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 ).
Publisher Copyright:
© 2018 Elsevier B.V.
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
UR - http://www.scopus.com/inward/record.url?scp=85058373531&partnerID=8YFLogxK
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 -