Reduced-order modeling of neutron transport separated in axial and radial space by Proper Generalized Decomposition with applications to nuclear reactor physics

TL;DR

The paper tackles the high dimensionality of steady-state neutron transport in tall, extruded reactor geometries by introducing Proper Generalized Decomposition (PGD) to separate the axial coordinate $z$ (and optionally the polar angle) into low-rank 2D/1D representations. It develops six 2D/1D PGD reduced-order models (ROMs) by arranging energy dependence across radial and axial modes (or sharing it between them) and by using axial-polar or axial decompositions, with detailed derivations of the corresponding submodels and cross-section constructions. Numerical experiments on Takeda LWR and C5G7 pin-cell benchmarks demonstrate that the PGD ROMs are convergent, with group-wise energy allocation often offering the best accuracy-per-time balance, and show potential advantages over conventional 2D/1D methods by avoiding isotropic leakage approximations and enabling multi-mode convergence to 3D solutions. The work suggests that PGD ROMs can enable faster, more scalable full-core simulations and lays out concrete avenues for extensions to k-eigenvalue problems, advanced PGD variants, and HPC-enabled implementations.

Abstract

In this article, we demonstrate the novel use of Proper Generalized Decomposition (PGD) to separate the axial and, optionally, polar dimensions of neutron transport. Doing so, the resulting Reduced-Order Models (ROMs) can exploit the fact that nuclear reactors tend to be tall, but geometrically simple, in the axial direction $z$, and so the 3D neutron flux distribution often admits a low-rank "2D/1D" approximation. Through PGD, this approximation is computed by alternately solving 2D and 1D sub-models, like in existing 2D/1D models of reactor physics. However, the present methodology is more general in that the decomposition is arbitrary-rank, rather than rank-one, and no simplifying approximations of the transverse leakage are made. To begin, we derive two original models: that of axial PGD -- which separates only $z$ and the sign of the polar angle $α\in\{-1,+1\}$ -- and axial-polar PGD -- which separates both $z$ and the full polar angle $μ$ from the radial domain. Additionally, we grant that the energy dependence $E$ may be ascribed to either radial or axial modes, or both, bringing the total number of candidate 2D/1D ROMs to six. To assess performance, these PGD ROMs are applied to two few-group benchmarks characteristic of Light Water Reactors. Therein, we find both the axial and axial-polar ROMs are convergent and that the latter are often more economical than the former. Ultimately, given the popularity of 2D/1D methods in reactor physics, we expect a PGD ROM which achieves a similar effect, but perhaps with superior accuracy, a quicker runtime, and/or broader applicability, would be eminently useful, especially for full-core problems.

Figures (10)

• Figure 1: The six-dimensional phase space of steady-state radiation transport consists of all possible spatial positions (a), angular trajectories (b), and energies (c).
• Figure 2: Spatial decomposition of a 3D mesh (a) into 2D/1D meshes (b), as for Proper Generalized Decomposition.
• Figure 3: Extruded geometry of the Takeda Light Water Reactor (a); within each axial layer (1, 2) every radial area (A, B, C) corresponds to a homogeneous material region (b).
• Figure 4: Axial-polar, or 1D($\mu$), (a) and axial, or 1D, (b) coordinate systems with gray arrows indicating particle direction.
• Figure 5: Angular quadratures of full- (a) and reduced-order (c/b) models by axial/-polar Proper Generalized Decomposition.