Goal-based h-adaptivity of the 1-D diamond difference discrete ordinate method

Jeffers, R. S.; Kópházi, J.; Eaton, M.D.; Févotte, François; Hülsemann, Frank; Ragusa, Jean C.

doi:10.1016/j.jcp.2017.01.037

Cited by 6 publications

(15 citation statements)

References 29 publications

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“…In 2011, Lathouwers combined the forward residuals and adjoint error to give accurate error measures for the effective multiplication factor in two-dimensional reactor problems [10]. For the diamond difference (DD) scheme, Jeffers derived DWR method to estimate the error in the QoI arising from the spatial discretization for fixed source and eigenvalue neutron transport problems [11].…”

Section: Science and Technology Of Nuclear Installationsmentioning

confidence: 99%

Analysis of Spatial Discretization Error Estimators Implemented in ARES Transport Code for SN Equations

Zhang

Liu

et al. 2018

Science and Technology of Nuclear Installations

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The discrete ordinates method (S N ) is one of the mainstream methods for neutral particle transport calculations. Assessing the quality of the numerical solution and controlling the discrete error are essential parts of large-scale high-fidelity simulations of nuclear systems. Three error estimators, a two-mesh estimator, a residual-based estimator, and a dual-weighted residual estimator, are derived and implemented in the ARES transport code to evaluate the error of zeroth-order spatial discretization for S N equations. The difference in scalar fluxes on coarse and fine meshes is adopted to indicate the error in the two-mesh method. To avoid zero residual in zeroth-order discretization, angular fluxes within one cell are reconstructed by Legendre polynomials. The error is estimated by inverting the discrete transport operator using the estimated directional residual as an anisotropic source. The inner product of the forward directional residual and the adjoint angular flux is employed to quantify the error in quantities of interest which can be denoted by a linear functional of forward angular flux. Method of Manufactured Solutions (MMS) is adopted to generate analytical solutions for S N equation with scattering and the determined true error is used to evaluate the effectivity of these estimators. Promising results are obtained in the numerical results for both homogeneous and heterogeneous cases. The larger error region is well captured and the average effectivity index for the local error estimation is less than unity. For the series test problems, the estimated goal quantity error can be contained within an order of magnitude around the exact error.

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Section: Science and Technology Of Nuclear Installationsmentioning

confidence: 99%

Analysis of Spatial Discretization Error Estimators Implemented in ARES Transport Code for SN Equations

Zhang

Liu

et al. 2018

Science and Technology of Nuclear Installations

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“…where K = W A. A represents the streaming, absorption and scattering operators (Jeffers et al 2017a) and W is a diagonal matrix consisting of the angular weights. In this work, we search for a strong solution (as opposed to a weak or classical solution (Oden and Reddy 1976)) to Equation (3), meaning Equation 5holds in the following integral sense:…”

Section: The 1d One-group Neutron Transport Equation and S N Discretimentioning

confidence: 99%

“…A dual weighted residual (DWR) goal based error estimator for the 1-D DD scheme has previously been derived to estimate the error in a quantity of interest (QoI) Jeffers et al (2017a). Local DWR calculations over individual cells provided error indicators for goal-oriented adaptive mesh refinement (GO-AMR) Jeffers et al (2017a). The aim of the GO-AMR was to give an optimal mesh for a given QoI Jeffers et al (2017a).…”

Section: Introductionmentioning

confidence: 99%

“…Others, such as Duo, Azmy, and Zikatanov (2009) use adjoint arguments in the derivation of an error estimator, but the adjoint solution is replaced by bounds to that term in terms of the forward solution. A more detailed literature review can be found in Jeffers et al (2017a) and Jeffers (2017).…”

Section: Introductionmentioning

confidence: 99%

“…However, it can be considered as a weighted residual (WR) method or a Petrov-Galerkin method where the trial and test function spaces (and basis functions) are different from one another (Jeffers et al 2017a;Pitkäranta 1978;Schunert 2013). p GO-AMR with a constant mesh size is tested, where the adaptivity is driven by the DD-S N goal-based error indicators described in Jeffers et al (2017a). The combination of the DWR refinement indicator with locally calculated merit functions is used to drive hp GO-AMR.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Goal-Based Error Estimation, Functional Correction, h, p and hp Adaptivity of the 1-D Diamond Difference Discrete Ordinate Method

Jeffers

Kópházi

Eaton

et al. 2017

Journal of Computational and Theoretical Transport

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This paper uses local dual weighted residual (DWR) error indicators to flag cells for goal-based refinement in a 1-D diamond difference (DD) discretisation of the discrete ordinate (SN) neutron transport equations. Goal-orientated adaptive mesh refinement (GO-AMR) aims to produce a mesh that is optimal for a given goal or QoI (Quantity of Interest). h, p and hp refinement is implemented and applied to various test cases. A merit function is derived for the combined hp algorithm and is calculated for each refinement option within each flagged cell. The refinement option with the highest merit function is chosen for a given flagged cell. This paper also investigates the use of the DWR error estimation as a correction term for the originally calculated QoI. If error correction and GO-AMR are combined the DWR error indicators do not always give an optimal mesh for the corrected value of the QoI. Therefore, refinement indicators based on the error in the error correction term are used and tested in this work

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An a posteriori residual-based spatial discretization error estimator for SN neutron transport

Hart

Azmy

Duo³

2020

Annals of Nuclear Energy

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Goal-based h-adaptivity of the 1-D diamond difference discrete ordinate method

Cited by 6 publications

References 29 publications

Analysis of Spatial Discretization Error Estimators Implemented in ARES Transport Code for SN Equations

Analysis of Spatial Discretization Error Estimators Implemented in ARES Transport Code for SN Equations

Goal-Based Error Estimation, Functional Correction, h, p and hp Adaptivity of the 1-D Diamond Difference Discrete Ordinate Method

An a posteriori residual-based spatial discretization error estimator for SN neutron transport

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