  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.
