zis¸ik ◽  
W. H. Sutton
Keyword(s):  
Radiative Transfer ◽  
Source Function ◽  
Transfer Problem ◽  
Integral Form ◽  
Computer Time ◽  
Algebraic Equations ◽  
Function Expansion ◽  
Expansion Technique ◽  
Reflecting Boundaries ◽  
Special Case

The radiative heat transfer problem for an isotropically scattering slab with specularly reflecting boundaries is reduced to the solution of a set of algebraic equations by expanding the source function in Legendre polynomials in the space variable in the integral form of the equation of radiative transfer. The lowest order S-1 analysis requires very little computer time for calculations, is easy to apply and yields results which are sufficiently accurate. For an absorbing, emitting, isotropically scattering medium with small and intermediate optical thickness (i.e., τ = 2), which is of great interest in engineering applications, and for which the P-1 and P-3 solutions of the P-N method are not sufficiently accurate, the S-1 solution yields highly accurate results. In the case of a slab with diffusely reflecting boundaries, the problem is split up into a set of simpler problems each of which is solved with the source function expansion technique as a special case of the general problem considered.

The Galerkin Method for Solving Radiation Transfer in Plane-Parallel Participating Media

1982 ◽  
Vol 104 (2) ◽  
pp. 351-354 ◽  
Author(s):  
M. N. O¨zis¸ik ◽  
Y. Yener
Keyword(s):  
Galerkin Method ◽  
Integral Form ◽  
Computer Time ◽  
Incident Radiation ◽  
Algebraic Equations ◽  
Participating Media ◽  
Plane Parallel ◽  
Reflecting Boundaries ◽  
Expansion Coefficients ◽  
The Galerkin Method

The Galerkin method is applied to solve radiative heat transfer in an isotropically scattering, absorbing, and emitting plane-parallel medium with diffusely reflecting boundaries. In this approach, the integral form of the equation of radiative transfer is transformed into a set of algebraic equations for the determination of the expansion coefficients associated with the representation of the incident radiation in a power series in the space variable. The method is easy and straightforward to apply and requires relatively little computer time for the computations, since explicit analytical expressions are obtainable for the expansion coefficients.

An Iterative Solution for Anisotropic Radiative Transfer in a Slab

1979 ◽  
Vol 101 (4) ◽  
pp. 695-698 ◽  
Author(s):  
W. H. Sutton ◽  
M. N. O¨zis¸ik
Keyword(s):  
Radiative Transfer ◽  
Iterative Method ◽  
Integral Form ◽  
Iterative Solution ◽  
Single Scattering ◽  
Anisotropic Scattering ◽  
Initial Guess ◽  
Isotropic Scattering ◽  
Wide Range ◽  
Reflecting Boundaries

An iterative method is applied to solve the integral form of the equation of radiative transfer for the cases of isotropic scattering, highly forward, and backward anisotropic scattering in plane-parallel slab with reflecting boundaries. Calculations are performed for the values of single scattering albedo from ω = 0.7 to 1.0 where the convergence was previously reported to be poor. It is found that the convergence is significantly improved for most cases if the P-1 approximation of the spherical harmonics method is used for the initial guess. Results are presented for the hemispherical reflectivity and transmissivity of the slab over a wide range of parameters.

Linear iterative refinement method for the rapid simulation of borehole nuclear measurements: Part I — Vertical wells

Geophysics ◽  
2010 ◽  
Vol 75 (1) ◽  
pp. E9-E29 ◽  
Author(s):  
Alberto Mendoza ◽  
Carlos Torres-Verdín ◽  
Bill Preeg
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Numerical Technique ◽  
Integral Form ◽  
Computer Time ◽  
Iterative Refinement ◽  
Detector Response ◽  
Layer By Layer ◽  
Response Factors ◽  
Refinement Method

As a result of its high numerical accuracy and versatility to include complex tool configurations and arbitrary spatial distributions of material properties, the Monte Carlo method is the foremost numerical technique used to simulate borehole nuclear measurements. Although recent advances in computer technology have considerably reduced the computer time required by Monte Carlo simulations of borehole nuclear measurements, the efficiency of the method is still not sufficient for estimation of layer-by-layer properties or combined quantitative interpretation with other borehole measurements. We develop and successfully test a new linear iterative refinement method to simulate nuclear borehole measurements accurately and rapidly. The approximation stems from Monte Carlo-derived geometric response factors, referred to as flux sensitivity functions (FSFs), for specific density and neutron-tool configurations. Our procedure first invokes the integral representation of Boltzmann’s transport equation to describe the detector response from the flux of particles emitted by the radioactive source. Subsequently, we use theMonte Carlo N-particle (MCNP) code to calculate the associated detector response function and the particle flux included in the integral form of Boltzmann’s equation. The linear iterative refinement method accounts for variations of the response functions attributable to local perturbations when numerically simulating neutron and density porosity logs. We quantify variations in the FSFs of neutron and density measurements from borehole environmental effects and spatial variations of formation properties. Simulations performed with the new approximations yield errors in the simulated value of density of less than [Formula: see text] with respect to Monte Carlo-simulated logs. Moreover, for the case of radial geometric factor of density, we observe a maximum shift of [Formula: see text] at 90% of the total sensitivity as a result of realistic variations of formation density. For radial variation of neutron properties (migration length), the maximum change in the radial length of investigation is [Formula: see text]. Neutron porosity values simulated with the new approximation differ by less than 10% from Monte Carlo simulations. The approximations enable the simulation of borehole nuclear measurements in seconds of CPU time compared to several hours with MCNP.

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