Reduced-order modeling of neutron transport by proper generalized decomposition

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Dominesey, Kurt A.
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Electronic thesis
Nuclear engineering and science
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The numerical simulation of neutron transport within a nuclear reactor—for brevity, reactor physics—is the foundation on which analysts: design new reactors; license existing designs; prolong the licenses of operating plants; optimize loading patterns of fuel and burnable absorbers; perform fuel cycle analyses; evaluate a reactor’s ability to self-regulate in response to perturbations; model reactivity control systems; design adequate shielding; and predict the isotopic composition of spent fuel. It is, in short, how humans understand the neutronic inner workings of a nuclear reactor, second only to physical experimentation. It is also, unfortunately, extremely resource-intensive to deterministically compute at high fidelities, being described as a six- or seven-dimensional integro-differential equation. These dimensions refer to the neutron field’s position in space r⃗ ≡ (x,y,z), direction of travel (in angular coordinates μ and ω), speed (energy) E, and instant in time t (for transients). As such, discretizing the neutron transport equation with even a scant ten unknowns in each dimension would yield a seven-dimensional mesh containing ten million degrees-of-freedom. Unsurprisingly, in practical simulations of large reactors, this number often runs into the billions or trillions, requiring either vast High Performance Computing (HPC) resources or drastic simplifications. This phenomenon is known generally as the “curse of dimensionality” and is common to many fields of numerical analysis. Presently, we aim to circumvent the curse of dimensionality by seeking a separable, low-rank approximation of the neutron flux, or solution. Moreover, unlike a posteriori, or data-driven, Reduced-Order Models (ROMs), this decomposition will be computed progressively by way of a greedy algorithm, eliminating the need for full-order reference solutions. Specifically, the a priori model order reduction technique here applied to reactor physics is Proper Generalized Decomposition (PGD). Because PGD approximates the solution to high-dimensional problems like radiation transport as a finite series of M products of low(er)-dimensional modes, one avoids solving the high-dimensional (full-order) problem entirely. Instead, only M nonlinear systems of low-dimensional subproblems need be solved; as such, the PGD ROM may be drastically cheaper to compute than the original problem, especially if few modes M are needed. Particularly, we here separate energy (yielding spatio-angular and energetic subproblems) and axial space (yielding 2D and 1D subproblems). Both ROMs are then validated in prototypical reactor physics benchmarks. Our first application—model order reduction in energy by PGD—is motivated by the extreme (ultrafine) resolution required to resolve individual nuclear resonances—sharp peaks across a narrow range of energy—in the neutron interaction probabilities, or cross sections, of nuclear fuels and other materials. This regime of fidelity is so demanding as to be typically impractical for geometries larger than a 1D or 2D slice across a single fuel pin—let alone the hundreds of pins comprising an assembly (or the hundreds of assemblies comprising a core). For now, however, we consider the more modest energy meshes (70 to 361 groups) usually employed for infinite lattices of pins or assemblies (that is, lattice physics), such that it is tractable to compute the full-order solution for comparison. Benchmark cases are taken to be representative light water reactor (LWR) pins of UO2 or Mixed Oxide fuel with CASMO-70, XMAS-172, and SHEM-361 energy meshes. To begin, we establish that both the Galerkin and Minimax PGD ROMs are able to compute the flux at sufficient precision (0.36% L2 error or less) in a tractable number of modes (M = 50). Next, we apply the ROM to cross section generation—often the objective of lattice physics—achieving results comparable to a homogenized, infinite medium model with 1 to 3 modes and comparable to the full-order model by 10 to 20 modes, as assessed by the error of the coarse-group model. Subsequently, we compare the coarse(ned)-group flux against that given by cross section condensation, finding similar L2 errors (0.5%) with 10 modes. Given additional modes, the ROM is able to converge below this threshold. Finally, this ROM is extended to criticality (eigen)problems by means of an original algorithm, which achieves k-eigenvalue errors less than 2 × 10−4 by M = 50. Further, the eigenvalue ROM again compares favorably to the coarse-group model with as few as 10 to 20 modes. Based on these results, we anticipate this PGD ROM may be able to calculate detailed flux distributions and cross sections more economically than the full-order model, at a marginal or negligible detriment to accuracy. Moreover, the ROM presents an alternative means of approximation to cross section condensation, preferable in that it introduces neither a loss of fidelity nor irrecoverable error. Secondly, we apply PGD to separate the axial and (optionally) polar dimensions of neutron transport. As nuclear reactors (especially LWRs) tend to be tall, but geometrically simple, in the axial, or z direction, we expect this ROM may save substantial effort and rapidly converge to a low-rank approximation. Moreover, we anticipate this approach may compare favorably to the methodologically distinct, but practically analogous 2D/1D methods already practiced in reactor physics. First, we derive two original models: that of axial PGD—which separates only z and the axial streaming direction v ∈ {−1,+1}—and axial-polar PGD—which separates both z and 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 then applied to two few-group benchmarks characteristic of LWRs. Therein, we find each ROM to be convergent and the axial-polar PGD to be often more economical than the axial PGD. 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. Finally, we discuss the neutron transport software developed to implement both the the full-order and PGD models, Aether. More specifically, in order to meaningfully apply these ROMs it was necessary to first establish a basic set of features—namely, unstructured mesh geometry, spatial discretization by finite elements, and hyperbolic transport with matrix-freesweeps. Since no software was available that met these requirements, we here develop an original, C++ library, in turn using the deal.II finite element package. Despite the specialized research objectives above, the software is organized such that the particularities of PGD do not appear in the full-order model, but rather are implemented as wrappers around or modifications of it. This allows the library to serve as a general-purpose research tool for deterministic radiation transport, even outside of applications in PGD. Ultimately, we intend to release Aether as a permissively open-source software library, such that others can use and modify this implementation at will. Moreover, while some outstanding features (preconditioning, parallelism) would be practically required, we envision with a modest effort, Aether could be made a useful application for end-users, not just developers, akin to OpenMC or OpenMOC.
August 2022
School of Engineering
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Rensselaer Polytechnic Institute, Troy, NY
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