Finite Element Solution of the Self-Adjoint Angular Flux Equation for Coupled Electron-Photon Transport

Liscum‐Powell, Jennifer; Prinja, Anil K.; Morel, Jim E.; Lorence, L.J.

doi:10.13182/nse142-270

Cited by 11 publications

(8 citation statements)

References 9 publications

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“…This was expected as both variational formulations look very much alike (see Eqs. (11) and (15)). For instance, the error at the first spatial point for the most refined mesh approaches 10 −2 cm −2 -s −1 in both cases.…”

Section: Dog Leg Void Duct Problemmentioning

confidence: 94%

Globally Conservative, Hybrid Self-Adjoint Angular Flux and Least-Squares Method Compatible with Voids

Labouré

McClarren

Wang

2017

Nuclear Science and Engineering

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In this paper, we derive a method for the second-order form of the transport equation that is both globally conservative and compatible with voids, using Continuous Finite Element Methods (CFEM). The main idea is to use the Least-Squares (LS) form of the transport equation in the void regions and the Self-Adjoint Angular Flux (SAAF) form elsewhere. While the SAAF formulation is globally conservative, the LS formulation need a correction in void. The price to pay for this fix is the loss of symmetry of the bilinear form. We first derive this Conservative LS (CLS) formulation in void. Second we combine the SAAF and CLS forms and end up with an hybrid SAAF-CLS method, having the desired properties. We show that extending the theory to near-void regions is a minor complication and can be done without affecting the global conservation of the scheme. Being angular discretization agnostic, this method can be applied to both discrete ordinates (S N ) and spherical harmonics (P N ) methods. However, since a globally conservative and void compatible second-order form already exists for S N (Wang et al. 2014), but is believed to be new for P N , we focus most of our attention on that latter angular discretization. We implement and test our method in Rattlesnake within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. Results comparing it to other methods are presented.

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Section: Dog Leg Void Duct Problemmentioning

confidence: 94%

Globally Conservative, Hybrid Self-Adjoint Angular Flux and Least-Squares Method Compatible with Voids

Labouré

McClarren

Wang

2017

Nuclear Science and Engineering

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“…where the integral over angle has been split into two half range integrals according to equation (18). Therefore, we have that…”

Section: Source Iteration Compatible Variational Formulation Of the Smentioning

confidence: 99%

“…al. applied the discrete ordinates (S N ) form of the SAAF equation to coupled electron-photon transport and derived a consistent DSA scheme [18]. They utilised a continuous Bubnov-Galerkin spatial discretisation with linear FEs and applied this to one-dimensional (1D) radiation transport problems.…”

Section: Introductionmentioning

confidence: 99%

A geometry conforming isogeometric method for the self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (SN) angular discretisation

Latimer

Kópházi

Eaton

et al. 2020

Annals of Nuclear Energy

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This paper presents the application of isogeometric analysis (IGA) to the spatial discretisation of the multi-group, self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (S N) angular discretisation. The IGA spatial discretisation is based upon non-uniform rational B-spline (NURBS) basis functions for both the test and trial functions. In addition a source iteration compatible maximum principle is used to derive the IGA spatially discretised SAAF equation. It is demonstrated that this maximum principle is mathematically equivalent to the weak form of the SAAF equation. The rate of convergence of the IGA spatial discretisation of the SAAF equation is analysed using a method of manufactured solutions (MMS) verification test case. The results of several nuclear reactor physics verification benchmark test cases are analysed. This analysis demonstrates that for higher-order basis functions, and for the same number of degrees of freedom, the FE based spatial discretisation methods are numerically less accurate than IGA methods. The difference in numerical accuracy between the IGA and FE methods is shown to be because of the higher-order continuity of NURBS basis functions within a NURBS patch as well as the preservation of both the volume and surface area throughout the solution domain within the IGA spatial discretisation. Finally, the numerical results of applying the IGA SAAF method to the OECD/NEA, seven-group, two-dimensional C5G7 quarter core nuclear reactor physics verification benchmark test case are presented. The results, from this verification benchmark test case, are shown to be in good agreement with solutions of the first-order form as well as the second-order even-parity form of the neutron transport equation for the same order of discrete ordinate (S N) angular approximation.

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“…The steady-state NTE is 44) with the appropriate boundary conditions. It is often important to obtain the neutron multiplication factor by solving the following generalized eigenvalue problem 45) with the homogeneous boundary condition, i.e.…”

Section: Neutron Transport Equationmentioning

confidence: 99%

“…It is noted that those conditions apply to all later numerical weak forms. Then the form of different formulations including SAAF (self-adjoint angular flux) [42,43,44,45,46] and LS (least-square) [47,48,49] suitable for spatial discretization with CFEM are derived. The simularity and difference between SAAF and LS are briefly discussed.…”

Section: Weak Forms Of the Multigroup Radiation Transport Equationsmentioning

confidence: 99%

Rattlesnake Theory Manual

Wang

Labouré

2018

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Rattlesnake is built on the MOOSE framework. Rattlesnake manages meshes with libMesh [25] and solves the system equation with PETSc [26] through MOOSE. Support of unstructured meshes, parallelization with domain decomposition, various time integration schemes, most of nonlinear solvers features, multiphysics coupling are all inherited from the framework. Rattlesnake is responsible for setting up the discretization of RTE with interfaces provided by MOOSE. Because Rattlesnake is built on a general framework, it can perform calculations in 1D, 2D and 3D unstructured meshes via MPI (message passing interface) [27] and multi-threading. The weak forms can be directly translated into code under the MOOSE framework, which results in codes easy to maintain and extend. Multiple developers can co-work seamlessly on Rattlesnake. This development pattern allows splitting the work into the physics-related items residing in Rattlesnake, and physics-irrelevant items residing in the framework. The benefit is twofold: Rattlesnake can push the development on the framework side, which can benefit all the other MOOSE-based applications and any improvements in MOOSE can improve Rattlesnake. For example, the common software development procedure is managed by MOOSE, e.g. the version control, regression tests, on-line documentation. In addition, coupling Rattlesnake with other MOOSE-based applications dealing with other physics is straightforward because these applications are based on the same framework. As an example, the reactor physics application MAMMOTH [2, 28] leverages components spread over Rattlesnake, MOOSE and its modules, Bison [29, 30, 31], and Relap-7 [32]. In summary, being built on MOOSE makes Rattlesnake powerful and unique. There indeed are constraints imposed by the framework, but this is not insurmountable and outweighted by all the advantages provided by this development pattern. In Section 2, we first state the RTE for various particles. Currently, we only consider two particles: neutron and thermal radiation but Rattlesnake can be extended easily to other types. Particles mostly share the same time derivative,

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Finite Element Solution of the Self-Adjoint Angular Flux Equation for Coupled Electron-Photon Transport

Cited by 11 publications

References 9 publications

Globally Conservative, Hybrid Self-Adjoint Angular Flux and Least-Squares Method Compatible with Voids

Globally Conservative, Hybrid Self-Adjoint Angular Flux and Least-Squares Method Compatible with Voids

A geometry conforming isogeometric method for the self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (SN) angular discretisation

Rattlesnake Theory Manual

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