A variational method for generating $n$-cross fields using higher-order $Q$-tensors

An $n$-cross field is a locally-defined orthogonal coordinate system invariant with respect to the cubic symmetry group. Cross fields are finding wide-spread use in mesh generation, computer graphics, and materials science among many applications. We consider the problem of generating an $n$-cross field using a higher-order $Q$-tensor theory that is constructed out of tensored projection matrices. It is shown that by a Ginzburg-Landau relaxation, one can reliably generate an $n$-cross field on arbitrary Lipschitz domains.