Discrete-ordinates quadratures based on linear and quadratic discontinuous finite elements over spherical quadrilaterals
Conference Paper
Overview
Identity
Additional Document Info
View All
Overview
abstract
Copyright (2015) by the American Nuclear Society. We present LDFE/QDFE-SQ discrete-ordinates quadratures based on linear and quadratic discontinuous finite elements (LDFEIQDFE) over spherical quadrilaterals (SQ) on the unit sphere. The LDFE-SQ quadratures are an extension of the Jarrell-Adams LDFE-ST quadratures which use spherical triangles (ST). The use of SQ instead of ST produces more uniform quadrature ordinate distributions reducing local integration variability. The QDFE-SQ quadratures demonstrate higher-order (i.e., quadratic) basis functions can be used within the discontinuous finite-element based quadrature methodology. The LDFE-SQ (resp. QDFE-SQ) quadratures place four (resp. nine) ordinates in each SQ. The weight of each ordinate is the integral of its basis function over the SQ surface. The LDFE/QDFE-SQ quadratures exactly integrate all 2nd-order spherical harmonics and higher orders tested (up to 6th-order) with 4th-order accuracy. The LDFE/QDFE-SQ quadratures also integrate the scalar flux for a simple one-cell problem with 4th-order accuracy - significantly better than Level Symmetric (1.5-order) and GaussChebyshev (2nd-order) quadratures and on-par with the Quadruple Range quadrature (4th-order). The LDFE/QDFE-SQ error convergence becomes more complicated for the Kobayashi benchmark showing between 2nd and 3rd-order accuracy at intermediate refinement and rapid convergence at high refinement. Locally-refined LDFE-SQ quadratures show much fewer ordinates are needed for error reduction when refinement is confined to the required cone of angle. The LDFE/QDFE-SQ quadratures are well-suited for use in adaptive discrete-ordinates since they are locally refinable, have strictly positive weights, and can be generated for large numbers of directions.
