### On the approximation of the Signorini problem with friction

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This note deals with contact shape optimization for problems involving “floating” structures. The boundedness of solutions to state problems with respect to admissible domains, which is the basic step in the existence analysis, is a consequence of Korn’s inequality in coercive cases. In semicoercive cases (meaning that floating bodies are admitted), the Korn inequality cannot be directly applied and one has to proceed in another way: to use a decomposition of kinematically admissible functions and...

The paper deals with a class of optimal shape design problems for elastic bodies unilaterally supported by a rigid foundation. Cost and constraint functionals defining the problem depend on contact stresses, i.e. their control is of primal interest. To this end, the so-called reciprocal variational formulation of contact problems making it possible to approximate directly the contact stresses is used. The existence and approximation results are established. The sensitivity analysis is carried out....

Finite element analysis of unilateral problems with obstacles on the boundary is given. Provided the exact solution is smooth enough, we obtain the rate of convergence $0\left(h\right)$ for the case of one and two (lower and upper) obstacles on the boundary. At the end of this paper the proof of convergence without any regularity assumptions on the exact solution $u$ is given.

The problem of a thin elastic plate, deflection of which is limited below by a rigid obstacle is solved. Using Ahlin's and Ari-Adini's elements on rectangles, the convergence is established and SOR method with constraints is proposed for numerical solution.

The approximation of a mixed formulation of elliptic variational inequalities is studied. Mixed formulation is defined as the problem of finding a saddle-point of a properly chosen Lagrangian $\mathcal{2}$ on a certain convex set $Kx\phantom{\rule{4pt}{0ex}}\Lambda $. Sufficient conditions, guaranteeing the convergence of approximate solutions are studied. Abstract results are applied to concrete examples.

The paper deals with numerical realization of contact problems with friction obeying the Coulomb law. The original problem is formulated as the fixed-point problem for a certain operator generated by the variational inequality. This inequality is transformed to a system of variational nonlinear equations generating other operators, in a sense "close" to the above one. The fixed-point problem of these operators is solved by the least-square method in which equations and the corresponding quadratic...

The dual variational formulation of some free boundary value problem is given and its approximation by finite element method is studied, using piecewise linear elements with non-positive divergence.

This contribution deals with a mixed variational formulation of 3D contact problems with the simplest model involving friction. This formulation is based on a dualization of the set of admissible displacements and the regularization of the non-differentiable term. Displacements are approximated by piecewise linear elements while the respective dual variables by piecewise constant functions on a dual partition of the contact zone. The rate of convergence is established provided that the solution...

Optimal shape design problem for a deformable body in contact with a rigid foundation is studied. The body is made from material obeying a nonlinear Hooke’s law. We study the existence of an optimal shape as well as its approximation with the finite element method. Practical realization with nonlinear programming is discussed. A numerical example is included.

Contact problems with given friction and the coefficient of friction depending on their solutions are studied. We prove the existence of at least one solution; uniqueness is obtained under additional assumptions on the coefficient of friction. The method of successive approximations combined with the dual formulation of each iterative step is used for numerical realization. Numerical results of model examples are shown.

The paper presents the analysis, approximation and numerical realization of 3D contact problems for an elastic body unilaterally supported by a rigid half space taking into account friction on the common surface. Friction obeys the simplest Tresca model (a slip bound is given a priori) but with a coefficient of friction $\mathcal{F}$ which depends on a solution. It is shown that a solution exists for a large class of $\mathcal{F}$ and is unique provided that $\mathcal{F}$ is Lipschitz continuous with a sufficiently small modulus of...

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