Tetrahedral Frame Fields via Constrained Third-Order Symmetric Tensors

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.