Tetrahedral frame fields via constrained third order symmetric tensors

TL;DR

This paper introduces a novel method for constructing smooth tetrahedral frame fields in three-dimensional domains by using constrained third order symmetric tensors and a Ginzburg-Landau-type functional, with applications to nematic liquid crystals.

Contribution

It develops a new tensor-based framework and a determinant maximization method for generating tetrahedral frame fields with boundary conditions.

Findings

Successfully recovers globally defined tensors outside singular sets

Produces smooth frame fields with filamentary singularities

Demonstrates the effectiveness of the Ginzburg-Landau approach in this context

Abstract

Tetrahedral frame fields have applications to certain classes of nematic liquid crystals and frustrated media. We consider the problem of constructing a tetrahedral frame field in three dimensional domains in which the boundary normal vector is included in the frame on the boundary. To do this we identify an isomorphism between a given tetrahedral frame and a symmetric, traceless third order tensor under a particular nonlinear constraint. We then define a Ginzburg-Landau-type functional which penalizes the associated nonlinear constraint. Using gradient descent, one retrieves a globally defined limiting tensor outside of a singular set. The tetrahedral frame can then be recovered from this tensor by a determinant maximization method, developed in this work. The resulting numerically generated frame fields are smooth outside of one dimensional filaments that join together at triple junctions.