|
|
向列相液晶流的一种二阶全离散格式
|
Abstract:
在本文中，我们对向列相液晶流的Ginzburg-Landau模型提出了一种二阶、线性、耦合的格式，证明了该格式在离散条件下的能量稳定性，最后，通过数值模拟展示了四奇异点和旋转流的湮没过程，并且验证了格式的数值精度。结果表明：该格式具有能量稳定性，且具有比较好的数值精度。
In this paper, we mainly propose and analyze a second-order, linear, coupled scheme for the Ginzburg-Landau model of the nematic liquid crystal flow, and prove its energy stability under discrete condition. Finally, we demonstrate the annihilation process of four singularities and rotating flows through numerical simulations, and verify the numerical accuracy of the scheme. The results show that the scheme has energy stability and good numerical accuracy.
| [1] | Oseen, C.W. (1933) The Theory of Liquid Crystals. Transactions of the Faraday Society, 29, 883-899.
https://doi.org/10.1039/tf9332900883 |
| [2] | Frank, F.C. (1958) Liquid Crystals. On the Theory of Liquid Crystals. Discussions of the Faraday Society, 25, 19-28.
https://doi.org/10.1039/df9582500019 |
| [3] | Adler, J.H., Atherton, T.J., Benson, T.R., Emerson, D.B. and Maclachlan, S.P. (2015) Energy Minimization for Liquid Crystal Equilibrium with Electric and Flexoelectric Effects. SIAM Journal on Scientific Computing, 37, S157-S176.
https://doi.org/10.1137/140975036 |
| [4] | Adler, J.H., Atherton, T.J., Emerson, D.B. and Maclachlan, S.P. (2015) An Energy-Minimization Finite-Element Approach for the Frank-Oseen Model of Nematic Liquid Crystals. SIAM Journal on Numerical Analysis, 53, 2226-2254.
https://doi.org/10.1137/140956567 |
| [5] | de Gennes, P.G. (1975) The Physics of Liquid Crystals. Physics Today, 28, 54-55. https://doi.org/10.1063/1.3069010 |
| [6] | Zhao, J. and Wang, Q. (2016) Semi-Discrete Energy-Stable Schemes for a Tensor-Based Hydrodynamic Model of Nematic Liquid Crystal Flows. Journal of Scientific Computing, 68, 1241-1266.
https://doi.org/10.1007/s10915-016-0177-x |
| [7] | Zhao, J., Yang, X., Gong, Y. and Wang, Q. (2017) A Novel Linear Second Order Unconditionally Energy Stable Scheme for a Hydrodynamic Q-Tensor Model of Liquid Crystals. Computer Methods in Applied Mechanics and Engineering, 318, 803-825. https://doi.org/10.1016/j.cma.2017.01.031 |
| [8] | Cai, Y., Shen, J. and Xu, X. (2017) A Stable Scheme and Its Convergence Analysis for a 2D Dynamic Q-Tensor Model of Nematic Liquid Crystals. Mathematical Models & Methods in Applied Sciences, 27, 1459-1488.
https://doi.org/10.1142/S0218202517500245 |
| [9] | Ericksen, J.L. (1961) Conservation Laws for Liquid Crystals. Transactions of the Society of Rheology, 5, 23-34.
https://doi.org/10.1122/1.548883 |
| [10] | Leslie, F.M. (1992) Continuum Theory for Nematic Liquid Crystals. Continuum Mechanics & Thermodynamics, 4, 167-175. https://doi.org/10.1007/BF01130288 |
| [11] | Ericksen, J.L. (1991) Liquid Crystals with Variable Degree of Orientation. Archive for Rational Mechanics and Analysis, 113, 97-120. https://doi.org/10.1007/BF00380413 |
| [12] | Badia, S., Guillen-Gonzalez, F. and Gutierrez-Santacreu, J.V. (2011) An Overview on Numerical Analyses of Nematic Liquid Crystal Flows. Archives of Computational Methods in Engineering, 18, Article No. 285.
https://doi.org/10.1007/s11831-011-9061-x |
| [13] | Becker, R., Feng, X. and Prohl, A. (2008) Finite Element Approximations of the Ericksen-Leslie Model for Nematic Liquid Crystal Flow. SIAM Journal on Numerical Analysis, 46, 1704-1731. https://doi.org/10.1137/07068254X |
| [14] | Badia, S. and Gutierrez-Santacreu, J.V. (2011) Finite Element Approximation of Nematic Liquid Crystal Flows Using Saddle-Point Structure. Journal of Computational Physics, 230, 1686-1706.
https://doi.org/10.1016/j.jcp.2010.11.033 |
| [15] | Labovsky, A., Layton, W.J., Manica, C.C., Neda, M. and Rebholz, L.G. (2009) The Stabilized Extrapolated Trapezoidal Finite-Element Method for the Navier-Stokes Equations. Computer Methods in Applied Mechanics and Engineering, 198, 958-974. https://doi.org/10.1016/j.cma.2008.11.004 |
| [16] | Guillen-Gonzalez, F.M. and Gutierrez-Santacreu, J.V. (2013) A Linear Mixed Finite Element Scheme for a Nematic Ericksen-Leslie Liquid Crystal Model. ESAIM: Mathematical Modelling and Numerical Analysis, 47, 1433-1464.
https://doi.org/10.1051/m2an/2013076 |
| [17] | Cabrales, R.C., Guillen-Gonzalez, F. and Gutierrez-Santacreu, J.V. (2015) A Time-Splitting Finite-Element Stable Approximation for the Ericksen-Leslie Equations. SIAM Journal on Scientific Computing, 37, B261-B282.
https://doi.org/10.1137/140960979 |
| [18] | Liu, C. and Walkington, N.J. (2000) Approximation of Liquid Crystal Flows. SIAM Journal on Numerical Analysis, 37, 725-741. https://doi.org/10.1137/S0036142997327282 |
| [19] | Ping, L., Liu, C. and Hui, Z. (2007) An Energy Law Preserving C0 Finite Element Scheme for Simulating the Kinematic Effects in Liquid Crystal Dynamics. Journal of Computational Physics, 227, 1411-1427.
https://doi.org/10.1016/j.jcp.2007.09.005 |
