# Computing cross fields -- A PDE approach based on the Ginzburg-Landau theory

## TL;DR

This paper introduces a PDE-based method using Ginzburg-Landau theory to compute crossfields on surfaces, aiding in quadrangular mesh generation by identifying well-distributed critical points.

## Contribution

It applies Ginzburg-Landau functional to crossfield computation on arbitrary surfaces, providing a novel PDE approach validated through finite element discretization.

## Key findings

- Validated on the unit disk with analytical comparison
- Successfully applied to the 2-sphere with critical points at anticube vertices
- Demonstrated the method's effectiveness for mesh generation

## Abstract

This paper proposes a method to compute crossfields based on the Ginzburg-Landau theory. The Ginzburg-Landau functional has two terms: the Dirichlet energy of the distribution and a term penalizing the mismatch between the fixed and actual norm of the distribution. Directional fields on surfaces are known to have a number of critical points, which are properly identified with the Ginzburg-Landau approach: the asymptotic behavior of Ginzburg-Landau problem provides well-distributed critical points over the 2-manifold, whose indices are as low as possible. The central idea in this paper is to exploit this theoretical background for crossfield computation on arbitrary surfaces. Such crossfields are instrumental in the generation of meshes with quadrangular elements. The relation between the topological properties of quadrangular meshes and crossfields are hence first recalled. It is then shown that a crossfield on a surface can be represented by a complex function of unit norm with a number of critical points, i.e., a nearly everywhere smooth function taking its values in the unit circle of the complex plane. As maximal smoothness of the crossfield is equivalent with minimal energy, the crossfield problem is equivalent to an optimization problem based on Ginzburg-Landau functional. A discretization scheme with Crouzeix-Raviart elements is applied and the correctness of the resulting finite element formulation is validated on the unit disk by comparison with an analytical solution. The method is also applied to the 2-sphere where, surprisingly but rightly, the computed critical points are not located at the vertices of a cube, but at those of an anticube.