A thermoelastic contact problem with a phase transition

^** Email: hoemberg{at}wias-berlin.de^*** Email: khlud{at}hydro.nsc.ru We investigate a thermomechanical contact problem with phasetransitions. The system of equations consists of a quasistaticmomentum balance and a semilinear energy balance. The phasetransition is described by an ordinary differential equation.Different mechanical properties of the respective phases aretaken care of by a mixture ansatz. We prove the existence ofa weak solution and a uniqueness result, the latter only beingvalid in one space dimension.     

Forward dynamical problem for the Timoshenko beam

The initial boundary value problem

A quasi-static Signorini contact problem for a thermoviscoelastic beam

This paper is concerned with the existence, uniqueness and numerical solution of a system of equations modelling the evolution of a quasi-static thermoviscoelastic beam that may be in contact with two rigid obstacles. A finite element approximation is proposed and analysed and some numerical results are given. Work partially supported by the Brazilian institution CNPq.     

Numerical approximation and error control for a thermoelastic contact problem

A numerical method using finite elements for the spatial discretization and the Crank–Nicolson scheme for the time stepping is applied to a partial differential equation problem involving thermoelastic contact. The Crank–Nicolson scheme is interpreted as a low order continuous Galerkin method. By exploiting the variational framework inherent in this approach, an a posteriori error estimate is derived. This estimate gives a bound on the approximation error that depends on computable quantities such as the mesh parameters, time step and numerical solution. In this paper, the a posteriori estimate is used to develop a time step refinement strategy. Several computational examples are included that demonstrate the performance of the method and validity of the estimate.     

Well posedness and stability result for a thermoelastic laminated Timoshenko beam with distributed delay term

In this work, we consider a linear thermoelastic laminated timoshenko beam with distributed delay, where the heat conduction is given by cattaneoâs law. we establish the well posedness of the system. For stability results, we prove exponential and polynomial stabilities of the system for the cases of equal and nonequal speeds of wave propagation.     

A contact problem for a viscoelastic plate and an elastic beam

Under consideration is the problem of contact of a viscoelastic plate with an elastic beam. To characterize the viscoelastic deformation of the plate, the hereditary integrals are used. The differential formulation of the problem with the conditions in the form of a system of equalities and inequalities in the domain of possible contact is presented, and its equivalence to a variational inequality is proved. The unique solvability of the problem is proved as well as the existence of the time derivative of the solution. A limit problem is also considered as the bending rigidity of the plate tends to infinity.     

Global existence and decay for the semilinear thermoelastic contact problem

In this paper a class of semilinear thermoelastic contact problems is considered and the existence and exponential decay of the weak solutions are obtained.     

Global existence and exponential stability for a contact problem between two thermoelastic beams

In this paper we analyze a dynamic unilateral contact problem between two thermoelastic beams. We establish the existence of a weak global-in-time solution, by a penalization method. Moreover, we study the asymptotic behavior of such a solution proving that the energy associated to the system decays exponentially to zero, as time goes to infinity.     

Numerical analysis of a quasi-static contact problem for a thermoviscoelastic beam

In this paper we revisit a quasi-static contact problem of a thermoviscoelastic beam between two rigid obstacles which was recently studied in [1]. The variational problem leads to a coupled system, composed of an elliptic variational inequality for the vertical displacement and a linear variational equation for the temperature field. Then, its numerical resolution is considered, based on the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Error estimates are proved from which, under adequate regularity conditions, the linear convergence is derived. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution.     

New general and explicit stability result for a thermoelastic Timoshenko system

Ricerche di Matematica - The goal of this article is to study a thermoelastic Timoshenko system with viscoelastic law acting on the shear force and thermoelastic dissipation acting on the bending...     

A locking-free scheme for the LQR control of a Timoshenko beam

In this paper we analyze a locking-free numerical scheme for the LQR control of a Timoshenko beam. We consider a non-conforming finite element discretization of the system dynamics and a control law constant in the spatial dimension. To solve the LQR problem we seek a feedback control which depends on the solution of an algebraic Riccati equation. An optimal error estimate for the feedback operator is proved in the framework of the approximation theory for control of infinite dimensional systems. This estimate is valid with constants that do not depend on the thickness of the beam, which leads to the conclusion that the method is locking-free. In order to assess the performance of the method, numerical tests are reported and discussed.     

Finite element approximation to a contact problem for a nonlinear thermoviscoelastic beam

In this paper we propose and analyze a finite element method to the solution of a quasi-static contact problem between a nonlinear beam and a rigid obstacle. Error estimates and energy decay are obtained and some numerical simulations described.     

Optimal chattering modes in the problem of the control of a Timoshenko beam

The linear problem of the control in a plane of the motion of a Timoshenko beam, one end of which is clamped to a rotating disc is considered. The angular acceleration of the disc serves as the control. It is proved that, in the problem of the quenching of the first mode, the optimal control has an infinite number of switchings in a finite time interval (a chattering control). The construction of a suboptimal control with a finite number of switchings is described.     

Stabilization of a viscoelastic Timoshenko beam

A viscoelastic Timoshenko beam is investigated. We prove an exponential decay of solutions for a large class of kernels with weaker conditions than the existing ones in the literature. This will allow the use of other types of viscoelastic material for Timoshenko type beams than the usually used ones.     

Stabilization of a nonlinear Timoshenko beam

We study the asymptotic behavior of the system governing the nonlinear vibrations of a Timoshenko beam.     

A numerical algorithm for the nonlinear Timoshenko beam system

An initial boundary value problem is considered for the dynamic beam system Its solution is found by means of an algorithm, the constituent parts of which are the finite element method, the implicit symmetric difference scheme used to approximate the solution with respect to the spatial and time variables, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. Errors of three parts of the algorithm are estimated and, as a result, its total error estimate is obtained. A numerical example is solved.     

A nonlinear transmission problem for a compound plate with thermoelastic part

In this paper, we study a nonlinear transmission problem for a plate that consists of thermoelastic and isothermal parts. The problem generates a dynamical system in a suitable Hilbert space. The main result is the proof of the asymptotic smoothness of this dynamical system. We also prove the existence of a compact global attractor in special cases when the nonlinearity is of Berger type or scalar. Copyright © 2012 John Wiley & Sons, Ltd.     

A new general decay for a transmission problem of viscoelastic Timoshenko systems

In this paper, we investigate the long-time behavior for a transmission problem of viscoelastic Timoshenko systems with different speeds of wave propagation. By constructing a new Lyapunov functional and combining the technique of perturbation energy with some precise estimates for multipliers, we establish a general uniform decay estimates for the energy.