Discontinuous isogeometric analysis methods for the first-order form of the neutron transport equation with discrete ordinate (SN) angular discretisation

In this paper two discontinuous Galerkin isogeometric analysis methods are developed and applied to the first-order form of the neutron transport equation with a discrete ordinate ( S N ) angular discretisation. The discontinuous Galerkin projection approach was taken on both an element level and the patch level for a given Non-Uniform Rational B-Spline (NURBS) patch. This paper describes the detailed dispersion analysis that has been used to analyse the numerical stability of both of these schemes. The convergence of the schemes for both smooth and non-smooth solutions was also investigated using the method of manufactured solutions (MMS) for multidimensional problems and a 1D semi-analytical benchmark whose solution contains a strongly discontinuous first derivative. This paper also investigates the challenges posed by strongly curved boundaries at both the NURBS element and patch level with several algorithms developed to deal with such cases. Finally numerical results are presented both for a simple pincell test problem as well as the C5G7 quarter core MOX/UOX small Light Water Reactor (LWR) benchmark problem. These numerical results produced by the isogeometric analysis (IGA) methods are compared and contrasted against linear and quadratic discontinuous Galerkin finite element (DGFEM) S N based methods.

[1]  Matthew D. Eaton,et al.  The application of isogeometric analysis to the neutron diffusion equation for a pincell problem with an analytic benchmark , 2012 .

[2]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[3]  S. Sherwin,et al.  Mesh generation in curvilinear domains using high‐order elements , 2002 .

[4]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[5]  Shawn D. Pautz,et al.  Discontinuous Finite Element SN Methods on Three-Dimensional Unstructured Grids , 2001 .

[6]  Ben Q. Li.  Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer , 2005 .

[7]  Jean-Yves Moller.  Eléments finis courbes et accélération pour le transport de neutrons , 2012 .

[8]  C. D. Boor,et al.  On Calculating B-splines , 1972 .

[9]  Jean C. Ragusa,et al.  On the Convergence of DGFEM Applied to the Discrete Ordinates Transport Equation for Structured and Unstructured Triangular Meshes , 2009 .

[10]  M. Y. Hussaini,et al.  An Analysis of the Discontinuous Galerkin Method for Wave Propagation Problems , 1999 .

[11]  Yuri Bazilevs,et al.  Computational Fluid-Structure Interaction: Methods and Applications , 2013 .

[12]  Kord Sterling Smith,et al.  Spatial homogenization methods for light water reactor analysis , 1980 .

[13]  James S. Warsa,et al.  A Continuous Finite Element-Based, Discontinuous Finite Element Method for SN Transport , 2008 .

[14]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[15]  P. George,et al.  Mesh Generation: Application to Finite Elements , 2007 .

[16]  Les A. Piegl,et al.  The NURBS book (2nd ed.) , 1997 .

[17]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

[18]  Spencer J. Sherwin,et al.  Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods , 2015, J. Comput. Phys..

[19]  James S. Warsa,et al.  Analytical SN solutions in heterogeneous slabs using symbolic algebra computer programs , 2002 .

[20]  Patrick Knupp,et al.  Code Verification by the Method of Manufactured Solutions , 2000 .

[21]  N. Anders Petersson,et al.  Detecting Translation Errors in CAD Surfaces and Preparing Geometries for Mesh Generation , 2001, IMR.

[22]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[23]  E. Lewis,et al.  Computational Methods of Neutron Transport , 1993 .

[24]  Alfred V. Aho,et al.  Data Structures and Algorithms , 1983 .

[25]  E. Lewis,et al.  Benchmark specification for Deterministic 2-D/3-D MOX fuel assembly transport calculations without spatial homogenisation (C5G7 MOX) , 2001 .

[26]  M. Cox.  The Numerical Evaluation of B-Splines , 1972 .

[27]  Richard Sanchez,et al.  Assembly homogenization techniques for core calculations , 2009 .

[28]  G. Karniadakis,et al.  Spectral/hp Element Methods for Computational Fluid Dynamics , 2005 .

[29]  J. Lautard,et al.  MINARET, A DETERMINISTIC NEUTRON TRANSPORT SOLVER FOR NUCLEAR CORE CALCULATIONS , 2010 .

[30]  T. Hughes,et al.  Efficient quadrature for NURBS-based isogeometric analysis , 2010 .