The “virtual density” theory of neutronics

Abstract

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...

Method of Characteristics for 3D, Full-Core Neutron Transport on Unstructured Mesh

“…While CSG is ideal for ray tracing, it can be difficult to use for representing complicated geometries and geometric deformation. 28 An alternative is unstructured mesh. Most commonly used in finite element solvers, unstructured mesh builds geometry out of discrete "elements" that are made of simple shapes such as triangles, quadrilaterals, hexahedra, and tetrahedra.…”

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

“…Geometric perturbation theory can be applied to deterministic methods and Monte Carlo methods. For the former, Reed 27 developed a virtual density theory by altering the material density instead of explicitly changing the geometry in 2014. Zheng et al 29 then used this theory in a deterministic neutron transport code in 2017.…”

“…Zheng et al 29 then used this theory in a deterministic neutron transport code in 2017. Reed et al 29,30 discussed the analytic equivalence between the virtual density theory and the boundary perturbation theory in 2018. For Monte Carlo methods, Yamamoto and Sakamoto 31 also developed a virtual density algorithm based on the Monte Carlo differential operator method 32,33 and the correlated sampling method and successfully applied the algorithm to analyze uniform isotropic and anisotropic geometry changes of boundary surfaces in a multigroup Monte Carlo code in 2018.…”

“Virtual density” and traditional boundary perturbation theories: Analytic equivalence and numeric comparison

“…Our original "virtual density" paper numerically validated non-uniform anisotropic virtual density theory against "virtual mesh" diffusion reference cases [1]. Now we seek to ascertain how well virtual density theory performs relative to traditional boundary perturbation theory.…”

“…Showing that our new theory works is nice, but showing how it compares to previous methods could "seal the deal". This paper assumes that the reader is familiar with the original virtual density paper and refers back to that paper multiple times [1].…”