Projection-Free Method for the Full Frank-Oseen Model of Liquid Crystals
CoRR(2024)
摘要
Liquid crystals are materials that experience an intermediate phase where the
material can flow like a liquid, but the molecules maintain an orientation
order. The Frank-Oseen model is a continuum model of a liquid crystal. The
model represents the liquid crystal orientation as a vector field and posits
that the vector field minimizes some elastic energy subject to a pointwise unit
length constraint, which is a nonconvex constraint. Previous numerical methods
in the literature assumed restrictions on the physical constants or had
regularity assumptions that ruled out point defects, which are important
physical phenomena to model. We present a finite element discretization of the
full Frank-Oseen model and a projection free gradient flow algorithm for the
discrete problem in the spirit of Bartels (2016). We prove Gamma-convergence of
the discrete to the continuous problem: weak convergence of subsequences of
discrete minimizers and convergence of energies. We also prove that the
gradient flow algorithm has a desirable energy decrease property. Our analysis
only requires that the physical constants are positive, which presents
challenges due to the additional nonlinearities from the elastic energy.
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