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We present an analysis and numerical study of an optimal control problem for the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a crucial component in modern technology. They exhibit long range orientational order in their nematic phase, which is represented by a tensor-valued (spatial) order parameter $Q = Q(x)$. Equilibrium LC states correspond to $$Q$$ functions that (locally) minimize an LdG energy functional. Thus, we consider an $L^2$-gradient flow of the LdG energy that allows for finding local minimizers and leads to a semi-linear parabolic PDE, for which we develop an optimal control framework. We then derive several a priori estimates for the forward problem, including continuity in space-time, that allow us to prove existence of optimal boundary and external ``force'' controls and to derive optimality conditions through the use of an adjoint equation. Next, we present a simple finite element scheme for the LdG model and a straightforward optimization algorithm. We illustrate optimization of LC states through numerical experiments in two and three dimensions that seek to place LC defects (where $Q(x) = 0$) in desired locations, which is desirable in applications.
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Modelling and simulation of the cholesteric Landau-de Gennes model
This paper discusses modeling and numerical issues in the simulation of the Landau--de Gennes (LdG) model of nematic liquid crystals (LCs) with cholesteric effects. We propose a fully-implicit, (weighted) $L^2$ gradient flow for computing energy minimizers of the LdG model, and note a time-step restriction for the flow to be energy decreasing. Furthermore, we give a mesh size restriction, for finite element discretizations, that is critical to avoid spurious numerical artifacts in discrete minimizers, particularly when simulating cholesteric LCs that exhibit ``twist.'' Furthermore, we perform a computational exploration of the model and present several numerical simulations in 3-D, on both slab geometries and spherical shells, with our finite element method. The simulations are consistent with experiments, illustrate the richness of the cholesteric model, and demonstrate the importance of the mesh size restriction.
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- Award ID(s):
- 2111474
- PAR ID:
- 10524447
- Publisher / Repository:
- The Royal Society
- Date Published:
- Journal Name:
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 480
- Issue:
- 2292
- ISSN:
- 1364-5021
- Subject(s) / Keyword(s):
- liquid crystals, Landau-de Gennes, finite-element method, cholesteric, gradient flow
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1098/rspa.2023.0813
