Stanisław Migórski

Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction

In this article we examine an evolution problem, which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and we prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.In this article we examine an evolution problem, which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and we prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.

Optimal Control of Parabolic Hemivariational Inequalities

In this paper we study the optimal control of systems driven by parabolic hemivariational inequalities. First, we establish the existence of solutions to a parabolic hemivariational inequality which contains nonlinear evolution operator. Introducing a control variable in the second member and in the multivalued term, we prove the upper semicontinuity property of the solution set of the inequality. Then we use this result and the direct method of the calculus of variations to show the existence of optimal admissible state–control pairs.

Quasi-Static Hemivariational Inequality via Vanishing Acceleration Approach

We study an abstract second order nonlinear evolution inclusion in a framework of evolution triple of spaces. We consider time-dependent possibly nonconvex nonsmooth functions and their Clarke subdifferentials operating on the unknown function. First we prove the existence of a weak solution. Then we study the asymptotic behavior of a sequence of solutions when a small parameter in the inertial term tends to zero. We prove that the limit function is a solution of a parabolic hemivariational inequality. Finally, we give an application of the abstract theorem to a quasi-static viscoelastic contact problem.

A CLASS OF VARIATIONAL-HEMIVARIATIONAL INEQUALITIES WITH APPLICATIONS TO FRICTIONAL CONTACT PROBLEMS

A class of variational-hemivariational inequalities is studied in this paper. An inequality in the class involves two nonlinear operators and two nondifferentiable functionals, of which at least one is convex. An existence and uniqueness result is proved for a solution of the inequality. Continuous dependence of the solution on the data is shown. Convergence is established rigorously for finite element solutions of the inequality. An error estimate is derived which is of optimal order for the linear finite element method under appropriate solution regularity assumptions. Finally, the results are applied to a variational-hemivariational inequality arising in the study of some frictional contact problems.

Evolution Hemivariational Inequality for a Class of Dynamic Viscoelastic Nonmonotone Frictional Contact Problems

In this paper, we consider a class of hyperbolic hemivariational inequalities modeling the frictional contact between a viscoelastic body and a rigid foundation. The friction condition is described by a multivalued subdifferential relation which includes both the slip displacement and the slip rate dependent friction, and the Tresca models. The existence of weak solutions to the problem is proved by exploiting a surjectivity result for a class of pseudomonotone mappings.

Optimal Shape Design Problems for a Class of Systems Described by Hemivariational Inequalities

Optimal shape design problems for systems governed by an elliptic hemivariational inequality are considered. A general existence result for this problem is established by the mapping method.

A class of dynamic frictional contact problems governed by a system of hemivariational inequalities in thermoviscoelasticity

Abstract In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the nonlinear constitutive viscoelastic law with a long-term memory, which includes the thermal effects and considers the general nonmonotone and multivalued subdifferential boundary conditions for the contact, friction and heat flux. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using recent results from the theory of hemivariational inequalities and a fixed point argument.

Weak solvability of a piezoelectric contact problem

We consider a mathematical model which describes the frictional contact between a piezoelectric body and a foundation. The material behaviour is modelled with a non-linear electro-elastic constitutive law, the contact is bilateral, the process is static and the foundation is assumed to be electrically conductive. Both the friction law and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system of two coupled hemi-variational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proof is based on an abstract result on operator inclusions in Banach spaces.

A hybrid algorithm for solving inverse problems in elasticity

Abstract The paper offers a new approach to handling difficult parametric inverse problems in elasticity and thermo-elasticity, formulated as global optimization ones. The proposed strategy is composed of two phases. In the first, global phase, the stochastic hp-HGS algorithm recognizes the basins of attraction of various objective minima. In the second phase, the local objective minimizers are closer approached by steepest descent processes executed singly in each basin of attraction. The proposed complex strategy is especially dedicated to ill-posed problems with multimodal objective functionals. The strategy offers comparatively low computational and memory costs resulting from a double-adaptive technique in both forward and inverse problem domains. We provide a result on the Lipschitz continuity of the objective functional composed of the elastic energy and the boundary displacement misfits with respect to the unknown constitutive parameters. It allows common scaling of the accuracy of solving forward and inverse problems, which is the core of the introduced double-adaptive technique. The capability of the proposed method of finding multiple solutions is illustrated by a computational example which consists in restoring all feasible Young modulus distributions minimizing an objective functional in a 3D domain of a photo polymer template obtained during step and flash imprint lithography.

Multi-deme, twin adaptive strategy hp-HGS

The article presents a twin adaptive, effective stochastic strategy for solving difficult inverse problems formulated as global optimization ones. It is especially dedicated to multimodal, noisy problems. The hp-hierarchic genetic strategy (hp-HGS) strategy offers two ways to decrease computational and memory costs. Firstly, it decreases the number of objective evaluations by using an adaptation of the inverse problem accuracy (HGS strategy). Secondly, the cost of the direct problem solution, necessary for objective evaluation, is decreased by the proper scaling of the finite element method error using the hp adaptation technique. The theory of hp-HGS mentioned in this article can guarantee its asymptotic correctness in the probabilistic sense i.e. the possibility of finding all solutions from a specified set. The theoretical results make the comparison of the expected computational cost of a single genetic epoch at a particular level of the hp-HGS tree possible. Moreover, Theorem 3.1 verifies the hp-HGS applicability condition (the Lipschitz continuity of the energy functional) for heat conduction problems. A simple computational test for the heat conduction inverse problem in an L-shape domain shows the hp-HGS behaviour in the case of objective bimodality.