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Local stability and a renormalized Newton Method for equilibrium liquid crystal director modeling

Gartland, Jr, E.C. and Ramage, Alison (2012) Local stability and a renormalized Newton Method for equilibrium liquid crystal director modeling. Working paper. University of Strathclyde, Glasgow.

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    Abstract

    We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange-Newton methods leads to problems with a double saddle-point form, for which we have previously proposed a preconditioned nullspace method as an effective solver [A. Ramage and E. C. Gartland, Jr., submitted]. The characterization of local stability of solutions is complicated by the double saddle-point structure, and here we develop efficiently computable criteria in terms of minimum eigenvalues of certain projected Schur complements. We also propose a modified outer iteration (“Renormalized Newton Method”) in which the orientation variables are normalized onto the constraint manifold at each iterative step. This scheme takes advantage of the special structure of these problems, and we prove that it is locally quadratically convergent. The Renormalized Newton Method bears some resemblance to the Truncated Newton Method of computational micromagnetics, and we compare and contrast the two.

    Item type: Monograph (Working paper)
    ID code: 40761
    Keywords: liquid crystals, reduced hessian method, saddle-point problems, unit-vector constraints, director models, local stability, renormalized newton method, equilibrium liquid crystal, director modeling, Mathematics
    Subjects: Science > Mathematics
    Department: Faculty of Science > Mathematics and Statistics
    Related URLs:
    Depositing user: Pure Administrator
    Date Deposited: 06 Aug 2012 15:56
    Last modified: 14 Dec 2012 12:03
    URI: http://strathprints.strath.ac.uk/id/eprint/40761

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