  In this paper we propose tools for high-order mesh optimization and
demonstrate their benefits in the context of multi-material Arbitrary
Lagrangian-Eulerian (ALE) compressible shock hydrodynamic applications. The
mesh optimization process is driven by information provided by the simulation
which uses the optimized mesh, such as shock positions, material regions, known
error estimates, etc. These simulation features are usually represented
discretely, for instance, as finite element functions on the Lagrangian mesh.
The discrete nature of the input is critical for the practical applicability of
the algorithms we propose and distinguishes this work from approaches that
strictly require analytical information. Our methods are based on node movement
through a high-order extension of the Target-Matrix Optimization Paradigm
(TMOP) of Knupp. The proposed formulation is fully algebraic and relies only on
local Jacobian matrices, so it is applicable to all types of mesh elements, in
2D and 3D, and any order of the mesh. We discuss the notions of constructing
adaptive target matrices and obtaining their derivatives, reconstructing
discrete data in intermediate meshes, node limiting that enables improvement of
global mesh quality while preserving space-dependent local mesh features, and
appropriate normalization of the objective function. The adaptivity methods are
combined with automatic ALE triggers that can provide robustness of the mesh
evolution and avoid excessive remap procedures. The benefits of the new
high-order TMOP technology are illustrated on several simulations performed in
the high-order ALE application BLAST.
