An adaptive, hanging-node, discontinuous isogeometric analysis method for the first-order form of the neutron transport equation with discrete ordinate (SN) angular discretisation

• Published in: Computer methods in applied mechanics and engineering 2017-05, Vol.318, p.215-241
• Main Authors: Owens, A.R., Welch, J.A., Kópházi, J., Eaton, M.D.
• Format: Article
• Language: English
• Subjects: Adaptive; Adaptive algorithms; Algorithms; Boundary element method; Discrete ordinates; Discretization; Finite element analysis; Finite element method; Galerkin method; Hanging-node; Heuristic; Hyperbolic equations; Isogeometric analysis; Mean free path; Neutron transport; Radiation shielding; Studies; Transport equations
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2017.01.036

In this paper a discontinuous, hanging-node, isogeometric analysis (IGA) method is developed and applied to the first-order form of the neutron transport equation with a discrete ordinate (SN) angular discretisation in two-dimensional space. The complexities involved in upwinding across curved element boundaries that contain hanging-nodes have been addressed to ensure that the scheme remains conservative. A robust algorithm for cycle-breaking has also been introduced in order to develop a unique sweep ordering of the elements for each discrete ordinates direction. The convergence rate of the scheme has been verified using the method of manufactured solutions (MMS) with a smooth solution. Heuristic error indicators have been used to drive an adaptive mesh refinement (AMR) algorithm to take advantage of the hanging-node discretisation. The effectiveness of this method is demonstrated for three test cases. The first is a homogeneous square in a vacuum with varying mean free path and a prescribed extraneous unit source. The second test case is a radiation shielding problem and the third is a 3×3 “supercell” featuring a burnable absorber. In the final test case, comparisons are made to the discontinuous Galerkin finite element method (DGFEM) using both straight-sided and curved quadratic finite elements. •A DG hanging-node IGA discretisation is developed for hyperbolic equations.•A conservative upwinding scheme across the curved element boundaries is established.•Optimal convergence rates are achieved when the underlying solution is smooth.•Adaptive formulation outperforms uniform refinement for a variety of test cases.•Comparisons made to DG finite elements.