{"title":"对tresca型问题的敏感性分析导致了Signorini条件","authors":"S. Adly, L. Bourdin, F. Caubet","doi":"10.1051/cocv/2022025","DOIUrl":null,"url":null,"abstract":"The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from the (vectorial) Tresca friction problem found in contact mechanics. The weak formulation of the considered problem leads to a variational inequality of the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex subdifferential of the friction functional on the Sobolev space H^1(Omega) and show that all its subgradients satisfy a PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the twice epi-differentiability, concept introduced and thoroughly studied by R.T. Rockafellar,\nwe show the differentiability of the solution to the parameterized Tresca-type problem and that its derivative satisfies a Signorini-type problem. Some numerical simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first time that the concept of twice epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2022-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Sensitivity analysis of a Tresca-type problem leads to Signorini's conditions\",\"authors\":\"S. Adly, L. Bourdin, F. Caubet\",\"doi\":\"10.1051/cocv/2022025\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from the (vectorial) Tresca friction problem found in contact mechanics. The weak formulation of the considered problem leads to a variational inequality of the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex subdifferential of the friction functional on the Sobolev space H^1(Omega) and show that all its subgradients satisfy a PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the twice epi-differentiability, concept introduced and thoroughly studied by R.T. Rockafellar,\\nwe show the differentiability of the solution to the parameterized Tresca-type problem and that its derivative satisfies a Signorini-type problem. Some numerical simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first time that the concept of twice epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.\",\"PeriodicalId\":50500,\"journal\":{\"name\":\"Esaim-Control Optimisation and Calculus of Variations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2022-04-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Esaim-Control Optimisation and Calculus of Variations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1051/cocv/2022025\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"AUTOMATION & CONTROL SYSTEMS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Control Optimisation and Calculus of Variations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1051/cocv/2022025","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
Sensitivity analysis of a Tresca-type problem leads to Signorini's conditions
The present paper investigates the sensitivity analysis, with respect to right-hand source term perturbations, of a scalar Tresca-type problem. This simplified, but nontrivial, model is inspired from the (vectorial) Tresca friction problem found in contact mechanics. The weak formulation of the considered problem leads to a variational inequality of the second kind depending on the perturbation parameter. The unique solution to this problem is then characterized by using the proximal operator of the corresponding nondifferentiable convex integral friction functional. We compute the convex subdifferential of the friction functional on the Sobolev space H^1(Omega) and show that all its subgradients satisfy a PDE with a boundary condition involving the convex subdifferential of the integrand. With the aid of the twice epi-differentiability, concept introduced and thoroughly studied by R.T. Rockafellar,
we show the differentiability of the solution to the parameterized Tresca-type problem and that its derivative satisfies a Signorini-type problem. Some numerical simulations are provided in order to illustrate our main theoretical result. To the best of our knowledge, this is the first time that the concept of twice epi-differentiability is applied in the context of mechanical contact problems, which makes this contribution new and original in the literature.
期刊介绍:
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