A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency

TL;DR

This paper develops a 3D mesh adaptivity method for minimizing the Landau-de Gennes free energy in nematic liquid crystals, analyzing its efficiency and impact on finite element simulations.

Contribution

It introduces a novel 3D mesh adaptivity scheme driven by a posteriori error estimates for Landau-de Gennes minimization, enhancing simulation accuracy and computational performance.

Findings

Algorithm convergence time varies with parameter choices.

Mesh adaptivity improves simulation accuracy.

Enhanced open source finite element software with 3D meshing.

Abstract

This paper presents a 3D mesh adaptivity strategy on unstructured tetrahedral meshes by a posteriori error estimates based on metrics, studied on the case of a nonlinear finite element minimization scheme for the Landau-de Gennes free energy functional of nematic liquid crystals. Newton's iteration for tensor fields is employed with steepest descent method possibly stepping in. Aspects relating the driving of mesh adaptivity within the nonlinear scheme are considered. The algorithmic performance is found to depend on at least two factors: when to trigger each single mesh adaptation, and the precision of the correlated remeshing. Each factor is represented by a parameter, with its values possibly varying for every new mesh adaptation. We empirically show that the time of the overall algorithm convergence can vary considerably when different sequences of parameters are used, thus posing a question about optimality. The extensive testings and debugging done within this work on the simulation of systems of nematic colloids substantially contributed to the upgrade of an open source finite element-oriented programming language to its 3D meshing possibilities, as also to an outer 3D remeshing module.