Energy dependent mesh adaptivity of discontinuous isogeometric discrete ordinate methods with dual weighted residual error estimators

摘要

Inthispaperahanging-node,discontinuousGalerkin,isogeometricdiscretisationofthemultigroup,discreteordinates (SN) equations is presented in which each energy group has its own mesh. The equations are discretised using NonUniform Rational B-Splines (NURBS), which allows the coarsest mesh to exactly represent the geometry for a wide rangeofengineeringproblemsofinterest; thiswouldnotbethecaseusingstraight-sided???niteelements. Information istransferredbetweenmeshesviatheconstructionofasupermesh. Thisisanon-trivialtaskfortwoarbitrarymeshes, but is signi???cantly simpli???ed here by deriving every mesh from a common coarsest initial mesh. In order to take full advantage of this ???exible discretisation, goal-based error estimators are derived for the multigroup, discrete ordinates equations with both ???xed (extraneous) and ???ssion sources, and these estimators are used to drive an adaptive mesh re???nement (AMR) procedure. The method is applied to a variety of test cases for both ???xed and ???ssion source problems. The error estimators are found to be extremely accurate for linear NURBS discretisations, with degraded performance for quadratic discretisations owing to a reduction in relative accuracy of the ???exact??? adjoint solution required to calculate the estimators. Nevertheless, the method seems to produce optimal meshes in the AMR process for both linear and quadratic discretisations, and is ??? ??100 more accurate than uniform re???nement for the same amount of computational e???ort for a 67 group deep penetration shielding problem.