Meshfree Solution of Q-tensor Equations of Nematostatics Using the MLPG Method

• Published in: Computer modeling in engineering & sciences Vol. 13; no. 2; pp. 91 - 102
• Main Authors: Pecher, Radek, Elston, Steve, Raynes, Peter
• Format: Journal Article
• Language: English
• Published: Henderson Tech Science Press 2006
• Subjects: Crystal defects; Energy conservation; Finite difference method; Finite element method; Galerkin method; Liquid crystals; Mathematical models; Meshless methods; Nematic crystals; Partial differential equations; Tensors
• ISSN: 1526-1492; 1526-1506
• DOI: 10.3970/cmes.2006.013.091

Meshfree techniques for solving partial differential equations in physics and engineering are a powerful new alternative to the traditional mesh-based techniques, such as the finite difference method or the finite element method. The elimination of the domain mesh enables, among other benefits, more efficient solutions of nonlinear and multi-scale problems. One particular example of these kinds of problems is a Q-tensor based model of nematic liquid crystals involving topological defects. This paper presents the first application of the meshless local Petrov-Galerkin method to solving the Q-tensor equations of nematostatics. The theoretical part introduces the Landau -- de Gennes free-energy functional and its meshfree minimisation subject to the given boundary constraints. The theory is followed by two example models with simple distortion profiles, including a twisted chiral nematic. The resulting profiles exhibit large local gradients and a high degree of continuity even for few semi-regularly distributed nodes, indicating the high accuracy of the meshfree approach used.