Phase-field approximations of the Willmore functional and flow

Elie Bretin; Simon Masnou; Edouard Oudet

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arXiv · 1305.5435

Phase-field approximations of the Willmore functional and flow

Elie Bretin·Simon Masnou·Edouard Oudet

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Abstract

We discuss in this paper phase-field approximations of the Willmore functional and the associated L2-flow. After recollecting known results on the approximation of the Willmore energy and its L1-relaxation, we derive the expression of the flows associated with various approximations, and we show their behavior by formal arguments based on matched asymptotic expansions. We introduce an accurate numerical scheme, whose local convergence can be proved, to describe with more details the behavior of two flows, the classical and the flow associated with an approximation model due to Mugnai. We propose a series of numerical simulations in 2D and 3D to illustrate their behavior in both smooth and singular situations.

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Elie Bretin, Simon Masnou, Edouard Oudet. 2013-05-23. Phase-field approximations of the Willmore functional and flow. https://arxiv.org/abs/1305.5435

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