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

Authors:
Mark A. Goffin;Christopher M. J. Baker;Andrew G. Buchan;Christopher C. Pain;Matthew D. Eaton;Paul N. Smith
Affiliations:
Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK;Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK;Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK;Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK;Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK;ANSWERS Software Service, AMEC, Kimmeridge House, Dorset Green Technology Park, Winfrith Newburgh, Dorchester, Dorset DT2 8ZB, UK
Venue:
Journal of Computational Physics
Year:
2013

Citing 10
Cited 0

A posteriori error estimation and adaptive mesh-refinement techniques

ICCAM'92 Proceedings of the fifth international conference on Computational and applied mathematics
An adaptive mesh refinement algorithm for the radiative transport equation

Journal of Computational Physics
Adaptive Galerkin finite element methods for partial differential equations

Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
Adjoint and defect error bounding and correction for functional estimates

Journal of Computational Physics
A new adaptive mesh refinement strategy for numerically solving evolutionary PDE's

Journal of Computational and Applied Mathematics
Mesh Generation: Application to Finite Elements

Mesh Generation: Application to Finite Elements
Adjoint A Posteriori Error Measures for Anisotropic Mesh Optimisation

Computers & Mathematics with Applications
Lepp-bisection algorithms, applications and mathematical properties

Applied Numerical Mathematics
A two-mesh adaptive mesh refinement technique for SN neutral-particle transport using a higher-order DGFEM

Journal of Computational and Applied Mathematics
Standard and goal-oriented adaptive mesh refinement applied to radiation transport on 2D unstructured triangular meshes

Journal of Computational Physics

Quantified Score

Hi-index	31.45

Visualization

Abstract

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"e"f"f, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for k"e"f"f with directional dependence. General error estimators are derived for any given functional of the flux and applied to k"e"f"f 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"e"f"f goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.