In this talk we discuss our experience with iterative methods for relaxation of high-order curvilinear meshes, which aim to improve the quality of the curved mesh elements through successive iterates for the coordinates of the high-order mesh nodes (control points). The motivation for our work is the need for new mesh optimization algorithms in the development of high-order finite element discretizations of shock hydrodynamics problems in the Arbitrary Lagrangian-Eulerian (ALE) framework [1], but the need for mesh improvement is present also in other areas of computational science, and is of general interest. The goal of mesh relaxation is to define a new computational mesh that is better than the current mesh with respect to some metric related to mesh distortion. We will review several such metrics, obtained by defining an appropriate high-order topological ``mesh Laplacian'' operator. We will focus on harmonic-type mesh relaxation, where a simple iterative scheme is applied to a linear system involving the mesh Laplacian and the high-order node coordinates. We will also investigate different smoothers/preconditioners for this iteration, such as the recently proposed polynomial method [2], as well as the effect of different choices for the high-order basis. The harmonic approach will be compared with other local and global high-order optimization methods available in the Mesquite mesh quality improvement toolkit [3].
[1] BLAST: High-order curvilinear finite elements for shock hydrodynamics, http://www.llnl.gov/CASC/blast.
[2] M.Berndt and N.Carlson, "Using Polynomial Filtering for Rezoning in ALE", Talk at the International Conference on Numerical Methods for Multi-Material Fluid Flows, Archachon, France, Sep 5-9, 2011, LA-UR 11-05015.
[3] Mesquite: Mesh Quality Improvement Toolkit, http://www.cs.sandia.gov/optimization/knupp/Mesquite.html.