International Conference «Mathematical and Information Technologies, MIT-2016»

28 August – 5 September 2016

Vrnjacka Banja, Serbia – Budva, Montenegro

Smolekho I.V.   Sadovskaya O.V.   Sadovskii V.M.  

Numerical Analysis of Acoustic Waves in a Liquid Crystal Taking into Account Couple-Stress Interaction

Reporter: Smolekho I.V.

Liquid crystals are unique materials because of unusual combination of the properties of elastic anisotropy, inherent to crystals, and fluidity, typical for liquids. A material transforms into the state of a liquid crystal in a certain temperature range under weak external perturbations. The mobility of the liquid crystal molecules allow external forces to change the orientation of crystals and, thus, to control their properties.

One of the approaches to the construction of a mathematical model to describe the behavior of liquid crystals is based on the representation of a liquid crystal medium as a fine-dispersed continuum. At each point of this continuum, the domains of a liquid crystal can move in accordance with laws of the dynamics of viscous or inviscid liquid and can rotate relative to a liquid, encountering resistance to rotation.

In the framework of acoustic approximation, the model of a liquid crystal without taking into account the couple stresses is described in [1, 2]. The system of equations of this model includes the equations of translational and rotational motion, the equation for the angle of rotation, the constitutive equations for pressure and tangential stress, as well as the equation of anisotropic heat conduction with variable coefficients. Parallel computational algorithm for the solution of this system is represented in [3, 4].

The present work is devoted to numerical solution of the second-order differential equations for tangential stress and angular velocity. These equations are derived from the system of equations describing the thermomechanical behavior of a liquid crystal taking into account the couple-stress interactions in two-dimensional case. Computational algorithm for numerical solution of the system of equations of the second order under given initial data and boundary conditions is worked out. The explicit finite-difference scheme “cross” of the second-order approximation is used. The stability condition for this scheme is obtained. The algorithm is implemented as a parallel program in the С language using the CUDA technology for computer systems with graphic accelerators. Computations are performed on the GPU, which is a coprocessor to the CPU. Graphical device consists of a large number of threads, each of which is associated with a mesh of the difference grid. In parallel mode, the threads of a graphic device perform operations of the same type in the meshes of grid on the calculation of solution at each time step.

A series of numerical calculations was carried out on the high-performance computational server Flagman with eight graphic solvers Tesla C2050 (448 CUDA cores on each GPU) of the Institute of Computational Modeling SB RAS to demonstrate the efficiency of proposed parallel program.

This work was partially supported by the Russian Foundation for Basic Research (grants no. 14-01-00130, 16-31-00078) and the Complex Fundamental Research Program no. 18 “Algorithms and Software for Computational Systems of Superhigh Productivity” of the Presidium of RAS.


[1] V. M. Sadovskii and O. V. Sadovskaya, “On the Acoustic Approximation of Thermomechanical Description of a Liquid Crystal”, Phys. Mesomech., 16(4), 310–316 (2013).

[2] V. M. Sadovskii, “Equations of the Dynamics of a Liquid Crystal under the Influence of Weak Mechanical and Thermal Perturbations”, AIP Conf. Proc., 1629, 311–318 (2014).

[3] O. V. Sadovskaya, “Numerical Simulation of the Dynamics of a Liquid Crystal in the Case of Plane Strain Using GPUs”, AIP Conf. Proc., 1629, 303–310 (2014).

[4] I. V. Smolekho, “Parallel Implementation of the Algorithm for Description of Thermoelastic Waves in Liquid Crystals”, Young Scientist, 11(91), 107–112 (2015) [in Russian].

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