Goal Oriented Angular Adaptive Algorithm For Ray Effects Mitigation

Accurate shielding calculation method is an important foundation of shielding design of nuclear system.Discrete ordinates method(SN)is one of the main shielding calculation methods.Angular discretization error inherent in the discrete ordinates method is a major problem,especially for localized source problems and problems with large volume of void regions or long duct.The ray effects greatly affect the calculation accuracy and the reliability of the shielding design.This dissertation proposes an angular adaptive algorithm together with a goal oriented error estimate to solve the SN equations.This method eliminates the ray effects and effectively controls the angular discretization errors.Angular adaptive refinement is based on the error estimation of the objective function.Local angular discretization errors are weighted by the importance of a given angular region toward the computational goal,providing an appropriate goal oriented angular adaptivity.We compare an interpolated angular flux value against a calculated value to generate local angular discretization errors.The importance of the objective function is obtained by solving the adjoint transport equation.Due to the fact that the angular flux in the highest energy group tends to be the peakiest and unsmooth,we test for refinement only in this group.The method divides the spatial domain into quadrature regions,which are allowed to have differently refined levels of quadrature sets,but all cells within one quadrature region have the same quadrature sets.The local angular discretization error is only calculated on the boundary of each quadrature region.The mapping schemes are applied to the transition of angular flux solution between adjacent spatial regions with different quadrature sets during the source iteration process.Since the high frequency errors are attenuated more rapidly than the low frequency errors,damped angular multigrid acceleration methods use higher order transport calculations to quickly eliminate the high frequency errors,with two layers of low order transport calculation to eliminate the low frequency errors and the diffusion equation to eliminate the lower frequency errors.A sequence of coarse and fine angular grids is used to remove different modes of errors from the estimate of the solution.We tested the accuracy and efficiency of the proposed goal oriented angular adaptive algorithm within the ARES code for a number of benchmark problems.The adaptive method generates the most effective discrete angle distribution and eliminates the influence of ray effects on the shielding calculation.The Kobayashi benchmarks provide significant challenges to the traditional algorithm,with deep penetration,narrow gap,and significant material discontinuity.The algorithm was very efficient for the Kobayashi problems,achieving the same accuracy with approximately 1.5 orders of magnitude fewer unknowns than uniform refinement,greatly reducing the amount of computation.However,the partition of the quadrature region can affect the adaptive process and cannot reflect the angular discretization errors of the internal meshes.The approximate solution of adjoint transport equation results in the failure to accurately evaluate the objective function calculation error.The simplification is still valid if we only focus on the adaptive refinement guidelines.The mapping schemes accurately transfer the angular flux solution and preserve angular flux shape and flux moments as far as possible.If the incoming angular flux is more anisotropic,the mapping methods distort the distribution of angular flux and affect the particle conservation of the equilibrium equation.The damped angular multigrid has further improved the calculation efficiency and the damped parameter has improved the stability of the acceleration method.Fusion neutron streaming experiment demonstrates the engineering application of the proposed method.The algorithms obtain a given accuracy for much lower computational unknowns and memory requirements than can be obtained with fixed quadrature sets.The research of this topic can eliminate the ray effects and improve the reliability of the shielding calculation.Thus,adaptive refinement is a viable approach to investigate difficult particle transport problems.