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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2412.15926 |
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| _version_ | 1866911032754044928 |
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| author | Bretin, Elie Chambolle, Antonin Masnou, Simon |
| author_facet | Bretin, Elie Chambolle, Antonin Masnou, Simon |
| contents | We investigate a new phase field model for representing non-oriented interfaces, approximating their area and simulating their area-minimizing flow. Our contribution is related to the approach proposed in arXiv:2105.09627 that involves ad hoc neural networks. We show here that, instead of neural networks, similar results can be obtained using a more standard variational approach that combines a Cahn-Hilliard-type functional involving an appropriate non-smooth potential and a Willmore-type stabilization energy. We give a $Γ$-convergence analysis of this phase field model in dimension $1$ and, for radially symmetric functions, in arbitrary dimension. We also propose a simple numerical scheme to approximate its $L^2$-gradient flow. We illustrate numerically that the new flow approximates fairly well the mean curvature flow of codimension $1$ or $2$ interfaces in dimensions $2$ and $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15926 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Cahn--Hilliard--Willmore phase field model for non-oriented interfaces Bretin, Elie Chambolle, Antonin Masnou, Simon Optimization and Control 74N20, 35A35, 53E10, 53E40, 65M32, 35A15 We investigate a new phase field model for representing non-oriented interfaces, approximating their area and simulating their area-minimizing flow. Our contribution is related to the approach proposed in arXiv:2105.09627 that involves ad hoc neural networks. We show here that, instead of neural networks, similar results can be obtained using a more standard variational approach that combines a Cahn-Hilliard-type functional involving an appropriate non-smooth potential and a Willmore-type stabilization energy. We give a $Γ$-convergence analysis of this phase field model in dimension $1$ and, for radially symmetric functions, in arbitrary dimension. We also propose a simple numerical scheme to approximate its $L^2$-gradient flow. We illustrate numerically that the new flow approximates fairly well the mean curvature flow of codimension $1$ or $2$ interfaces in dimensions $2$ and $3$. |
| title | A Cahn--Hilliard--Willmore phase field model for non-oriented interfaces |
| topic | Optimization and Control 74N20, 35A35, 53E10, 53E40, 65M32, 35A15 |
| url | https://arxiv.org/abs/2412.15926 |