Least-Squares and Other Residual Based Techniques for Radiation Transport Calculations
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In this dissertation, we develop several novel methods based on or related to least-squares transport residual for solving deterministic radiation transport problems. For the first part of this dissertation a nonlinear spherical harmonics (PN) closure (TPN) was developed based on analysis of the least-squares residual for time-dependent PN equations in 1D slab geometry. The TPN closure suppresses the oscillations induced by Gibbs phenomenon in time-dependent transport calculations effectively. Simultaneously, a nonlinear viscosity term based on the spatial and temporal variations is realized and used in the extension to filtered PN method (NFPN). NFPN determines the angular viscosity on the fly and potentially fixed the issue existed in linear FPN that filtering strength needs to be predefined by iteratively solving the problem. We further developed another type of NFPN and demonstrate both of the two NFPN preserve the thick diffusion limit for thermal radiative transfer problems theoretically and numerically. We also developed several novel methods along with error analyses for steady-state neutron transport calculations based on least-squares methods. Firstly, a relaxed L1 finite element method was developed based on nonlinearly weighting the least-squares formulation by the pointwise transport residual. In problems such as void and near-void situations where least-squares accuracy is poor, the L1 method improves the solution. Further, a non-converged RL1 still can present comparable accuracy. We then developed a least-squares method based on a novel contiguous-discontinuous functional. A proof is provided for the conservation preservation for such a method, which is significant for problems such as k-eigenvalue calculations. Also, a second order accuracy is observed with much lower error magnitudes in several quantities of interest for heterogeneous problems compared with self-adjoint angular flux (SAAF) solution. Lastly, we extended the CD methodology with 1/σt-weighted least-squares functional to derive a CD-SAAF method and developed a SN-PN angular hybrid scheme. The hybrid scheme can employ high order SN in regions with strong transport feature to couple with low order PN in regions with diffusive flux. In k-eigenvalue calculations, it shows superb accuracy with low degrees of freedom.
Zheng, Weixiong (2016). Least-Squares and Other Residual Based Techniques for Radiation Transport Calculations. Doctoral dissertation, Texas A & M University. Available electronically from