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Minimizing Oseen-Frank energy for nematic liquid crystals: Algorithms and numerical results. (English) Zbl 0911.35007

This paper presents a family of new algorithms for computing equivalent configurations in a nematic liquid crystal occupying a bounded region \(\Omega\) in \(\mathbb{R}^3\), when a strong anchoring condition obtains at the bounding surface \(\partial\Omega\). The problem is formulated in terms of seeking solutions for the unit vector field \(u\) (the director) that yield local minima of the energy integral \[ E(u)=\iiint_VW(u,\nabla u)dV,\tag{*} \] where \(W\) is the Oseen-Frank energy density function given by \[ 2W=K_1(\nabla\cdot u)^2+K_2\bigl(u\cdot(\nabla\times u)\bigr)^2+K_3\bigl| u\times(\nabla\times u)\bigr|^2+(K_2+K_4)\bigl(tr(\lambda u)^2-(\nabla\cdot u)^2\bigr). \] The authors use the fact that \(K_4\) does not appear in (*) to obtain results for the full physical range of the other Frank constants \(K_1, K_2\) and \(K_3\). By first constructing a sequence \(u^n(x)\) such that \(| u^n(x)|=1\) for a.e. \(x\in\Omega\), \(u^n(x)=\varphi(x)\) for a.e. \(x\in\partial\Omega\) and \(u^n(x)\) converges to a local minimiser of (*), where \(\varphi(x)\) is the prescribed value of \(u\) on \(\partial\Omega\), the authors use a ‘one step’ procedure to develop a family of algorithms for determining equilibrium configurations in a continuous setting. They then employ a framework of finite differences to discretize the algorithms so that these configurations can be computed numerically on a workstation. Numerical results are presented pictorially for the cases in which one of the constants \(K_1, K_2, K_3\) is either preponderant or negligible with the other two being equal, as well as for the materials PAA and MBBA. The stability of the hedgehog solution with respect to variations in each of the constants in discussed, and from the results presented it would seem that, in general, the classical relation (the Hélein condition) is not optimal. Finally, in an appendix, the authors establish the elliptic nature of the Euler equations related to this optimisation problem.

MSC:

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
76A15 Liquid crystals
49M25 Discrete approximations in optimal control
35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
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