Sensitivity analysis of a Tresca-type problem leads to Signorini's conditions

IF 1.3 3区 数学 Q4 AUTOMATION & CONTROL SYSTEMS Esaim-Control Optimisation and Calculus of Variations Pub Date : 2022-04-07 DOI:10.1051/cocv/2022025
S. Adly, L. Bourdin, F. Caubet
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引用次数: 2

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, 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.
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对tresca型问题的敏感性分析导致了Signorini条件
本文研究了一类标量tresca型问题关于右源项摄动的灵敏度分析。这个简化但不平凡的模型的灵感来自于接触力学中发现的(矢量)Tresca摩擦问题。所考虑的问题的弱形式导致了依赖于摄动参数的第二类变分不等式。然后利用相应的不可微凸积分摩擦泛函的近算子来表征该问题的唯一解。我们计算了Sobolev空间H^1(Omega)上的摩擦泛函的凸次微分,并证明了它的所有次梯度都满足包含被积函数凸次微分的边界条件的偏微分方程。利用R.T. Rockafellar引入并深入研究的二次外延可微性概念,证明了参数化tresca型问题解的可微性及其导数满足signorini型问题。为了说明我们的主要理论结果,给出了一些数值模拟。据我们所知,这是第一次在机械接触问题的背景下应用二次外延可微性的概念,这使得这一贡献在文献中新颖而原始。
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来源期刊
Esaim-Control Optimisation and Calculus of Variations
Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics
自引率
7.10%
发文量
77
期刊介绍: ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.
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