Shape Optimization in Contact Problems with Coulomb Friction and a Solution-Dependent Friction Coefficient

Beremlijski, Petr; Haslinger, Jaroslav; Outrata, Jiří V.; Pathó, Róbert

doi:10.1137/130948070

Cited by 14 publications

(13 citation statements)

References 10 publications

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“…We finally conclude that all regularised and penalised formulations (3.3), (3.4), (3.5), (3.6) and (3.7) can be written in full generality as non-linear variational formulation: 8) where the integrand j(u, v, n) is non-linear with respect to the solution u but linear with respect to the test function v.…”

Section: Regularisation Of the Friction Termmentioning

confidence: 82%

“…For this last model of friction, the uniqueness of the contact solution is not ensured for the continuous model and examples of non-uniqueness can be built. Consequently, in [7] and [8], the authors analyse the derivation of the discretised problem, which admits a unique solution for small friction coefficients, by using subgradient calculus. Eventually, a thorough review of other results in shape optimisation for contact problems can be found in [31].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Regularisation Of the Friction Termmentioning

confidence: 82%

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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“…(ii) Let (u * , α * , ρ * ) be a solution to (4). Let further τ * be the vector of Lagrangian multipliers for the inequality constraints associated with this solution.…”

Section: Theorem 33 ([1]mentioning

confidence: 99%

“…It now follows from the definition of the coderivative that D * Σ(f , y)(y * ) can be approximated by the set of vectors ξ such that (ξ, −y * ) belongs to the set (21). This type of approximation has been used in numerous applications of the ImP technique to MPECs with complicated variational systems on the lower level [3,4,18].…”

Section: Define Now the Associated Perturbation Mappingmentioning

confidence: 99%

Inverse truss design as a conic mathematical program with equilibrium constraints

Kočvara¹,

Outrata²

2017

Discrete &Amp; Continuous Dynamical Systems - S

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We formulate an inverse optimal design problem as a Mathematical Programming problem with Equilibrium Constraints (MPEC). The equilibrium constraints are in the form of a second-order conic optimization problem. Using the so-called Implicit Programming technique, we reformulate the bilevel optimization problem as a single-level nonsmooth nonconvex problem. The major part of the article is devoted to the computation of a subgradient of the resulting composite objective function. The article is concluded by numerical examples demonstrating, for the first time, that the Implicit Programming technique can be efficiently used in the numerical solution of MPECs with conic constraints on the lower level.

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“…Moreover, in [13], convergence analysis with respect to the discretization parameter is performed. In the same spirit, we finally mention the series of papers [1,2,14] dedicated to shape optimization for the contact problem with Coulomb friction, which is much more cumbersome. There, the authors manage to characterize an outer approximation of the shape subdifferential of the functional to minimize, then use a bundle algorithm.…”

Section: Introductionmentioning

confidence: 99%

Shape derivatives for an augmented Lagrangian formulation of elastic contact problems

Bastien

Deteix

2021

ESAIM: COCV

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This work deals with shape optimization of an elastic body in sliding contact (Signorini) with a rigid foundation. The mechanical problem is written under its augmented Lagrangian formulation, then solved using a classical iterative approach. For practical reasons we are interested in applying the optimization process with respect to an intermediate solution produced by the iterative method. Due to the projection operator involved at each iteration, the iterate solution is not classically shape differentiable. However, using an approach based on directional derivatives, we are able to prove that it is conically differentiable with respect to the shape, and express sufficient conditions for shape differentiability. Finally, from the analysis of the sequence of conical shape derivatives of the iterative process, conditions are established for the convergence to the conical derivative of the original contact problem.

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Shape Optimization in Contact Problems with Coulomb Friction and a Solution-Dependent Friction Coefficient

Cited by 14 publications

References 10 publications

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

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

Inverse truss design as a conic mathematical program with equilibrium constraints

Shape derivatives for an augmented Lagrangian formulation of elastic contact problems

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