Optimal control of an elastic contact problem involving Tresca friction law

Amassad, Amina; Chenais, Denise

doi:10.1016/s0362-546x(00)00241-8

Cited by 36 publications

(44 citation statements)

References 6 publications

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“…On the boundary part Γ C , the Tresca friction law will be imposed. As in, a nonseparation condition on Γ C is assumed leading to the following space of admissible displacement fields In, an evolutionary Tresca friction model is set‐up in the framework of rate independent processes. It is based on the following energy functional

scriptE

and dissipation potential

scriptR

scriptE MathClass-open(t MathClass-punc, v MathClass-close) MathClass-punc: MathClass-rel= \int_{Ω} W MathClass-open(MathClass-rel\nabla v MathClass-close) MathClass-bin- f MathClass-open(t MathClass-close) v normald x MathClass-punc, W MathClass-open(MathClass-rel\nabla v MathClass-close) MathClass-punc: MathClass-rel= \frac{1}{2} boldC ϵ MathClass-open(v MathClass-close) MathClass-punc: ϵ MathClass-open(v MathClass-close) MathClass-punc,

…”

Section: Examples and Extensionsmentioning

confidence: 99%

“…On the basis of ,Ch. 2, the Tresca‐type evolution law from can equivalently be reformulated as follows: Given

f MathClass-rel\in H^{1} MathClass-open(0 MathClass-punc, T MathClass-punc; L^{2} MathClass-open(Ω MathClass-punc, R^{d} MathClass-close) MathClass-close)

and

u_{0} MathClass-rel\in V MathClass-bin\cap scriptS MathClass-open(0 MathClass-close)

, find u ∈ H 1 (0, T ; V ) with u (0) = u 0 such that

u MathClass-open(t MathClass-close) MathClass-rel\in scriptS MathClass-open(t MathClass-close)

for all t and

scriptE MathClass-open(t_{1} MathClass-punc, u MathClass-open(t_{1} MathClass-close) MathClass-close) MathClass-bin+ \int_{t_{0}}^{t_{1}} scriptR MathClass-open(\partial_{t} u MathClass-open(τ MathClass-close) MathClass-close) normald τ MathClass-rel= scriptE MathClass-open(t_{0} MathClass-punc, u MathClass-open(t_{0} MathClass-close) MathClass-close) MathClass-bin+ \int_{t_{0}}^{t_{1}} \int_{Ω} MathClass-bin- \partial_{t} f MathClass-open(τ MathClass-close) u MathClass-open(τ MathClass-close) normald x normald τ

…”

Section: Examples and Extensionsmentioning

confidence: 99%

“…for all t 0 , t 1 ∈ [0, T ]. According to, for every

f MathClass-rel\in H^{1} MathClass-open(0 MathClass-punc, T MathClass-punc; L^{2} MathClass-open(Ω MathClass-punc, R^{d} MathClass-close) MathClass-close)

and

u_{0} MathClass-rel\in V MathClass-bin\cap scriptS MathClass-open(0 MathClass-close)

, there exists a unique function u ∈ H 1 (0, T ; V ) satisfying the aforementioned conditions.…”

Section: Examples and Extensionsmentioning

confidence: 99%

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Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

Knees

Schröder

2012

Math Methods in App Sciences

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A global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D), the displacement fields are regular. The proof relies on a difference quotient argument for the directions tangential to the crack. To obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann (2002) is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. On the basis of Falk's approximation theorem for variational inequalities, convergence rates for finite element discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.

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scriptE

and dissipation potential

scriptR

scriptE MathClass-open(t MathClass-punc, v MathClass-close) MathClass-punc: MathClass-rel= \int_{Ω} W MathClass-open(MathClass-rel\nabla v MathClass-close) MathClass-bin- f MathClass-open(t MathClass-close) v normald x MathClass-punc, W MathClass-open(MathClass-rel\nabla v MathClass-close) MathClass-punc: MathClass-rel= \frac{1}{2} boldC ϵ MathClass-open(v MathClass-close) MathClass-punc: ϵ MathClass-open(v MathClass-close) MathClass-punc,

…”

Section: Examples and Extensionsmentioning

confidence: 99%

“…On the basis of ,Ch. 2, the Tresca‐type evolution law from can equivalently be reformulated as follows: Given

f MathClass-rel\in H^{1} MathClass-open(0 MathClass-punc, T MathClass-punc; L^{2} MathClass-open(Ω MathClass-punc, R^{d} MathClass-close) MathClass-close)

and

u_{0} MathClass-rel\in V MathClass-bin\cap scriptS MathClass-open(0 MathClass-close)

, find u ∈ H 1 (0, T ; V ) with u (0) = u 0 such that

u MathClass-open(t MathClass-close) MathClass-rel\in scriptS MathClass-open(t MathClass-close)

for all t and

scriptE MathClass-open(t_{1} MathClass-punc, u MathClass-open(t_{1} MathClass-close) MathClass-close) MathClass-bin+ \int_{t_{0}}^{t_{1}} scriptR MathClass-open(\partial_{t} u MathClass-open(τ MathClass-close) MathClass-close) normald τ MathClass-rel= scriptE MathClass-open(t_{0} MathClass-punc, u MathClass-open(t_{0} MathClass-close) MathClass-close) MathClass-bin+ \int_{t_{0}}^{t_{1}} \int_{Ω} MathClass-bin- \partial_{t} f MathClass-open(τ MathClass-close) u MathClass-open(τ MathClass-close) normald x normald τ

…”

Section: Examples and Extensionsmentioning

confidence: 99%

“…for all t 0 , t 1 ∈ [0, T ]. According to, for every

f MathClass-rel\in H^{1} MathClass-open(0 MathClass-punc, T MathClass-punc; L^{2} MathClass-open(Ω MathClass-punc, R^{d} MathClass-close) MathClass-close)

and

u_{0} MathClass-rel\in V MathClass-bin\cap scriptS MathClass-open(0 MathClass-close)

, there exists a unique function u ∈ H 1 (0, T ; V ) satisfying the aforementioned conditions.…”

Section: Examples and Extensionsmentioning

confidence: 99%

See 1 more Smart Citation

Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

Knees

Schröder

2012

Math Methods in App Sciences

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show abstract

“…[50] used a bundle algorithm to perform shape optimisation. Another way to get optimality conditions, see [6] and [3], is to introduce a regularised problem, depending on a small regularisation parameter, study its optimality conditions and pass to the limit when the regularisation parameter tends to zero. This approach, A. Maury, G. Allaire, et al called penalisation, was used in various numerical shape optimisation works: for example, with the SIMP method in [13], or with parameterised shapes (using splines) in [52] and [35].…”

Section: Introductionmentioning

confidence: 99%

“…In [58], for the Tresca model (also called the prescribed friction model), a conical derivative is computed, merely in two-space dimensions and for specific directions of differentiation. Once again, penalised and regularised formulations can be used as in [34], [3] and [61]. Theoretical results are also given for normal compliance model in [38] and [39] and for Coulomb friction model in [23].…”

Section: Introductionmentioning

confidence: 99%

Shape optimisation with the level set method for contact problems in linearised elasticity

Maury¹,

Allaire²,

Jouve³

2017

The SMAI Journal of Computational Mathematics

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Abstract. This article is devoted to shape optimisation of contact problems in linearised elasticity, thanks to the level set method. We circumvent the shape non-differentiability, due to the contact boundary conditions, by using penalised and regularised versions of the mechanical problem. This approach is applied to five different contact models: the frictionless model, the Tresca model, the Coulomb model, the normal compliance model and the Norton-Hoff model. We consider two types of optimisation problems in our applications: first, we minimise volume under a compliance constraint, second, we optimise the normal force, with a volume constraint, which is useful to design compliant mechanisms. To illustrate the validity of the method, 2D and 3D examples are performed, the 3D examples being computed with an industrial software.Math. classification. 74P05, 75P10, 74P15, 74M10, 74M15, 49Q10, 49Q12, 35J85.

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Optimal control of friction coefficient in Signorini contact problems

Essoufi

Zafrar

2021

Optim Control Appl Methods

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The optimal control of friction coefficient in Signorini contact problem is handled. First, we give the Tikhonov regularized optimal control problem subject to quasivariational inequality of the second kind. Since the relation control‐state is weak, we introduce another penalized control problem based algorithm of Bensoussan–Lions type applied to the quasi‐variational inequality. In order to get the optimality condition, and since the cost function is not differentiable, the quasi‐variational inequality is regularized to be a variational equality.

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Optimal control of an elastic contact problem involving Tresca friction law

Cited by 36 publications

References 6 publications

Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

Shape optimisation with the level set method for contact problems in linearised elasticity

Optimal control of friction coefficient in Signorini contact problems

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