On the use of local maximum entropy approximants for Cahn–Hilliard phase-field models in 2D domains and on surfaces

Fatemeh Amiri, Saeed Ziaei-Rad, Navid Valizadeh, Timon Rabczuk

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

We apply a local maximum entropy (LME) approximation scheme to the fourth order phase-field model for the traditional Cahn–Hilliard theory. The discretization of the Cahn–Hilliard equation by Galerkin method requires at least C1-continuous basis functions. This requirement can be fulfilled by using LME shape functions which are C-continuous. In this case, the primal variational formulations of the fourth-order partial differential equation are well defined and integrable. Hence, there is no need to split the fourth-order partial differential equation into two second-order partial differential equations; this splitting scheme is a common practice in mixed finite element formulations with C0-continuous Lagrange shape functions. Furthermore, we use a general and simple numerical method such as statistical manifold learning technique that allows dealing with general point set surfaces avoiding a global parameterization, which can be applied to tackle surfaces of complex geometry and topology, and to solve Cahn–Hilliard equation on general surfaces. Finally, the flexibility and robustness of the presented methodology is demonstrated for several representative numerical examples.

Original languageEnglish
Pages (from-to)1-24
Number of pages24
JournalComputer Methods in Applied Mechanics and Engineering
Volume346
DOIs
Publication statusPublished - 2019 Apr 1

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partial differential equations
Partial differential equations
Entropy
shape functions
entropy
formulations
Galerkin method
Galerkin methods
Parameterization
parameterization
learning
Numerical methods
flexibility
topology
Topology
methodology
requirements
Geometry
geometry
approximation

Keywords

  • Cahn–Hilliard equation
  • Dimensionality reduction methods
  • Local maximum entropy
  • Nonlinear manifold learning method
  • Phase-field models

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications

Cite this

On the use of local maximum entropy approximants for Cahn–Hilliard phase-field models in 2D domains and on surfaces. / Amiri, Fatemeh; Ziaei-Rad, Saeed; Valizadeh, Navid; Rabczuk, Timon.

In: Computer Methods in Applied Mechanics and Engineering, Vol. 346, 01.04.2019, p. 1-24.

Research output: Contribution to journalArticle

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