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 hyperoctahedral symmetry group (cubic for n = 3). Cross fields are finding widespread use in mesh generation, computer graphics, and materials science among many applications. It was recently shown in [A. Chemin et al., International Meshing Roundtable, 2019, pp. 89-108] that 3-cross fields can be embedded into the set of symmetric fourth-order tensors. The concurrent work [D. Palmer, D. Bommes, and J. Solomon, Algebraic Representations for Volumetric Frame Fields, preprint, arXiv:1908.05411 (2019)] further develops a relaxation of this tensor field via a certain set of varieties. In this paper, we consider the problem of generating an arbitrary n-cross field using a fourth-order Q-tensor theory that is constructed out of tensored projection matrices. We establish that by a Ginzburg-Landau relaxation towards a global projection, one can reliably generate an n-cross field on arbitrary Lipschitz domains. Our work provides a rigorous approach that offers several new results including porting the tensor framework to arbitrary dimensions, providing a new relaxation method that embeds the problem into a global steepest descent, and offering a relaxation scheme for aligning the cross field with the boundary. Our approach is designed to fit within the classical Ginzburg-Landau PDE theory, offering a concrete road map for the future careful study of singularities of energy minimizers.