Discrete Ordinates Transport Calculation In Thick Diffusion Limit With High-Order Finite Element Spatial Discretization

Thick diffusion limit neutron transport calculation inside thermonuclear devices is one of the research focuses for numerical simulation of high-energy-density systems.The growing intricacy of optical thick installations’ geometric and design schemes requires numerical methods that can exhibit the property of asymptotic diffusion limit in strongly discontinuous problems but also for matching the high-order curved mesh.Meanwhile,the extensive amount of calculations for practical applications puts a strain on the computing resources and simulation efficiency.The optical thick models have multiple scales,strong discontinuity,and strong scattering characteristics.Based on the general discrete ordinates neutron transport method,this dissertation studies the arbitrary-order spatial discretization schemes appropriate for multi-coordinates,affine mapping algorithm for high-order curved mesh and optimized source iteration algorithm.These efforts aim to improve the confidence of neutron transport calculation in thick diffusion limit.The arbitrary-order discontinuous finite element spatial discretization scheme based on Lagrangian basis function is studied.From the weak form of the discrete ordinates transport equation in a two-dimensional Cartesian coordinate,a mapping relationship between the physical space and the reference space is established by the affine transformation between the physical element and the reference element.The arbitrary-order finite element space is used to construct the discontinuous finite element scheme with the asymptotic preservation property of the thick diffusion limit.A isoparametric mapping algorithm based on Lagrangian interpolation polynomials is proposed to approximate the coordinates in the reference space to establish the correspondence between the reference space and the actual physical space for improving the accuracy and efficiency of complex geometric description.The high-order discontinuous finite element discretization scheme on curvilinear mesh in Cartesian coordinate is extended to cylindrical coordinate.According to the continuity condition of the angular space,three kinds of angular flux recurrence approximation schemes are studied to maintain the particle conservation of the spatial grids.The Lagrangian basis functions are used to derive the corresponding elementary matrices to realize the finite element space discretization of the neutron transport equation in the conserved form.A goal-oriented source iteration optimization algorithm is developed for the strong spatial heterogeneity problems with optical thickness media to adaptively determine the partitioning of the geometry and dynamically change the angular quadrature sets in iterations.The importance function based on the adjoint and forward transport calculations is solved to obtain the response function to get a problem-dependent,goaloriented spatial decomposition.The difference in the scalar fluxes from one high-order quadrature set to a lower one provides the error estimation as a driving force behind the dynamic quadrature.The transport simulation and numerical analyses are performed for the thick diffusion limit problems.The arbitrary-order discontinuous finite element scheme based on Lagrangian basis function is robust and preserve transport solutions in the thick diffusion limit for strong spatial heterogeneity problems containing scattering optical thickness medium either in Cartesian or cylindrical coordinates.The convergence rates with respect to the model degrees of freedom obtained using curved mesh refinement of O((p+l)/2)on problems with smooth solutions are observed.However,the strong anisotropy of the angular flux in optically thick problems with material discontinuity leads to spatially unsmooth real solutions,which may bring about a decrease in the simulation accuracy of various spatial discretization methods and even to negative flux that have no physical meaning.The goal-oriented source iterative optimization algorithm can significantly reduce the total number of transport calculation unknowns and mitigate the discretization error for achieving the balance between the accuracy and efficiency of transport calculation to ensure the reliability of transport calculations.For benchmark problems and practical nuclear model with strong discontinuities and scattering characteristics,the goal-oriented optimization algorithm can effectively reduce the total number of calculation unknowns and demonstrate higher simulation accuracy and efficiency than the standard discrete ordinate calculation.The research has improved the discrete ordinate method for thick diffusion limit neutron transport problems,providing ideas and methods for numerical simulations and engineering applications of nuclear devices.