Wang, Yaqi (2009-05). Adaptive Mesh Refinement Solution Techniques for the Multigroup SN Transport Equation Using a Higher-Order Discontinuous Finite Element Method. Doctoral Dissertation. Thesis uri icon


  • In this dissertation, we develop Adaptive Mesh Refinement (AMR) techniques
    for the steady-state multigroup SN neutron transport equation using a higher-order
    Discontinuous Galerkin Finite Element Method (DGFEM). We propose two error estimations,
    a projection-based estimator and a jump-based indicator, both of which
    are shown to reliably drive the spatial discretization error down using h-type AMR.
    Algorithms to treat the mesh irregularity resulting from the local refinement are
    implemented in a matrix-free fashion. The DGFEM spatial discretization scheme
    employed in this research allows the easy use of adapted meshes and can, therefore,
    follow the physics tightly by generating group-dependent adapted meshes. Indeed,
    the spatial discretization error is controlled with AMR for the entire multigroup SNtransport
    simulation, resulting in group-dependent AMR meshes. The computing
    efforts, both in memory and CPU-time, are significantly reduced. While the convergence
    rates obtained using uniform mesh refinement are limited by the singularity
    index of transport solution (3/2 when the solution is continuous, 1/2 when it is discontinuous),
    the convergence rates achieved with mesh adaptivity are superior. The
    accuracy in the AMR solution reaches a level where the solution angular error (or ray
    effects) are highlighted by the mesh adaptivity process. The superiority of higherorder
    calculations based on a matrix-free scheme is verified on modern computing architectures.
    A stable symmetric positive definite Diffusion Synthetic Acceleration (DSA)
    scheme is devised for the DGFEM-discretized transport equation using a variational
    argument. The Modified Interior Penalty (MIP) diffusion form used to accelerate the
    SN transport solves has been obtained directly from the DGFEM variational form of
    the SN equations. This MIP form is stable and compatible with AMR meshes. Because
    this MIP form is based on a DGFEM formulation as well, it avoids the costly
    continuity requirements of continuous finite elements. It has been used as a preconditioner
    for both the standard source iteration and the GMRes solution technique
    employed when solving the transport equation. The variational argument used in
    devising transport acceleration schemes is a powerful tool for obtaining transportconforming
    diffusion schemes.
    xuthus, a 2-D AMR transport code implementing these findings, has been developed
    for unstructured triangular meshes.

publication date

  • May 2009