Nonlinear Diffusion Acceleration in Voids for the Least-Squares Transport Equation

Abstract

In this dissertation we present advances to the nonlinear diffusion acceleration for void regions using second order forms of the transport equation. We consider the weighted least-squares and the self-adjoint angular flux transport equations. We show that these two equations are closely related through the definition of the weight function and how the nonlinear diffusion acceleration can be extended to hold in void regions. Using a Fourier analysis we show the convergence properties of our method for homogeneous and heterogeneous problems. We use several problems to study the numerical behavior and the influence of different discretization schemes.

Second order forms of the transport equation allow the use of continuous finite elements (CFEM). CFEM discretization is computationally cheaper and easier to implement on unstructured meshes, allowing more detailed geometries. The selfadjoint transport operators result normally in symmetric positive-definite matrices which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage. In this dissertation we study the weighted least-squares and compare to the self-adjoint angular flux transport equations with void treatment, both well defined in voids.

The nonlinear diffusion acceleration (NDA) is an effective scheme to increase convergence for highly diffusive problems, but can also ensure conservation for nonconservative transport schemes. However, for second order transport equations, the scheme was not yet defined in voids. In this dissertation we derived modifications to the NDA to handle problems containing void regions. A Fourier analysis showed that the newly developed modifications accelerates unconditionally for scattering ratios smaller than one. Extensive testing on various parameters was performed to ensure that the modifications are stable and efficient.

Numerical tests with Reed’s problem showed that the NDA scheme results in a non-constant flux shape in the void regions. Further investigations revealed that this coarse mesh problem is caused by the interface coupling between void and material regions. The separation of the low-order equation at the interface ameliorates these problems. We give a proof-of-concept for a high-order CFEM/low-order DFEM scheme as well as for an artificial diffusion scheme to restore causality and obtain an improved scalar flux solution in the void.

The NDA void modifications were then tested on a modified C5G7 problem, a challenging reactor physics benchmark. The results were compared to first order transport and the self-adjoint angular flux equation with void treatment. The results indicated that the weighted least-squares equations give adequate results while maintaining a symmetric positive-definite matrix.