### 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 C^{1}-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 C^{0}-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 language | English |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 346 |

DOIs | |

Publication status | Published - 2019 Apr 1 |

### 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

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## Cite this

*Computer Methods in Applied Mechanics and Engineering*,

*346*, 1-24. https://doi.org/10.1016/j.cma.2018.11.023