[Paper Review] A Deflation Technique for Detecting Multiple Liquid Crystal Equilibrium States

James H. Adler, D. B. Emerson+2 more|arXiv (Cornell University)|Jan 27, 2016

Liquid Crystal Research Advancements29 references3 citations

TL;DR

This paper introduces a deflation technique integrated with nested iteration and multigrid solvers to systematically detect multiple distinct equilibrium configurations in nematic and cholesteric liquid crystals using the Frank-Oseen free-energy model. The method efficiently locates local and global minimizers, including complex defect structures and helical phases, by modifying the nonlinear system to exclude previously found solutions while preserving finite-element sparsity and enabling reuse of existing multigrid preconditioners.

ABSTRACT

Multiple equilibrium states arise in many physical systems, including various types of liquid crystal structures. Having the ability to reliably compute such states enables more accurate physical analysis and understanding of experimental behavior. This paper adapts and extends a deflation technique for the computation of multiple distinct solutions arising in the context of modeling equilibrium configurations of nematic and cholesteric liquid crystals. The deflation method is applied as part of an overall free-energy variational approach and is modified to fit the framework of optimization of a functional with pointwise constraints. It is shown that multigrid methods designed for the undeflated systems may be applied to efficiently solve the linear systems arising in the application of deflation. For the numerical algorithm, the deflation approach is interwoven with nested iteration, creating a dynamic and efficient method that further enables the discovery of distinct solutions. Finally, four numerical experiments are performed demonstrating the efficacy and accuracy of the algorithm in detecting important physical phenomena, including bifurcation and disclination behaviors. The final numerical experiment expands the algorithm to model cholesteric liquid crystals and illustrates the full discovery power of the deflation process.

Motivation & Objective

• To develop a robust numerical method for detecting multiple distinct equilibrium states in liquid crystal systems, which are often missed by standard minimization techniques due to multiple local extrema.
• To adapt the deflation technique—originally for PDEs and complementarity problems—to the constrained variational framework of liquid crystal free-energy minimization with pointwise unit-length constraints on the director field.
• To integrate deflation with nested iteration and existing multigrid solvers to enhance computational efficiency and convergence robustness in large-scale simulations.
• To demonstrate the method’s capability in resolving complex physical phenomena such as bifurcations, disclinations, and chiral helical structures in both nematic and cholesteric phases.
• To enable accurate and systematic discovery of energetically favorable configurations, including those with high energy barriers, through dynamic solution search.

Proposed method

• The deflation technique modifies the nonlinear system by introducing a pole-like correction term that suppresses previously computed solutions, enabling sequential discovery of distinct equilibrium states.
• The method is embedded within a finite-element variational formulation of the Frank-Oseen free-energy model, subject to the pointwise constraint ||n|| = 1.
• Deflated linear systems arising in Newton iterations are solved using multigrid methods originally designed for the undeflated system, preserving sparsity and convergence rates.
• Nested iteration is interwoven with deflation to dynamically refine initial guesses at each mesh level, improving convergence and reducing computational cost.
• The approach uses Braess-Sarazin relaxation within multigrid solvers and tracks solution progress across a hierarchy of grids to enhance robustness and efficiency.
• A generalized tunneling strategy is considered for future work to overcome energy barriers between separated minima valleys.

Experimental results

Research questions

• RQ1 Can deflation be effectively adapted to the constrained optimization framework of liquid crystal free-energy minimization with unit-length director constraints?
• RQ2 How can deflation be integrated with nested iteration and multigrid solvers to maintain computational efficiency while discovering multiple solutions?
• RQ3 To what extent can the deflation method resolve complex physical phenomena such as disclinations and helical structures in nematic and cholesteric liquid crystals?
• RQ4 How does the method perform in locating solutions with high energy barriers, particularly when initial guesses are far from the true minimum?
• RQ5 Can the deflation process be enhanced with adaptive or dynamic initial guess strategies to improve convergence in challenging configurations?

Key findings

• The deflation method successfully locates multiple distinct equilibrium states in nematic liquid crystals, including configurations with disclinations and bifurcated structures, even when standard solvers converge to only one solution.
• For the cholesteric system, the method discovered a left-handed helical configuration with a 2π rotation about the y-axis, achieving a free energy of 2.984×10⁻⁸, matching analytical predictions.
• On a 256×256 grid, the algorithm required only 253 Newton iterations to discover the final solution, with 63 iterations attributed to the deflation process, indicating effective solution isolation.
• The average multigrid iteration count on the finest grid for the deflated system was 63.0, demonstrating that multigrid convergence remains robust despite deflation-induced modifications.
• The method achieved a work unit (WU) count of 493.0 for the cholesteric solution, significantly higher than the 100.7 for the first solution, reflecting the computational cost of discovering multiple states.
• The algorithm demonstrated high accuracy and efficiency, with free energy values converging to within 10⁻⁸ of analytical benchmarks, confirming the reliability of the computed equilibria.