Tetrahedral Frame Fields via Constrained Third-Order Symmetric Tensors

Dmitry Golovaty, Matthias Kurzke, Jose Alberto Montero, Daniel Spirn

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number48
JournalJournal of Nonlinear Science
Issue number3
StatePublished - Jun 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


  • Ginzburg-Landau functional
  • Liquid crystal
  • Tetrahedral frame
  • Third-order tensor


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