Tetrahedral Frame Fields via Constrained Third-Order Symmetric Tensors

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

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.

Citation

Golovaty, D., Kurzke, M., Montero, J. A., & Spirn, D. (2023). Tetrahedral Frame Fields via Constrained Third-Order Symmetric Tensors. Journal of Nonlinear Science, 33(3), Article 48. https://doi.org/10.1007/s00332-023-09898-x

• Journal Article Type: Article
• Acceptance Date: Feb 15, 2023
• Online Publication Date: Mar 31, 2023
• Publication Date: 2023-06
• Deposit Date: Mar 3, 2023
• Publicly Available Date: Apr 1, 2024
• Journal: Journal of Nonlinear Science
• Print ISSN: 0938-8974
• Electronic ISSN: 1432-1467
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 33
• Issue: 3
• Article Number: 48
• DOI: https://doi.org/10.1007/s00332-023-09898-x
• Keywords: Tetrahedral frame · Third-order tensor · Liquid crystal · Ginzburg-Landau functional
• Public URL: https://nottingham-repository.worktribe.com/output/17945368
• Publisher URL: https://link.springer.com/article/10.1007/s00332-023-09898-x
• Additional Information: Received: 4 October 2022; Accepted: 15 February 2023; First Online: 31 March 2023; : ; : The authors declare that they have no conflict of interests.