From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics

Gwinner, Joachim; Ovcharova, Nina

doi:10.1080/02331934.2014.1001758

Cited by 24 publications

(13 citation statements)

References 30 publications

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“…Finally, we note that the existence of a solution to problem , respectively , relies on the pseudomonotonicity of the nonsmooth boundary functional and has been investigated in . We recall that the functional

φ : X \times X \to R

, where X is a real reflexive Banach space, is pseudomonotone if

u_{n} ⇀ u

(weakly) in X and

{liminf}_{n \to \infty} φ (u_{n}, u) \geq 0

imply

{limsup}_{n \to \infty} φ (u_{n}, v) \leq φ (u, v)

for all v ∈ X .…”

Section: Boundary Integral Operator Formulationmentioning

confidence: 99%

“…(6), relies on the pseudomonotonicity of the nonsmooth boundary functional and has been investigated in [21,36,37]. We recall that the functional ϕ : X × X → IR, where X is a real reflexive Banach space, is pseudomonotone if un u (weakly ) in X and lim inf…”

Section: N Ovcharovamentioning

confidence: 99%

“…The solvability of (P ε,h ) and the convergence of its solutions to a solution of the boundary hemivariational inequality (6) relies on the following general approximation result from [21].…”

Section: It Holds Thatmentioning

confidence: 99%

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On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

Ovcharova

2016

Math Methods in App Sciences

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In this paper, we couple regularization techniques of nondifferentiable optimization with the h-version of the boundary element method (h-BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality (HVI) with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by a h-BEM. We prove convergence of the h-BEM Galerkin solution of the regularized problem in the energy norm, provide an a-priori error estimate and give a numerical example.

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φ : X \times X \to R

, where X is a real reflexive Banach space, is pseudomonotone if

u_{n} ⇀ u

(weakly) in X and

{liminf}_{n \to \infty} φ (u_{n}, u) \geq 0

imply

{limsup}_{n \to \infty} φ (u_{n}, v) \leq φ (u, v)

for all v ∈ X .…”

Section: Boundary Integral Operator Formulationmentioning

confidence: 99%

Section: N Ovcharovamentioning

confidence: 99%

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On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

Ovcharova

2016

Math Methods in App Sciences

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“…For application of Griffith's fracture formula to solids and composites with non-linear cracks, we refer to [12,21,25,28] and references therein. We emphasize that these works develop constructive methods and numerical algorithms for finding optimal cracks and their advance.…”

Section: Remark 22mentioning

confidence: 99%

Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack

Lazarev

2015

Z. Angew. Math. Mech.

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The equilibrium problems for two-dimensional elastic body with a rigid delaminated inclusion are considered. In this case, there is a crack between the rigid inclusion and the elastic body. Non-penetration conditions on the crack faces are given in the form of inequalities. We analyze the dependence of solutions and derivatives of the energy functionals on the thickness of rigid inclusion. The existence of the solution to the optimal control problem is proved. For that problem the cost functional is defined by derivatives of the energy functional with respect to a crack perturbation parameter while the thickness parameter of rigid inclusion is chosen as the control function.

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“…Constraints arise in a variety of applications. The constraint operator (see (3)) may become: the trace operator under contact conditions [1][2][3], the jump operator for cracks and anticracks [4][5][6], the gradient operator in plasticity [7], the divergence operator under incompressibility conditions [8][9][10], a state-constraint in mathematical programs with equilibrium constraints [11,12], and the like. The constraint problems are related to parameter identification problems (see the theory in References [13][14][15] and application to biological systems in Reference [16]), to inverse problems by the mean of observation data used in mathematical physics [17,18] and in acoustics [19][20][21], to overdetermined and free-boundary problems [22,23].…”

Section: Introductionmentioning

confidence: 99%

On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

Granada

Gwinner

Kovtunenko

2018

Axioms

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This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet–Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method.

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From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics

Cited by 24 publications

References 30 publications

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

On the coupling of regularization techniques and the boundary element method for a hemivariational inequality modelling a delamination problem

Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack

On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

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