Tetrahedral frame fields via constrained third order symmetric tensors
arXiv:2210.00575 · doi:10.1007/s00332-023-09898-x
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.
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