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

TL;DR

This paper introduces a variational method using higher-order $Q$-tensors and Ginzburg-Landau relaxation to generate $n$-cross fields in arbitrary dimensions, with applications in mesh generation and materials science.

Contribution

It extends tensor-based $n$-cross field generation to arbitrary dimensions using a new relaxation scheme within the Ginzburg-Landau PDE framework.

Findings

Reliable generation of $n$-cross fields on Lipschitz domains.

New relaxation method embedded into a global steepest descent.

Boundary alignment scheme for cross fields.

Abstract

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. It was recently by other authors that $3$-cross fields can be embedded into the set of symmetric $4$th-order tensors. Another concurrent work 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.