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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.

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.

A novel node collocation approach for the application of radial basis function meshless methods in neutron diffusion equations is presented in this paper. By introducing ghost nodes, the number and position of external nodes can be flexibly controlled to investigate their impact on the ill-conditioning of the coefficient matrix and the solution accuracy. The steady-state single-group neutron diffusion equation is solved using the improved method, and the discretization method and solution process are provided. The boundary scale factor, which measures the distance of external nodes from the computational domain, shows a negative correlation with the error once it stabilizes. An optimal boundary scale factor exists regarding its influence on the matrix condition number; as the shape factor of the radial basis function increases, the optimal value also increases. An increase in the number of layers of external nodes slightly enhances solution accuracy but significantly raises the condition number of the matrix. For a shape factor of 0.01, the computational error decreases by 4.01 % when the number of layers is increased from 1 to 4. When the outer number factor is small, fluctuations in the solution error are observed, with stability achieved once it exceeds 0.5. Although increasing the outer number factor raises the matrix condition number, its impact is less pronounced than that of increasing the number of layers.

A novel node collocation approach for the application of radial basis function meshless methods in neutron diffusion equations is presented in this paper. By introducing ghost nodes, the number and position of external nodes can be flexibly controlled to investigate their impact on the ill-conditioning of the coefficient matrix and the solution accuracy. The steady-state single-group neutron diffusion equation is solved using the improved method, and the discretization method and solution process are provided. The boundary scale factor, which measures the distance of external nodes from the computational domain, shows a negative correlation with the error once it stabilizes. An optimal boundary scale factor exists regarding its influence on the matrix condition number; as the shape factor of the radial basis function increases, the optimal value also increases. An increase in the number of layers of external nodes slightly enhances solution accuracy but significantly raises the condition number of the matrix. For a shape factor of 0.01, the computational error decreases by 4.01 % when the number of layers is increased from 1 to 4. When the outer number factor is small, fluctuations in the solution error are observed, with stability achieved once it exceeds 0.5. Although increasing the outer number factor raises the matrix condition number, its impact is less pronounced than that of increasing the number of layers.