This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals.Three(alternating direction)minimization sch...This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals.Three(alternating direction)minimization schemes are applied to a structure preserving finite element discretization of the uniaxial model:a standard gradient descent method,the“heavy-ball”method,and Nesterov’s method.The performance of the schemes is measured in terms of the number of iterations required to obtain the equilibrium state,as well as the total computational time(wall time).The numerical experiments clearly show that the accelerated gradient descent schemes reduce the number of iterations and computational time significantly,despite the hard uniaxial constraint that is not“smooth”when defects are present.Moreover,our results show that accelerated schemes are not hindered when combined with an alternating direction minimization algorithm and are easy to implement.展开更多
Based on Landau–de Gennes theory and two-dimensional finite-difference iterative method, the spontaneous distortion in hybrid alignment nematic cells with M = ±1/2 disclination lines is investigated by establish...Based on Landau–de Gennes theory and two-dimensional finite-difference iterative method, the spontaneous distortion in hybrid alignment nematic cells with M = ±1/2 disclination lines is investigated by establishing two models. The fine structures of defect cores are described in the order space S^2/Z2. The joint action of elastic anisotropy（L2/L1） and biaxiality of defects induces the spontaneous twist distortion, accompanied by the movement of the defect center to the upper or lower plate. For each model, four mixed defect structures appear with the same energy, which are defined as energetically degenerated quadruple states.展开更多
The defect structures of s = ±1/2 twist disclinations in twisted nematic and twisted chiral liquid crystals have been investigated within the Landau-de Gennes theory numerically. Our results show that there exist...The defect structures of s = ±1/2 twist disclinations in twisted nematic and twisted chiral liquid crystals have been investigated within the Landau-de Gennes theory numerically. Our results show that there exists eigenvalue exchange across the defect core of both the two models. The defect core is essentially biaxial and never isotropic. The defect centre is uniaxial and is surrounded by a strong biaxial region.展开更多
We consider the stability of a specific nematic liquid crystal configuration under an applied magnetic field. We show that for some specific configuration there exist two critical values Hn and Hsh of applied magnetic...We consider the stability of a specific nematic liquid crystal configuration under an applied magnetic field. We show that for some specific configuration there exist two critical values Hn and Hsh of applied magnetic field. When the intensity of the magnetic field is smaller than Hn, the configuration of the energy is only global minimizer, when the intensity is between Hn and Hsh, the configuration is not global minimizer, but is weakly stable, and when the intensity is larger than Hsh, the configuration is instable. Moreover, we also examine the asymptotic behavior of the global minimizer as the intensity tends to the infinity.展开更多
基金supported in part by NSF grant DMS-1555222(CAREER).
文摘This paper illustrates the efficacy of using accelerated gradient descent schemes for minimizing a uniaxially constrained Landau-de Gennes model for nematic liquid crystals.Three(alternating direction)minimization schemes are applied to a structure preserving finite element discretization of the uniaxial model:a standard gradient descent method,the“heavy-ball”method,and Nesterov’s method.The performance of the schemes is measured in terms of the number of iterations required to obtain the equilibrium state,as well as the total computational time(wall time).The numerical experiments clearly show that the accelerated gradient descent schemes reduce the number of iterations and computational time significantly,despite the hard uniaxial constraint that is not“smooth”when defects are present.Moreover,our results show that accelerated schemes are not hindered when combined with an alternating direction minimization algorithm and are easy to implement.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11374087 and 11447179)the Key Subject Construction Project of Hebei Province University,China
文摘Based on Landau–de Gennes theory and two-dimensional finite-difference iterative method, the spontaneous distortion in hybrid alignment nematic cells with M = ±1/2 disclination lines is investigated by establishing two models. The fine structures of defect cores are described in the order space S^2/Z2. The joint action of elastic anisotropy（L2/L1） and biaxiality of defects induces the spontaneous twist distortion, accompanied by the movement of the defect center to the upper or lower plate. For each model, four mixed defect structures appear with the same energy, which are defined as energetically degenerated quadruple states.
文摘The defect structures of s = ±1/2 twist disclinations in twisted nematic and twisted chiral liquid crystals have been investigated within the Landau-de Gennes theory numerically. Our results show that there exists eigenvalue exchange across the defect core of both the two models. The defect core is essentially biaxial and never isotropic. The defect centre is uniaxial and is surrounded by a strong biaxial region.
文摘We consider the stability of a specific nematic liquid crystal configuration under an applied magnetic field. We show that for some specific configuration there exist two critical values Hn and Hsh of applied magnetic field. When the intensity of the magnetic field is smaller than Hn, the configuration of the energy is only global minimizer, when the intensity is between Hn and Hsh, the configuration is not global minimizer, but is weakly stable, and when the intensity is larger than Hsh, the configuration is instable. Moreover, we also examine the asymptotic behavior of the global minimizer as the intensity tends to the infinity.
