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

F. Amiri, S. Ziaei-Rad, N. Valizadeh and T. Rabczuk

Computer Methods in Applied Mechanics and Engineering, Volume 346, December 2018, Pages 1-24

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 is 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 techniques that allows dealing with general point set surfaces avoiding a global parametrization, 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.

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