A finite element approach for minimizing line and surface energies arising in the study of singularities in liquid crystals
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914228416282624 |
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| author | Stantejsky, Dominik |
| author_facet | Stantejsky, Dominik |
| contents | Motivated by a problem originating in the study of defect structures in nematic liquid crystals, we describe and study a numerical algorithm for the resolution of a Plateau-like problem. The energy contains the area of a two-dimensional surface $T$ and the length of its boundary $\partial T$ reduced by a prescribed curve to make our problem non-trivial. We additionally include an obstacle $E$ for $T$ and pose a surface energy on $E$. We present an algorithm based on the Alternating Direction Method of Multipliers that minimizes a discretized version of the energy using finite elements, generalizing existing TV-minimization methods. We study different inclusion shapes demonstrating the rich structure of minimizing configurations and provide physical interpretation of our findings for colloidal particles in nematic liquid crystal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24928 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A finite element approach for minimizing line and surface energies arising in the study of singularities in liquid crystals Stantejsky, Dominik Numerical Analysis Analysis of PDEs 49Q20, 65K10, 65N30, 76A15 Motivated by a problem originating in the study of defect structures in nematic liquid crystals, we describe and study a numerical algorithm for the resolution of a Plateau-like problem. The energy contains the area of a two-dimensional surface $T$ and the length of its boundary $\partial T$ reduced by a prescribed curve to make our problem non-trivial. We additionally include an obstacle $E$ for $T$ and pose a surface energy on $E$. We present an algorithm based on the Alternating Direction Method of Multipliers that minimizes a discretized version of the energy using finite elements, generalizing existing TV-minimization methods. We study different inclusion shapes demonstrating the rich structure of minimizing configurations and provide physical interpretation of our findings for colloidal particles in nematic liquid crystal. |
| title | A finite element approach for minimizing line and surface energies arising in the study of singularities in liquid crystals |
| topic | Numerical Analysis Analysis of PDEs 49Q20, 65K10, 65N30, 76A15 |
| url | https://arxiv.org/abs/2512.24928 |