Abstract
Neutral particle transport in media exhibiting large and complex material property spatial variation is modeled by representing cross sections as lognormal random functions of space and generated through a nonlinear memory-less transformation of a Gaussian process with covariance uniquely determined by the covariance of the cross section. A Karhunen-Loeve decomposition of the Gaussian process is implemented to effciently generate realizations of the random cross sections and Woodcock Monte Carlo used to transport particles on each realization and generate benchmark solutions for the mean and variance of the particle flux as well as probability densities of the particle reflectance and transmittance. A computationally effcient stochastic collocation method is implemented to directly compute the statistical moments such as the mean and variance, while a polynomial chaos expansion in conjunction with stochastic collocation provides a convenient surrogate model that also produces probability densities of output quantities of interest. Extensive numerical testing demonstrates that use of stochastic reduced-order modeling provides an accurate and cost-effective alternative to random sampling for particle transport in random media.
Highlights
Problem Statement and Stochastic ModelWhere x and μ are the particle spatial and angular variables and the label ω denotes an element in the sample space, i.e., a particular realization
The random transport equation of interest is given by μ ∂ψ(x, μ, ∂x ω) +Σt(x, ω)ψ(x, μ, ω) =Σs(x, ω) dμ ψ(x, μ, ω), −10 ≤ x ≤ L; − 1 ≤ μ ≤ 1 ψ(0, μ) = δ(1 − μ), μ > 0; ψ(L, μ) = 0, μ < 0, (1)where x and μ are the particle spatial and angular variables and the label ω denotes an element in the sample space, i.e., a particular realization
Realizations of the random cross sections are generated using a lognormal transformation of the KarhunenLoéve (KL) expansion truncated at expansion order K = 7 in which the eigenspectrum is solved using the Nystöm method (NM) with an NNy = 100 node discretization
Summary
Where x and μ are the particle spatial and angular variables and the label ω denotes an element in the sample space, i.e., a particular realization. Macroscopic total or scattering cross section r, r = {t, s}, is represented as the product. Of microscopic cross section r and the stochastically dependent material atom density Nat: Σr(x, ω) = σrNat(x, ω). The atomic density Nat(x, ω) is modeled as a lognormal random field derived from Gaussian random field g(x, ω), Nat(x, ω) = exp g(x, ω) ,. Yielding all macroscopic cross sections for the material and ensuring that these values are always positive
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