Optimal Control of Elastoplastic Processes: Analysis, Algorithms, Numerical Analysis and Applications

Herzog, Roland; Meyer, Christian; Wachsmuth, Gerd

doi:10.1007/978-3-319-05083-6_4

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…Optimal control problems governed by a static version of (2.4) have also been studied in recent years, yielding optimality conditions for its primal and dual variants [26,40,41], as well as numerical algorithms for solving the problem [42]. The quasi-static model has been considered in [61,63,64].…”

Section: Elastoplasticitymentioning

confidence: 99%

On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

Reyes

2018

GAMM-Mitteilungen

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Variational inequalities are an important mathematical tool for modelling free boundary problems that arise in different application areas. Due to the intricate nonsmooth structure of the resulting models, their analysis and optimization is a difficult task that has drawn the attention of researchers for several decades. In this paper we focus on a class of variational inequalities, called of the second kind, with a twofold purpose. First, we aim at giving a glance at some of the most prominent applications of these types of variational inequalities in mechanics, and the related analytical and numerical difficulties. Second, we consider optimal control problems constrained by these variational inequalities and provide a thorough discussion on the existence of Lagrange multipliers and the different types of optimality systems that can be derived for the characterization of local minima. The article ends with a discussion of the main challenges and future perspectives of this important problem class.

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Section: Elastoplasticitymentioning

confidence: 99%

On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

Reyes

2018

GAMM-Mitteilungen

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“…Optimal control of a problem of static plasticity in the infinite-dimensional setting is the subject of [HMW12] and [HMW13]. The results were used in [HMW14] to numerically solve a quasi-static control problem by time-discretization. Optimality conditions for time-continuous, infinitedimensional, rate-independent control problems of quasi-static plasticity type could be derived in [Wac12], [Wac15], [Wac16] by means of time-discretization.…”

Section: Introductionmentioning

confidence: 99%

Optimal control of reaction-diffusion systems with hysteresis

Münch

2018

ESAIM: COCV

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This paper is concerned with the optimal control of hysteresis-reaction-diffusion systems. We study a control problem with two sorts of controls, namely distributed control functions, or controls which act on a part of the boundary of the domain. The state equation is given by a reaction-diffusion system with the additional challenge that the reaction term includes a scalar stop operator. We choose a variational inequality to represent the hysteresis. In this paper, we prove first order necessary optimality conditions. In particular, under certain regularity assumptions, we derive results about the continuity properties of the adjoint system. For the case of distributed controls, we improve the optimality conditions and show uniqueness of the adjoint variables. We employ the optimality system to prove higher regularity of the optimal solutions of our problem. Finally, we derive regularity properties of the value function of a perturbed control problem when the set of controls is restricted. The specific feature of rate-independent hysteresis in the state equation leads to difficulties concerning the analysis of the solution operator. Non-locality in time of the Hadamard derivative of the control-to-state operator complicates the derivation of an adjoint system.

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“…The solution we choose, which was largely investigated in the framework of control theory for problems with hardening but not in the framework of shape optimization, is the use of a regularized penalized problem to get rid of the variational inequality. On this issue we mention, for the static case, [24], [26], [27], [11] (using a primal formulation), [28], [5] (for a second order optimality condition), for the quasi-static case [69] and for other plastic models [10] and [36].…”

Section: Introductionmentioning

confidence: 99%

Elasto-plastic Shape Optimization Using the Level Set Method

Maury¹,

Allaire²,

Jouve³

2018

SIAM J. Control Optim.

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This article is concerned with shape optimization of structures made of a material obeying Hencky's laws of plasticity, with the stress bound expressed by the von Mises effective stress. The ill-posedness of the model is circumvented by using two regularized versions of the mechanical problem. The first one is the classical Perzyna formulation which is regularized, the second one is a new regularized formulation proposed for the von Mises criterion. Shape gradients are calculated thanks to the adjoint method. The optimal shape is numerically computed by using the level set method. To illustrate the validity of the method, 2D examples are performed.

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Optimal Control of Elastoplastic Processes: Analysis, Algorithms, Numerical Analysis and Applications

Cited by 5 publications

References 24 publications

On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

Optimal control of reaction-diffusion systems with hysteresis

Elasto-plastic Shape Optimization Using the Level Set Method

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