ABSTRACT

Ideally, one would like to specify an arbitrary local condition that gives the local relationship between the two covariant tangent vectors. This is done in this chapter using functionals that depend on a quadratic integrand. The main advantage of the variational approach of this chapter is that the resulting Euler-Lagrange equations provide a least-squares fit to a given local condition. The chapter introduces two basic classes of functionals, based on the elliptic norm and the rank of two matrices, which lead to alignment and diagonalization grid generators. A nonsymmetric form of the diagonalization equations is derived for numerical purposes. The chapter shows how to construct weight functions for the diagonalization equations to attain least-square solutions to a given local condition on the tangent vectors. It gives a weighted grid generator whose local condition is stated in terms of rotation matrices. The chapter also shows how to combine ellipticity and alignment into a single functional.