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An energy law preserving \(C^0\) finite element scheme for simulating the kinematic effects in liquid crystal dynamics. (English) Zbl 1133.65077

Authors’ summary: We use finite element methods to simulate the hydrodynamical systems governing the motions of nematic liquid crystals in a bounded domain \(\Omega \). We reformulate the original model in the weak form which is consistent with the continuous dissipative energy law for the flow and director fields in \(\mathbf W^{1,2+\sigma} (\Omega )\) \((\sigma > 0\) is an arbitrarily small number). This enables us to use convenient conformal \(C^0\) finite elements in solving the problem. Moreover, a discrete energy law is derived for a modified midpoint time discretization scheme. A fixed iterative method is used to solve the resulted nonlinear system so that a matrix free time evolution may be achieved and velocity and director variables may be solved separately. A number of hydrodynamical liquid crystal examples are computed to demonstrate the effects of the parameters and the performance of the method.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76A15 Liquid crystals
82D25 Statistical mechanics of crystals
82-08 Computational methods (statistical mechanics) (MSC2010)
76A05 Non-Newtonian fluids

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References:

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