A moving mesh method for modelling defects in nematic liquid crystals

Modelling and computation of liquid crystals

“…Gradient flow is a dynamics driven by a free energy. There are quite a number of works devoted to obtaining the defect patterns by solving the gradient flow equation corresponding to different LC systems (Fukuda et al 2004, Ravnik and Žumer 2009, MacDonald et al 2020. For the LdG theory, the corresponding L 2 -gradient flow equation can be written as…”

“…One is the energy-minimization-based approach (Cohen et al 1987, Alouges 1997, Adler, Atherton, Emerson and MacLachlan 2015, Nochetto, Walker and Zhang 2017, Majumdar and Wang 2018, Gartland Jr, Palffy-Muhoray and Varga 1991, which is often numerically solved by Newton-type or quasi-Newton methods. The other approach is to follow the gradient flow dynamics driven by the free energy corresponding to specific models of LCs (Fukuda, Stark, Yoneya and Yokoyama 2004, Ravnik and Žumer 2009, Canevari, Majumdar and Spicer 2017, Wang, Canevari and Majumdar 2019, MacDonald, Mackenzie and Ramage 2020. Various efficient numerical methods have been developed to solve the gradient flow equations, including energy-stable numerical schemes such as convex splitting methods (Elliott and Stuart 1993), invariant energy quadratization methods (Yang 2016) and scalar auxiliary variable methods (Shen, Xu and Yang 2018).…”

Modeling and Computation of Liquid Crystals

“…Gradient flow is a dynamics driven by a free energy. There are quite a number of works devoted to obtain the defect patterns by solving the gradient flow equation corresponding to different LC systems (Fukuda et al 2004, Ravnik and Žumer 2009, MacDonald et al 2020. For the LdG theory, the corresponding L 2 -gradient flow equation can be written as…”

“…One is the energy-minimization based approach (Cohen, Hardt, Kinderlehrer, Lin and Luskin 1987, Alouges 1997, Adler, Atherton, Emerson and MacLachlan 2015, Nochetto, Walker and Zhang 2017, Majumdar and Wang 2018, Gartland Jr, Palffy-Muhoray and Varga 1991, which is often numerically solved by Newton-type or quasi-Newton method. The other approach is to follow the gradient flow dynamics driven by the free energy corresponding to individual model of LCs (Fukuda, Stark, Yoneya and Yokoyama 2004, Ravnik and Žumer 2009, Canevari, Majumdar and Spicer 2017, Wang, Canevari and Majumdar 2019, MacDonald, Mackenzie and Ramage 2020. Various efficient numerical methods have been developed to solve the gradient flow equations, including energy stable numerical schemes such as convex splitting method (Elliott and Stuart 1993), invariant energy quadratization method (Yang 2016), scalar auxiliary variable method , etc.…”