Ginzburg-Landau et champs de croix : apports liés aux applications de maillage

Details

Supervisors

Remacle, Jean-François ; Van Schaftingen, Jean

Faculty

Ecole polytechnique de Louvain

Degree label

Master [120] : ingénieur civil en mathématiques appliquées

Abstract

enMinimizers of so-called Ginzburg-Landau functionals are functions that minimize both their variations and the passage of their norms to values ​​different from 1. The arbitration between the importance of these two elements is done via a parameter epsilon. When epsilon is set to 0, the norm of the minimizers is 1 unless, if necessary, in a finite number of points. The presence of those points (called singular points) depends on the surface considered. The study of Ginzburg-Landau functionals is fashionable. In the context of mesh applications, these functionals are notably used for the construction of the most regular possible cross fields intended to be supports for the construction of quadrangular meshes as regular as possible. In this context, the singular points of minimizers of Ginzburg-Landau functionals can be mapped to singular nodes in the meshes we are trying to construct. Indeed, these meshes pursue the same two objectives as those involved in the Ginzburg-Landau functionals, namely, be as regular as possible and placing singular nodes in an optimized manner if singular nodes are necessary. In the case of non-constant curvature surfaces, the introduction of a covariant gradient can potentially make it possible to construct Ginzburg-Laudau functionals that are more suitable for these surfaces. A Ginzburg-Landau functional containing a covariant gradient will be studied and some results related to other Ginzburg-Landau functionals will be given. We will also see that the adequacy between mathematical results and numerical results is complicated especially when we try to obtain numerical results similar to those of a mathematical study involving an epsilon parameter tending towards 0.