Reference Jacobian optimization-based rezone strategies for arbitrary Lagrangian Eulerian methods

  • Authors:

  • Patrick Knupp;Len G. Margolin;Mikhail Shashkov

  • Affiliations:

  • Parallel Computing Sciences Department, Sandia National Laboratories, MS-0847, P.O.Box 5800, Albuquerque, New Mexico 87185-0847;Center for Nonlinear Studies, Los Alamos National Laboratory, MS-B258, Los Alamos, New Mexico 87545;Theoretical Division, T-7, Los Alamos National Laboratory, MS-B284, Los Alamos, New Mexico 87545

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2002

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Abstract

The philosophy of the arbitrary Lagrangian-Eulerian (ALE) methodology for solving multidimensional fluid flow problems is to move the computational grid, using the flow as a guide, to improve the accuracy and efficiency of the simulation. A principal element of ALE is the rezone phase in which a "rezoned" grid is created that is adapted to the fluid motion. We will describe a general rezone strategy that ensures the continuing geometric quality of the computational grid, while keeping the "rezoned" grid as close as possible to the Lagrangian grid at each time step. Although the methodology can be applied to more general grid types, here we restrict ourselves to logically rectangular grids in two dimensions. Our rezoning phase consists of two components: a sequence of local optimizations followed by a single global optimization. The local optimization defines a set of "reference" Jacobians which incorporates our definition of mesh quality at each point of the grid. The set of reference Jacobians then is used in the global optimization. At each node we form a local patch from the adjacent cells of the Lagrangian grid and construct a local realization of the Winslow smoothness functional on this patch. Minimization of this functional with respect to the position of the central node defines its "virtual" location (the node is not actually moved at this stage). By connecting this virtually moved node to its (stationary) neighbors, we define a reference Jacobian that represents the best locally achievable geometric grid quality. The "rezoned" grid results from a minimization (where the points are actually moved) of a global objective function that measures the distance (in a least-squares sense) between the Jacobian of the rezoned grid and the reference Jacobian. This objective function includes a "barrier" that penalizes grids whose cells are close to being inverted. The global objective function is minimized by direct optimization leading to the rezoned grid. We provide numerical examples to demonstrate the robustness and effectiveness of our methodology on model examples as well as for ALE calculations of Rayleigh-Taylor unstable flow.