An Hp-Angular Adaptivity Algorithm With The Discrete Ordinates Method For Shielding Calculation

The discrete ordinates method is one of the most classical deterministic methods in solving the Boltzmann transport equation,which is commonly used in shielding calculation and design for various nuclear installations.It is a challenge to the computational accuracy and efficiency of the discrete ordinates method due to the complexity and scale of shielding models,especially for shielding problems containing narrow duct or void featured heavy streaming and angular anisotropy,making the angular distribution of angular fluxes more unsmooth further.Insufficient accuracy of angular discretization in the discrete ordinates method can lead to excessive angular discretization errors,which significantly affects the reliability of shielding calculations for realistic nuclear devices.For large angular discretization errors in complex shielding problems with the narrow duct or void,this dissertation studies in an hp-angular adaptivity with the discrete ordinates method for shielding problems to significantly reduce angular discretization errors and improve the computational accuracy and efficiency.This hp-angular adaptivity uses the discontinuous finite element quadrature sets as its basis quadrature sets.Based on the discontinuous finite element method,these quadrature sets generate discrete quadrature points and weights by mapping from the surface of polyhedron into that of the unit sphere.The hierarchy of these quadrature ordinates can provide two kinds of refinement methods for angular adaptivity,increasing the angular mesh and rising the degree of discontinuous finite element basis functions.This adaptivity relies on the global or goal-based error estimates to update both angular mesh and the degree of the underlying discontinuous finite element basis functions.Moreover,the rotation technique uses the quaternion rotation and the geometric flexibility of discontinuous finite element quadrature sets to realize local refinement in some interested angular domains,which can guarantee the accuracy of goal-based error estimates in the presence of ray effects.The angular adaptivity allows different angular local refinement to be applied in space by decomposing shielding models into several quadrature regions.The coarse-fine and fine-coarse mapping algorithms derived by moment conservation are developed to pass the angular solutions between quadrature regions with different quadrature sets.For multi-group shielding problems,by calculating the anisotropy quantification factor based on the contributon transport theory,a multi-group angular adaptivity is studied to quantify the anisotropy of angular fluxes at different spatial positions for each energy group and determine the energy-spatial regions to be refined for a goal function.This anisotropy quantification factor combined with material characteristics of shielding models is also used to guide automatic decomposition of quadrature regions in the angular adaptivity,which can ensure the reliability of error estimates and good user-friendliness.A parallel solving method with angular adaptivity based on spatial-angular decomposition is studied to guarantee load balancing,improving the computational efficiency of large-scale complex shielding problems.Numerical results demonstrate that the hp-angular adaptivity can generate angular distributions in different energy-space dimensions for global or goal-based physical quantities to reduce effectively angular discrete errors,and minimize the calculation cost for a given accuracy.For the Kobayashi benchmarks,the hp-angular adaptivity has a reduction to approximately 1/50 in quadrature ordinates for a given accuracy compared with uniform angular discretization,and obtains faster convergence compared with h-angular adaptivity.The results in the double-group void shielding problem illustrate that the multi-group angular adaptivity can decompose quadrature regions automatically and has obvious advantages in reducing the number of discrete ordinates and computational time.For the IRI-TUB experiment with four kinds of ducts,the number of discrete ordinates and the calculation time using the multi-group parallel angular adaptivity drop by at least one order of magnitude with a satisfied accuracy for engineering requirements.The relative errors between the calculated results and measured values in various reaction rates at some interested locations for fast and thermal energy group are within 15%and 25%,respectively.This research is helpful to improve the computational accuracy and efficiency for the complex shielding problems with the narrow duct or void,and can further apply to shielding simulations for advanced nuclear installations.