On the Use of Symmetrized Transport Equation in Goal-Oriented Adaptivity

Hanuš, Milan; McClarren, Ryan G.

doi:10.1080/23324309.2016.1164722

Cited by 7 publications

(6 citation statements)

References 14 publications

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“…Furthermore, the computational cost of this procedure could be halved by solving a special neutron transport formulation that yields the primal scalar neutron flux as its solution which can be differentiated to retrieve the adjoint scalar neutron flux in a manner similar to the even/odd parity neutron flux equations [43].…”

Section: The Weighted Error Indicator (Wei)mentioning

confidence: 99%

“…Consequently, for a given number of dof, the computational effort required to calculate the WEI is roughly double that required to calculate the uniformly refined solution. If the methodology laid out in work by Hanus and McClarren was implemented then the WEI results could potentially be acquired at no extra cost [43]. The effects of using the FEI driven AMR algorithm and comparing it to the WEI driven AMR algorithm will be investigated in subsequent verification benchmark test cases.…”

Section: Regionmentioning

confidence: 99%

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Spatial adaptivity of the SAAF and Weighted Least Squares (WLS) forms of the neutron transport equation using constraint based, locally refined, isogeometric analysis (IGA) with dual weighted residual (DWR) error measures

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Kópházi

Eaton

et al. 2021

Journal of Computational Physics

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This paper describes a methodology that enables NURBS (Non-Uniform Rational B-spline) based Isogeometric Analysis (IGA) to be locally refined. The methodology is applied to continuous Bubnov-Galerkin IGA spatial discretisations of second-order forms of the neutron transport equation. In particular this paper focuses on the self-adjoint angular flux (SAAF) and weighted least squares (WLS) equations. Local refinement is achieved by constraining degrees of freedom on interfaces between NURBS patches that have different levels of spatial refinement. In order to effectively utilise constraint based local refinement, adaptive mesh refinement (AMR) algorithms driven by a heuristic error measure or forward error indicator (FEI) and a dual weighted residual (DWR) or goal-based error measure (WEI) are derived. These utilise projection operators between different NURBS meshes to reduce the amount of computational effort required to calculate the error indicators. In order to apply the WEI to the SAAF and WLS second-order forms of the neutron transport equation the adjoint of these equations are required. The physical adjoint formulations are derived and the process of selecting source terms for the adjoint neutron transport equation in order to calculate the error in a given quantity of interest (QoI) is discussed. Several numerical verification benchmark test cases are utilised to investigate how the constraint based local refinement affects the numerical accuracy and the rate of convergence of the NURBS based IGA spatial discretisation. The nuclear reactor physics verification benchmark test cases show that both AMR algorithms are superior to uniform refinement with respect to accuracy per degree of freedom. Furthermore, it is demonstrated that for global QoI the FEI driven AMR and WEI driven AMR produce similar results. However, if local QoI are desired then WEI driven AMR algorithm is more computationally efficient and accurate per degree of freedom.

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Section: The Weighted Error Indicator (Wei)mentioning

confidence: 99%

Section: Regionmentioning

confidence: 99%

Spatial adaptivity of the SAAF and Weighted Least Squares (WLS) forms of the neutron transport equation using constraint based, locally refined, isogeometric analysis (IGA) with dual weighted residual (DWR) error measures

Latimer

Kópházi

Eaton

et al. 2021

Journal of Computational Physics

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“…The secondorder forms do have issues in voids [36,37,38] that cause either inaccuracy or loss of the self-adjoint character of the equations. Adaptivity in space [39,40,41,42] and space-angle [43], as well as selective reduction of degrees of freedom [44] have all been explored as well.…”

Section: Introductionmentioning

confidence: 99%

A high-order / low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations

Peng,

McClarren

2020

Preprint

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Dynamical low-rank (DLR) approximation methods have previously been developed for time-dependent radiation transport problems. One crucial drawback of DLR is that it does not conserve important quantities of the calculation, which limits the applicability of the method. Here we address this conservation issue by solving a low-order equation with closure terms computed via a high-order solution calculated with DLR. We observe that the high-order solution well approximates the closure term, and the low-order solution can be used to correct the conservation bias in the DLR evolution. We also apply the linear discontinuous Galerkin method for the spatial discretization to obtain the asymptotic limit.We then demonstrate with the numerical results that this so-called high-order / low-order (HOLO) algorithm is conservative without sacrificing computational efficiency and accuracy.

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“…SPD linear systems of equations can be solved efficiently with preconditioned conjugate gradient matrix solution algorithms [8]. Amongst the second-order forms of the neutron transport equation the most widely used are the EP, self-adjoint angular flux (SAAF), and symmetrised neutron transport (ST) [9] forms. Such forms, of the neutron transport equation, have been used extensively because they are self-adjoint equations and extremum variational principles may be derived for them [2] which allows for upper and lower bounds of the solution to be determined [10].…”

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

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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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On the Use of Symmetrized Transport Equation in Goal-Oriented Adaptivity

Cited by 7 publications

References 14 publications

Spatial adaptivity of the SAAF and Weighted Least Squares (WLS) forms of the neutron transport equation using constraint based, locally refined, isogeometric analysis (IGA) with dual weighted residual (DWR) error measures

Spatial adaptivity of the SAAF and Weighted Least Squares (WLS) forms of the neutron transport equation using constraint based, locally refined, isogeometric analysis (IGA) with dual weighted residual (DWR) error measures

A high-order / low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations

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

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