TY - JOUR

T1 - Minimising the error in eigenvalue calculations involving the Boltzmann transport equation using goal-based adaptivity on unstructured meshes

AU - Goffin, Mark A.

AU - Baker, Christopher M.J.

AU - Buchan, Andrew G.

AU - Pain, Christopher C.

AU - Eaton, Matthew D.

AU - Smith, Paul N.

PY - 2013/6/1

Y1 - 2013/6/1

N2 - This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, keff, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for keff with directional dependence. General error estimators are derived for any given functional of the flux and applied to keff to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The keff goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.

AB - This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, keff, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for keff with directional dependence. General error estimators are derived for any given functional of the flux and applied to keff to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The keff goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.

KW - A posteriori error analysis

KW - Anisotropic mesh adaptivity

KW - Goal-based mesh refinement

KW - Neutron multiplication factor

KW - Neutron transport

UR - http://www.scopus.com/inward/record.url?scp=84875782624&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2012.12.035

DO - 10.1016/j.jcp.2012.12.035

M3 - Article

AN - SCOPUS:84875782624

SN - 0021-9991

VL - 242

SP - 726

EP - 752

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -