A practical guide to direct optimization for planar grid-generation

In this paper, we provide a guide to using the direct optimization formulation of variational grid-generation. Particular emphasis is placed on the smoothness, or length, functional; this is undoubtedly the most important functional in variational grid-generation, producing smooth grids and ensuring well-posedness of the minimization problem when combined with more unruly functionals. Unfortunately, in its most primary form, length can produce folded grids when used with nonconvex geometries. Historically, there have been two solutions to this dilemma: using an inverse mapping (the famous Winslow generator), or augmenting the functional with others that promote unicity. Both strategies have the drawback that the resulting minimization problem becomes complicated and expensive. As another alternative, we introduce a generalized strategy for length which does not use inverse mappings or auxiliary functionals, but makes strong use of reference grids. This strategy provides flexibility in controlling grid quality, and its minimization problems can be solved using a simple multigrid algorithm, yielding a robust grid-generation scheme with optimal complexity. We also survey recent developments in the use of variational grid-generation (in the form of direct optimization) for the alignment problem.