TY - JOUR

T1 - On the use of local maximum entropy approximants for Cahn–Hilliard phase-field models in 2D domains and on surfaces

AU - Amiri, Fatemeh

AU - Ziaei-Rad, Saeed

AU - Valizadeh, Navid

AU - Rabczuk, Timon

N1 - Funding Information:
Fatemeh Amiri and Saeed Ziaei-Rad would like to acknowledge the programme ‘Cooperation with Iranian non-resident scientists and experts’ by Iran’s National Elites Foundation for the financial support during the duration of this project. Navid Valizadeh and Timon Rabczuk gratefully acknowledge the support of the German Research Foundation (DFG project number 405890576 ).

PY - 2019/4/1

Y1 - 2019/4/1

N2 - 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.

AB - 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.

KW - Cahn–Hilliard equation

KW - Dimensionality reduction methods

KW - Local maximum entropy

KW - Nonlinear manifold learning method

KW - Phase-field models

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U2 - 10.1016/j.cma.2018.11.023

DO - 10.1016/j.cma.2018.11.023

M3 - Article

AN - SCOPUS:85058373531

VL - 346

SP - 1

EP - 24

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

ER -