Numerical solution of the problem of optimal control of the heating of a thermoelastic plate by internal heat sources

We give the statement and numerical solution of the problem of optimal control (in the sense of rapidity of response) of the heating of an unbounded plate by internal heat sources in the presence of restrictions on the maximal (in absolute value) thermal stresses. In constructing the solution of this nonlinear optimal control problem we use the method of the inverse heat conduction problem.     

Finite-difference solution of the problem of optimizing the speed of heating of inhomogeneous thermoelastic bodies by internal heat sources

We consider the problem of optimal control of one-dimensional nonstationary temperature regimes of inhomogeneous bodies with respect to speed of action with restrictions on the control and normal thermoelastic stresses ₂₂ of maximum absolute value (in cylindrical and spherical coordinates the circumferential stresses .Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 61–65.     

Speed-optimal control of heating of inhomogeneous bodies by internal heat sources with constraints on the parameters of the thermal process

We use the method of the inverse heat-conduction problem to state and construct a numerical solution of the problem of speed-optimal control of the one-dimensional nonstationary thermal regimes of inhomogeneous bodies using internal heat sources with constraints on the control and the parameters of the thermal process. The control problem is nonlinear since the maximal values of the constraining parameters are attained at internal points of the body whose position changes during heating. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 110–114.     

Sufficient conditions for time optimal heating of rigid bodies by internal heat sources

We derive sufficient conditions for the time optimality of the control of heating of rigid bodies by internal heat sources under constraints on phase coordinates. We numerically solve the optimal-control problem in the case of heating of an unbounded plats with constraints on temperature and thermoelastic stresses.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 56–60, 1987.     

Boundary control of a vibrating plate with internal damping

It is shown that finite energy states of a vibrating viscoelastic plate of the Kelvin–Voigt type are, in general, not exactly controllable by L₂-boundary controls. Accordingly, we present a result on approximative controllability. The method is general.     

Analyticity of a thermoelastic plate with variable coefficients

A thermoelastic plate with variable coefficients in the absence of dissipative boundary mechanisms is considered. Using C₀-semigroup theory and some techniques in partial differential equations, it is shown that the corresponding C₀-semigroup is analytic and hence the exponential stability of the system is concluded.     

Quasi-optimal control of axisymmetric temperature displacements in a given section of a body using internal heat sources

For a thermoelastic half-space we study the problem of constructing in the space of continuous functions a quasi-optimal control of the axisymmetric vertical displacements in a given section parallel to the boundary surface in the form of a sequence that minimizes the original optimality criterion. For a half-space heated over a circular region by a heat flux of constant intensity we carry out a numerical analysis of the behavior of the elements of the minimizing sequence. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 104–109.     

Exact controllability of thermoelastic plate equation with memory

In this paper, we consider exact control problem for a coupled system of plate with Gurtin‐Pipkin equation. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the dual system. Firstly, we obtain the observability inequality of the dual system by means of multiplier method. Then, we prove that the system is exactly controllable based on the Hilbert Unique Method.     

A study of regimes for convective heating of deformable bodies with heat sources

We discuss a numerical/analytic method of solving the problem of determining regimes for heating electrically conducting elastic bodies using convection and electromagnetic radiation so as to optimize energy consumption. We give the results of studies of the temperature fields and stresses, and also the optimal heating regimes for a hollow glass ball.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 160–165.     

Natural convection at a heated semiinfinite vertical plate with temperature dependent heat sources or sinks

Numerical results are presented for the transient and steady-state velocity field and the temperature field. These results were obtained by solving the partial differential equations describing the conservation of mass, momentum and energy by an explicit finite-difference method in time dependent form. It has been observed that the velocity componentsu, v (absolute) and the temperature0 have larger values in the presence of heat sources than in their absence. Opposite is the phenomenon in the presence of heat sinks. A worthwhile result obtained from the detailed examination of the problem is that the presence of heat sources delays the attainment of steady-state condition.     

Optimal heating of a piecewise-homogeneous cylindrical glass shell by the surrounding medium and heat sources

We state the problem of stress-optimization of regimes for heating a piecewise-homogeneous cylindrical glass shell from the surrounding medium and heat sources and give an analytic/numerical solution of the problem under given bounds on the parameters of the thermostressed state.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 154–159.     

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.     

Analysis of the thermoelastic model of a plate with non-linear shape memory alloy reinforcements

In this paper we study the questions of the existence and uniqueness of the solutions for a thermoelastic system of equations in a two-dimensional domain, where both the viscosity v and the rigidity D are positive. It seems that such a system has up to now been considered in a one-dimensional setting only. The change of dimensions enforces the growth conditions with respect to θ and the additional regularity of the data. The existence of solutions in the case of the Neumann boundary conditions for θ and some weak regularity of data is proved. Under stronger regularity conditions the uniqueness is also established. The system has an interpretation as a plate reinforced by a shape memory alloy (SMA) wire mesh.     

Optimization of vertical axisymmetric thermal displacements in a round plate using heat sources

We consider the problem of using heat sources for optimal control of the distribution of the vertical axisymmetric thermal displacements of a thin round plate with edge clamped at an angle of revolution. On the basis of the method of the inverse thermoelasticity problem we construct a solution of the control problem. For specific cases of heating the plate we carry out a numerical analysis of the behavior of the optimal control.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Poly a, Issue 36, 1992, pp. 105–110.     

On a thermoelastic plate equation in an exterior domain

Obtained are the existence of solutions and the local energy decay of a linear thermoelastic plate equation in a 3 dim. exterior domain. The thermoplate equation is formulated as a Sobolev equation in the abstract framework. Our proof of the existence theorem is based on an argument due to Goldstein (Semigroups of Linear Operators and Applications. Oxford University Press: New York, 1985). To obtain the local energy decay, we use the commutation method in order to treat the high‐frequency part and a precise expansion of the resolvent operator obtained by constructing the parametrix in order to treat the low‐frequency. Copyright © 2002 John Wiley & Sons, Ltd.     

Dynamics of rotating thermoelastic disks with stationary heat source

Rotating disks are important components of car brakes or sawing units. In both cases, heat effects to be induced via stationary local contacts between pads and disk or workpiece and saw blade, respectively, influence the dynamic behavior and raise interesting problems in theory and practice. Therefore, the discussion of dynamic thermoelasticity in rotating disks with stationary heat sources is of basic interest to understand the interaction of temperature and displacement or stress fields in such structural elements. It will be analyzed here in detail for the case of an elastic disk for which there is a full (but weak) coupling of strain and temperature within the two applications mentioned. As a relatively general case, the combined excitation by a mechanical load and a simultaneously acting heat source will be examined.     

Optimization of the thermoelastic state of an element of a T-joint under local heating

We give a mathematical formulation of the problem of optimizing the stressed state of a T-joint under local heating and propose a method of solution based on the finite element method. We give the results of numerical solution of specific problems. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol 39, No. 1, 1996, pp. 115–118.     

Two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating

In this work, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the theory of two-temperature generalized thermoelasticity. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating.     

Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions

In a bounded domain, we consider an Euler–Bernoulli-type thermoelastic plate equation with perturbed boundary conditions. The boundary conditions are such that when the perturbation parameter goes to infinity, we recover the hinged boundary conditions, while one recovers the clamped boundary conditions when the perturbation parameter goes to zero. Relying on resolvent estimates, we show that the underlying semigroup is uniformly, with respect to the perturbation parameter, analytic and exponentially stable. The main features of our proof are appropriate decompositions of the components of the system and the use of Lions? interpolation inequalities.