Least-Squares PN formulation of the transport equation using self-adjoint-angular-flux consistent boundary conditions

Vincent M. Laboure; Yaqi Wang; Mark D. DeHart

Least-Squares P_N formulation of the transport equation using self-adjoint-angular-flux consistent boundary conditions

Vincent M. Laboure
, Yaqi Wang
, Mark D. DeHart

Nuclear Reactor Technology

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

6 Scopus citations

Abstract

In this paper, we study the Least-Squares (LS) P_N form of the transport equation compatible with voids [ ] in the context of Continuous Finite Element Methods (CFEM). We first derive weakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment [ ], in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework [ ] using a spherical harmonics (P_N) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and find the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.

Original language	English
Title of host publication	Physics of Reactors 2016, PHYSOR 2016
Subtitle of host publication	Unifying Theory and Experiments in the 21st Century
Publisher	American Nuclear Society
Pages	3376-3385
Number of pages	10
ISBN (Electronic)	9781510825734
State	Published - 2016
Event	Physics of Reactors 2016: Unifying Theory and Experiments in the 21st Century, PHYSOR 2016 - Sun Valley, United States Duration: May 1 2016 → May 5 2016

Publication series

Name	Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century
Volume	5

Conference

Conference	Physics of Reactors 2016: Unifying Theory and Experiments in the 21st Century, PHYSOR 2016
Country/Territory	United States
City	Sun Valley
Period	05/1/16 → 05/5/16

Keywords

Boundary conditions
Least-Squares
P
Transport

Cite this

Laboure, V. M., Wang, Y., & DeHart, M. D. (2016). Least-Squares P_N formulation of the transport equation using self-adjoint-angular-flux consistent boundary conditions. In Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century (pp. 3376-3385). (Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century; Vol. 5). American Nuclear Society.

Laboure, Vincent M. ; Wang, Yaqi ; DeHart, Mark D. / Least-Squares P_N formulation of the transport equation using self-adjoint-angular-flux consistent boundary conditions. Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century. American Nuclear Society, 2016. pp. 3376-3385 (Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century).

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title = "Least-Squares PN formulation of the transport equation using self-adjoint-angular-flux consistent boundary conditions",

abstract = "In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids [ ] in the context of Continuous Finite Element Methods (CFEM). We first derive weakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment [ ], in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework [ ] using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and find the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.",

keywords = "Boundary conditions, Least-Squares, P, Transport",

author = "Laboure, \{Vincent M.\} and Yaqi Wang and DeHart, \{Mark D.\}",

note = "Funding Information: This work is supported by the U.S. Department of Energy, under DOE Idaho Operations Office Contract DE-AC07-05ID14517. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. This work was also partly funded by the National Science Foundation under Grant No. 1217170.; Physics of Reactors 2016: Unifying Theory and Experiments in the 21st Century, PHYSOR 2016 ; Conference date: 01-05-2016 Through 05-05-2016",

year = "2016",

language = "English",

series = "Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century",

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Laboure, VM, Wang, Y & DeHart, MD 2016, Least-Squares P_N formulation of the transport equation using self-adjoint-angular-flux consistent boundary conditions. in Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century. Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century, vol. 5, American Nuclear Society, pp. 3376-3385, Physics of Reactors 2016: Unifying Theory and Experiments in the 21st Century, PHYSOR 2016, Sun Valley, United States, 05/1/16.

Least-Squares P_N formulation of the transport equation using self-adjoint-angular-flux consistent boundary conditions. / Laboure, Vincent M.; Wang, Yaqi; DeHart, Mark D.
Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century. American Nuclear Society, 2016. p. 3376-3385 (Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century; Vol. 5).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Laboure, Vincent M.

AU - Wang, Yaqi

AU - DeHart, Mark D.

N1 - Funding Information: This work is supported by the U.S. Department of Energy, under DOE Idaho Operations Office Contract DE-AC07-05ID14517. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. This work was also partly funded by the National Science Foundation under Grant No. 1217170.

PY - 2016

Y1 - 2016

N2 - In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids [ ] in the context of Continuous Finite Element Methods (CFEM). We first derive weakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment [ ], in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework [ ] using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and find the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.

AB - In this paper, we study the Least-Squares (LS) PN form of the transport equation compatible with voids [ ] in the context of Continuous Finite Element Methods (CFEM). We first derive weakly imposed boundary conditions which make the LS weak formulation equivalent to the Self-Adjoint Angular Flux (SAAF) variational formulation with a void treatment [ ], in the particular case of constant cross-sections and a uniform mesh. We then implement this method in Rattlesnake with the Multiphysics Object Oriented Simulation Environment (MOOSE) framework [ ] using a spherical harmonics (PN) expansion to discretize in angle. We test our implementation using the Method of Manufactured Solutions (MMS) and find the expected convergence behavior both in angle and space. Lastly, we investigate the impact of the global non-conservation of LS by comparing the method with SAAF on a heterogeneous test problem.

KW - Boundary conditions

KW - Least-Squares

KW - P

KW - Transport

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M3 - Conference contribution

AN - SCOPUS:84992021828

T3 - Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century

SP - 3376

EP - 3385

BT - Physics of Reactors 2016, PHYSOR 2016

PB - American Nuclear Society

T2 - Physics of Reactors 2016: Unifying Theory and Experiments in the 21st Century, PHYSOR 2016

Y2 - 1 May 2016 through 5 May 2016

ER -

Laboure VM, Wang Y, DeHart MD. Least-Squares P_N formulation of the transport equation using self-adjoint-angular-flux consistent boundary conditions. In Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century. American Nuclear Society. 2016. p. 3376-3385. (Physics of Reactors 2016, PHYSOR 2016: Unifying Theory and Experiments in the 21st Century).