  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.
