A Renormalized Newton Method for Liquid Crystal Director Modeling

Eugene C. Gartland, Jr, Alison Ramage

Research output: Working paper

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 linear systems 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., SIAM J. Sci. Comput., 35 (2013), pp. B226–B247]. Here we propose and analyze 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. This brings to light some anomalies of the Truncated Newton Method.
LanguageEnglish
Place of PublicationGlasgow
PublisherUniversity of Strathclyde
Pages1-27
Number of pages27
Publication statusPublished - Dec 2014

Fingerprint

Truncated Newton Method
Newton Methods
Liquid Crystal
Modeling
Micromagnetics
Lagrange Method
Variational Model
Nonlinear Systems of Equations
Unit vector
Saddlepoint
Anomaly
Electric Field
Discretization
Linear Systems
Numerical Solution
Prototype
Iteration

Keywords

  • liquid crystals
  • director models
  • unit-vector constraints
  • saddle-point problems
  • reduced hessian method
  • renormalized newton method

Cite this

Gartland, Jr, E. C., & Ramage, A. (2014). A Renormalized Newton Method for Liquid Crystal Director Modeling. (pp. 1-27). Glasgow: University of Strathclyde.
Gartland, Jr, Eugene C. ; Ramage, Alison. / A Renormalized Newton Method for Liquid Crystal Director Modeling. Glasgow : University of Strathclyde, 2014. pp. 1-27
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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 linear systems 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., SIAM J. Sci. Comput., 35 (2013), pp. B226–B247]. Here we propose and analyze 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. This brings to light some anomalies of the Truncated Newton Method.",
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Gartland, Jr, EC & Ramage, A 2014 'A Renormalized Newton Method for Liquid Crystal Director Modeling' University of Strathclyde, Glasgow, pp. 1-27.

A Renormalized Newton Method for Liquid Crystal Director Modeling. / Gartland, Jr, Eugene C.; Ramage, Alison.

Glasgow : University of Strathclyde, 2014. p. 1-27.

Research output: Working paper

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N2 - 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 linear systems 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., SIAM J. Sci. Comput., 35 (2013), pp. B226–B247]. Here we propose and analyze 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. This brings to light some anomalies of the Truncated Newton Method.

AB - 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 linear systems 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., SIAM J. Sci. Comput., 35 (2013), pp. B226–B247]. Here we propose and analyze 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. This brings to light some anomalies of the Truncated Newton Method.

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KW - director models

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KW - renormalized newton method

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Gartland, Jr EC, Ramage A. A Renormalized Newton Method for Liquid Crystal Director Modeling. Glasgow: University of Strathclyde. 2014 Dec, p. 1-27.