Finite Element Numerical Solution of a Self-Adjoint Transport Equation for Coupled Electron-Photon Problems

Powell, Jennifer L.

doi:10.2172/760742

Cited by 5 publications

(2 citation statements)

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“…The second-order form of the BTE (the even-parity equation) often has some limitations on the cross sections or boundary conditions (Duderstadt and Martin 1979). Also the so-called SAAF second-order equation has not yet been shown to be as accurate as the first-order form of the BTE (Liscum-Powell 2000). This is why it is not clear that the second-order form is better than the first-order form and more studies have to be done on both of these.…”

Section: Introductionmentioning

confidence: 99%

“…Because of these different interactions, three coupled BTEs are needed to cover the transport of photons, electrons and positrons. The coupled system is rarely presented in the literature (Liscum-Powell 2000, Lorence et al 1989, Tervo and Kolmonen 2002 and usually the methods are described only for one type of particle, typically for neutrons and electrons (Larsen et al 1997, Martin and Duderstadt 1977, Martin et al 1981. In this paper, we describe the coupled system in detail and also consider its physical background.…”

Section: Introductionmentioning

confidence: 99%

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Modelling the transport of ionizing radiation using the finite element method

2004

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Radiation therapy treatment planning is based on the calculation of the absorbed dose in the patient domain. For exact dose calculations, the solution of three coupled Boltzmann transport equations (BTEs) is needed to cover the transport of photons, electrons and positrons. In many situations, however, two coupled systems for photons and electrons are enough. The use of numerical methods in finding the exact solution of the unknown particle fluxes is necessary. In the stationary case, the BTE has six variables, three spatial, two directional and one energy variable. In this paper, we describe an approach in which the finite element method (FEM) is used to solve the six-dimensional problem. For the coupled photon-electron system, the variational formulation and the existence and uniqueness of the solution are derived. We simulate the solution of two coupled BTEs describing the travelling of photons and electrons in two spatial dimensions. The results are compared to Monte Carlo calculations with good agreement.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Modelling the transport of ionizing radiation using the finite element method

2004

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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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