Eigenvalue sensitivity to system dimensions

Favorite, Jeffrey A.; Bledsoe, Keith C.

doi:10.1016/j.anucene.2010.01.004

Cited by 16 publications

(12 citation statements)

References 3 publications

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“…He developed the first higher order (variational) estimates for internal interface perturbations by combining elements of the Roussopoulos and Schwinger functionals [59,60]. Favorite and Bledsoe discussed the sensitivity of eigenvalue with respect to uniform expansions or contractions of surfaces [61], and they showed equivalence with Rahnema's earlier work [49].…”

Section: Higher Order Transport and Diffusion (1990s-2000s)mentioning

confidence: 93%

The “virtual density” theory of neutronics

Reed¹,

Smith

Forget

2018

Annals of Nuclear Energy

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Sustainable nuclear energy will likely require fast reactors to complement the current light water reactor paradigm. In particular, breed-and-burn sodium fast reactors (SFRs) offer a unique combination of fuel cycle and power density features. Unfortunately, large breed-and-burn SFRs are plagued by positive sodium void worth. In order to mitigate this drawback, one must quantify various sources of negative reactivty feedback, among which geometry distortions (bowing and flowering of fuel assemblies) are often dominant. These distortions arise mainly from three distinct physical phenomena: irradiation swelling, thermal swelling, and seismic events.Distortions are notoriously difficult to model, because they break symmetry and periodicity. Currently, no efficient and fully generic method exists for computing neutronic effects of distortions. Computing them directly via diffusion would require construction of exotic hyperfine meshes with continuous re-meshing. Many deterministic transport methods are geometrically flexible but would require tedious, intricate re-meshing or re-tracking to capture distortion effects. Monte Carlo offers the only high-fidelity approach to arbitrary geometry, but resolving minute reactivities and flux shift tallies within large heterogeneous cores requires CPU years per case and is thus prohibitively expensive. Currently, the most widely-used methods consist of various approximations involving weighting the uniform radial swelling reactivity coefficient by the power distribution. These approximations agree fairly well with experimental data for flowering in some cores, but they are not fully generic and cannot be trusted for arbitrary distortions. Boundary perturbation theory, developed in the 1980s, is fully general and mathematically rigorous, but it is inaccurate for coarse mesh diffusion and has apparently never been applied in industry.Our solution is the "virtual density" theory of neutronics, which alters material density (isotropically or anisotropically) instead of explicitly changing geometry. While geometry is discretized, material densities occupy a continuous domain; this allows density changes to obviate the greatest computational challenges of geometry changes. Although primitive forms of this theory exist in Soviet literature, they are only applicable to cases in which entire cores swell uniformly. Thus, we conceive a much more general and pragmatic form of "virtual density" theory to model non-uniform and localized geometry distortions via perturbation theory.In order to efficiently validate "virtual density" perturbation theory, we conceive the "virtual mesh" method for diffusion theory. This new method involves constructing a slightly perturbed "fake" mesh that produces correct first-order reactivity and flux shifts due to anisotropic swelling or expansion of individual mesh cells. First order reactivities computed on a "virtual mesh" agree with continuous energy Monte Carlo to within 1-uncertainty.We validate "virtual density" theory via the "virtual mesh" me...

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Section: Higher Order Transport and Diffusion (1990s-2000s)mentioning

confidence: 93%

The “virtual density” theory of neutronics

Reed¹,

Smith

Forget

2018

Annals of Nuclear Energy

View full text Add to dashboard Cite

show abstract

“…Favorite and Bledsoe 34 introduced the Heaviside step function to describe the cross section as a function of surface positions or system dimensions in the polar coordinate system to calculate the derivative.…”

Section: Iib Heaviside Step Functionmentioning

confidence: 99%

“…The fundamental difficulty in adjoint-weighted geometric sensitivity analyses is calculating the derivative of the cross sections with respect to the perturbing geometry variable. To overcome this difficulty, Favorite and Bledsoe 34 introduced the Heaviside step function to describe macroscopic cross sections in the neighborhood of a material interface in 2010. Favorite and Bledsoe 34 and Favorite 35 developed a firstorder adjoint-weighted algorithm, referred to here as the Favorite and Bledsoe algorithm, to calculate the k-eigenvalue sensitivities with respect to surface positions or system dimensions in a multigroup Monte Carlo code.…”

Section: Introductionmentioning

confidence: 99%

“…To overcome this difficulty, Favorite and Bledsoe 34 introduced the Heaviside step function to describe macroscopic cross sections in the neighborhood of a material interface in 2010. Favorite and Bledsoe 34 and Favorite 35 developed a firstorder adjoint-weighted algorithm, referred to here as the Favorite and Bledsoe algorithm, to calculate the k-eigenvalue sensitivities with respect to surface positions or system dimensions in a multigroup Monte Carlo code. Then, Kiedrowski et al 36 and Kiedrowski and Brown 37 incorporated the Favorite and Bledsoe algorithm into a continuousenergy Monte Carlo code.…”

Section: Introductionmentioning

confidence: 99%

“…The algorithm does not allow rotations, translations, or nonuniform expansions or contractions. 34,35 However, these geometric transformations are very common and important in reactivity calculations. The differential rotation worth of control drums in space reactors is an important example.…”

Section: Introductionmentioning

confidence: 99%

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Calculating the k-Eigenvalue Sensitivity to Typical Geometric Perturbations with the Adjoint-Weighted Method in the Continuous-Energy Reactor Monte Carlo Code RMC

Huang

et al. 2019

Nuclear Science and Engineering

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Geometric sensitivity analyses of the k-eigenvalue have many applications in analyses of geometric uncertainty, calculations of differential control rod worth, and searches for critical geometry. The adjoint-weighted first-order geometric sensitivity theory is widely used and has continuously evolved with the Monte Carlo methods. However, the existing adjoint-weighted algorithm can do only uniform isotropic expansions or contractions of surfaces. The adjoint-weighted algorithm also requires computation of adjoint-weighted scattering and fission reaction rates exactly at material interfaces, which has an infinitesimal probability in reality. This paper presents an improved geometry adjoint-weighted perturbation algorithm that is incorporated into the continuous-energy Reactor Monte Carlo (RMC) code. The improvement of the adjoint-weighted algorithm is decomposed into three steps for constructing a cross-section function of geometric parameters using logical expressions, calculating the derivative of the cross-section function, and estimating the adjoint-weighted surface reaction rates. The improved algorithm can accommodate common one-parameter geometric perturbations of internal interfaces or boundary surfaces as well as those of cells as long as the perturbed cells can be described by logical expressions of spatial surface equations. The perturbation algorithm is compared with a direct difference method, the linear least-squares fitting method with central differences, for several typical geometric perturbations including translation, fixed-axis rotation, and uniform isotropic/anisotropic expansion transformations of planar, spherical, cylindrical, and conical surfaces. The differences between the two methods are not more than 3% and not more than 3σ for the majority of the test examples. Even though the perturbation algorithm has higher figures of merit than the direct difference method for the majority of the test examples, there is no guarantee that the former can always be more efficient than the latter. The limitation in the efficiency of the perturbation algorithm was demonstrated by the totally reflecting light water reactor pin model.

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Adjoint-based constant-mass partial derivatives

Favorite

2017

Annals of Nuclear Energy

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Eigenvalue sensitivity to system dimensions

Cited by 16 publications

References 3 publications

The “virtual density” theory of neutronics

The “virtual density” theory of neutronics

Calculating the k-Eigenvalue Sensitivity to Typical Geometric Perturbations with the Adjoint-Weighted Method in the Continuous-Energy Reactor Monte Carlo Code RMC

Adjoint-based constant-mass partial derivatives

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