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Minimising the error in eigenvalue calculations involving the Boltzmann transport equation using goal-based adaptivity on unstructured meshes. (English) Zbl 1311.82041

Summary: 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, \(k_{\mathrm{eff}}\), 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 \(k_{\mathrm{eff}}\) 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 \(k_{\mathrm{eff}}\) goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.

MSC:

82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D75 Nuclear reactor theory; neutron transport

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