Shape optimization involving the Tresca friction law in a 2D linear elastic model
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910699731550208 |
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| author | Bourdin, Loïc Caubet, Fabien de Cordemoy, Aymeric Jacob |
| author_facet | Bourdin, Loïc Caubet, Fabien de Cordemoy, Aymeric Jacob |
| contents | The aim of this work is to analyse a shape optimization problem in a mechanical friction context. Precisely we perform a shape sensitivity analysis of a Tresca friction problem, that is, a boundary value problem involving the usual linear elasticity equations together with the (nonsmooth) Tresca friction law on a part of the boundary. We prove that the solution to the Tresca friction problem admits a directional shape derivative which moreover coincides with the solution to a boundary value problem involving tangential Signorini's unilateral conditions. Then an explicit expression of the shape gradient of the Tresca energy functional is provided (which allows us to provide numerical simulations illustrating our theoretical results). Our methodology is not based on any regularization procedure, but rather on the twice epi-differentiability of the (nonsmooth) Tresca friction functional which is analyzed thanks to a change of variables which is well-suited in the two-dimensional case. The obstruction in the higher-dimensional case is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10158 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shape optimization involving the Tresca friction law in a 2D linear elastic model Bourdin, Loïc Caubet, Fabien de Cordemoy, Aymeric Jacob Optimization and Control Analysis of PDEs 49Q10, 49Q12, 49J40, 74M10, 74M15, 74P10 The aim of this work is to analyse a shape optimization problem in a mechanical friction context. Precisely we perform a shape sensitivity analysis of a Tresca friction problem, that is, a boundary value problem involving the usual linear elasticity equations together with the (nonsmooth) Tresca friction law on a part of the boundary. We prove that the solution to the Tresca friction problem admits a directional shape derivative which moreover coincides with the solution to a boundary value problem involving tangential Signorini's unilateral conditions. Then an explicit expression of the shape gradient of the Tresca energy functional is provided (which allows us to provide numerical simulations illustrating our theoretical results). Our methodology is not based on any regularization procedure, but rather on the twice epi-differentiability of the (nonsmooth) Tresca friction functional which is analyzed thanks to a change of variables which is well-suited in the two-dimensional case. The obstruction in the higher-dimensional case is discussed. |
| title | Shape optimization involving the Tresca friction law in a 2D linear elastic model |
| topic | Optimization and Control Analysis of PDEs 49Q10, 49Q12, 49J40, 74M10, 74M15, 74P10 |
| url | https://arxiv.org/abs/2411.10158 |