Adaptive Discrete Ordinates Algorithm For Shielding Calculation

The discrete ordinates method is one of the mainstream deterministic methods for radiation shielding calculation of nuclear systems.However,angular fluxes distribution in spatial-angular space becomes unsmooth and discontinuous because of the strong heterogeneity in the radiation shielding models.Significant discretization errors,such as the ray effects and unphysical oscillations,can be observed when the spatial-angular discretization schemes cannot describe the exact angular fluxes distributions.And thus the reliability of shileding design can be influenced seriously.For the strong heterogeneity effects that exist in the discrete ordinates shielding calculations,this dissertation investigates the adaptive spatial-angular variable discretization schemes.Local refinement is performed in the spatial-angular domain under the guidance of error estimators,aiming to control the spatial-angular discretization errors in deterministic calculations and increase the efficiency of shielding calculation.Local spatial discretization error can be estimated by the two-mesh method and the residual-based method.For zeroth order spatial discretization schemes,the profiles of angular fluxes within each mesh cell are approximately constructed by Legendre polynomials.And thus the profiles of residual within each mesh cell can be obtained.For quantity of interest(Qol)in the radiation shielding calculations,the inner product of adjoint fluxes and forward residuals is adopted to estimate the Qol spatial error component,which stems from spatial discretization.Based on the tree-based hierarchical mesh cells,mesh refinement and coarsing algorithm is established to make local mesh variation practical.Recursion procedure is employed to conduct transport sweep over tree-based mesh structures.To maintain the spatial profiles of angular fluxes in the coarse-fine mapping procedure,the zeroth order spatial moment is preserved in the zeroth order schemes while the zeroth and first order spatial moments are preserved in finite element schemes.For multi-group problems,spatial adaptivity can be conducted in each energy group independently.And different spatial mesh cells can be generated in different energy groups according to the estimated loacal spatial discretization errors or importance functions to Qol.To deal with the excessive memory demands and reduce calculation time costs,parallel calculation is conducted by dividing local spatial regions into different processors and aggregating discrete angular directions into groups.Coupled spatial-angular adaptive algorithm is established to account for the coupling relationship bwtween spatial discretization and angular discretization.In the coupled algorithm,the angular adaptivity is only performed in the first group because of the most anisotropic angular fluxes occurs in the first group.And then the adaptive quadrature sets obtained in the first group are employed in the spatial adaptive iteration procedure for all groups.Angular adaptivity is proceeded based on the quadrature spatial regions.And the same quadrature sets are used within one quadrature spatial region.Polynomial weighting method and spherical harmonics expansion method are adopted to mapping angular fluxes between different angular refinement levels.Within the framework of ARES transport code system,the adaptive discrete ordinates radiation shielding algorithm is tested with a series of problems.The numerical results indicate that reasonable spatial-angular discretization schemes can be generated adaptively for the specific problems.And thus,the savings of spatial meshes and angular directions can be achieved.Spatial adaptive algorithm can effectively refine regions which contain material discontinuity,steep angular flux gradient,nonsmooth or discontinous angular fluxes.Also,spatial regions with larger values towards quantity of interest can be refined.The total amount of spatial mesh cells can be reduced by about 1 order of magnitude in the spatial adaptive algorithm.In the multi-group detector-response problem,about a seventieth spatial mesh cells and a fifteenth calculation time is required in the spatial adaptive algorithm compared to uniform refinement when the relative error of QoI reaches 0.1%.In deep-penetration shielding problems with straight duct,2 orders of magnitude savings of spatial-angular discretization unknowns can be achieved at a certain accuracy through coupled adaptive spatial-angular discretization algorithm.The accuracy of the coupled adaptive algorithm is verified by calculating the fast neutron fluxes in the IRI-TUB experimental straight duct models.The differences between calculated values and experimental values are less then 15%.The calculation of XAPR straight duct model demenstrates the engineering application of the proposed adaptive algorithm.The research in this dissertation can resolve the remarkable discretization errors originated from strong heterogeneity in the realistic shielding problems.Also,this algorithm can improve the reliability of radiation shielding calculation and have promising application prospects in nuclear engineering.