An energy law preserving C finite element scheme for simulating the kinematic effects in liquid crystal dynamics |
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Authors: | Ping Lin Chun Liu Hui Zhang |
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Affiliation: | aDepartment of Mathematics, The National University of Singapore, Singapore 117543, Singapore;bDepartment of Mathematics, Pennsylvania State University, University Park, PA 18601, USA;cSchool of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR China |
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Abstract: | In this paper, we use finite element methods to simulate the hydrodynamical systems governing the motions of nematic liquid crystals in a bounded domain Ω. 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 W1,2+σ(Ω) (σ > 0 is an arbitrarily small number). This enables us to use convenient conformal C0 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. |
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Keywords: | Liquid crystal flow Non-Newtonian fluids C0 finite element approximation Discrete energy law Singularity dynamics |
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