Ginzburg-Landau energy and placement of singularities in generated cross fields

TL;DR

This paper introduces a novel Ginzburg-Landau based method for generating cross fields with controlled singularities, which enhances mesh construction by allowing explicit placement of singularities and analyzing their energy contributions.

Contribution

It extends Ginzburg-Landau methods to impose inner singularities through domain perforation and provides a way to compute and interpret the energy related to these singularities.

Findings

Method allows explicit placement of singularities in cross fields.

Energy of cross fields converges as holes shrink, matching theoretical Ginzburg-Landau energy.

Provides insights into singularity degrees and their relation to topology.

Abstract

Cross field generation is often used as the basis for the construction of block-structured quadrangular meshes, and the field singularities have a key impact on the structure of the resulting meshes. In this paper, we extend Ginzburg-Landau cross field generation methods with a new formulation that allows a user to impose inner singularities. The cross field is computed via the optimization of a linear objective function with localized quadratic constraints. This method consists in fixing singularities in small holes drilled in the computational domain with specific degree conditions on their boundaries, which leads to non-singular cross fields on the drilled domain. We also propose a way to calculate the Ginzburg-Landau energy of these cross fields on the perforated domain by solving a Neumann linear problem. This energy converges to the energy of the Ginzburg-Landau functional as epsilon and the radius of the holes tend to zero. To obtain insights concerning the sum of the inner singularity degrees, we give: (i) an extension of the Ginzburg-Landau energy to the piecewise smooth domain allowing to identify the positions and degrees of the boundary singularities, and (ii) an interpretation of the Poincar\'e-Hopf theorem focusing on internal singularities.