Control of thermal stresses and displacements in thermoelastic bodies

By developing the method of the inverse thermoelastic and thermal conductivity problem we study problems of optimal control of thermal stresses and displacements in the case of one- and two-dimensional temperature fields. The solution of the optimization problems reduces to solving integral equations of first or second kind for determining the internal heat sources.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 18–23.     

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.     

Control of thermoelastic stresses in a cylinder in the case of a two-dimensional nonaxisymmetric temperature field

We obtain new functional relations among the components of thermal stresses, and also their dependence on the temperature field. Using these relations we find solutions of a control problem that we pose. We state necessary conditions for the existence of the required optimal control which must be satisfied by the given distribution function of the control of thermal stresses.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 56–60.     

Strong controllability and optimal control of the heat equation with a thermal source

In this paper we consider an optimal control system described byn-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. Here we assume there is no constraints on control. This problem is reduced to a moment problem.We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure approximated by a finite combination of atomic measures. This construction gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional.     

Speed-optimized control of heating of a thermoelastic plate with the help of internal heat sources

Speed-optimized control of the heating of a plate with restrictions on the thermal stresses is constructed as a junction of the distribution of internal heat sources by the method of the inverse problem of heat conduction.Translated from Matematicheskie Methody i Fiziko-mekhanicheskie Polya, No. 26, pp. 55–59, 1987.     

A nonlinear optimal control problem for the nonsteady temperature of a layer with constraints on the thermal stress

We propose a numerical method of constructing the optimal heating regime for a thermally stressed unbounded layer with constraints on the control and thermal stresses. Solving the nonlinear optimization problem for rapidity is reduced to solving the inverse problem of thermoelasticity. The results of numerical studies are presented. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Optimization of vertical axisymmetric displacements of a thin circular plate under a nonstationary thermal load

We study the problem of optimal control of the distribution of vertical axisymmetric thermal displacements of a thin circular plate fixed along its edge. The displacements are caused by a nonstationary heat load on one of the end surfaces. The thermal action on the other end surface is chosen as the control function. Using the Hankel and Laplace transforms in the space of continuous functions, we construct the solution of the inverse problem of thermoelasticity to which the original control problem has been reduced.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 148–153.     

Piecewise optimal distributed Controls for 2D Boussinesq equations

In this article, we study the dynamics of a piecewise (in time) distributed optimal control problem for the Boussinesq equations which model velocity tracking over time coupled to thermal dynamics. We also study the dynamics of semidiscrete approximation of this problem. We prove that the rates of velocity tracking coupled to thermal dynamics are exponential and that the difference between the solution of the semi‐discrete piecewise optimal control problem and the desired states in L² and H¹ norms decay to zero exponentially as n→∞. Copyright © 2000 John Wiley & Sons, Ltd.     

Evolutionary Algorithms for Approximate Optimal Control of the Heat Equation with Thermal Sources

In this paper, optimal control problem (OCP) governed by the heat equation with thermal sources is considered. The aim is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. To obtain an approximate solution of this problem, a partition of the time-control space is considered and the discrete form of the problem is converted to a quasi assignment problem. Then by using an evolutionary algorithm, an approximate optimal control function is obtained as a piecewise linear function. Numerical examples are given to show the proficiency of the presented algorithm.     

Optimization of axisymmetric thermal displacements in a given section of an unbounded layer

We study the problem of optimal control of the distribution of axisymmetric vertical or radial thermal displacements in a given section of an unbounded layer. Using the method of the inverse problem of thermoelasticity we construct the solution of the control problem. For specific cases of heating of the layer we give a numerical analysis of the behavior of the optimal control.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 55–61.     

Lp-Lq time decay estimate to the solution of the Cauchy problem of the system of equations for nonlocal model of hyperbolic thermoelasticity theory

The aim of this paper is to present a new system of equations describing nonlocal model of hyperbolic thermoelasticity theory. We used the Papkin and Gurtin approach based on the constitutive relations for internal energy e(x), and heat flux q(x), with integral terms. Such system of equations describes the propagation of thermal perturbation with finite velocity. Using the modified Cagniard–de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three–dimensional space. Basing on the constructed matrix of fundamental solutions in the explicit formula we represent the solution of the Cauchy problem to this system of equations in the form of some kind of convolutions. Next, applying the method of Sobolev spaces, we obtain the L^p−L^q time decay estimate to the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Problem of optimal control by the natural frequency of oscillations of an orthotropic shell of revolution and its finite-dimensional approximation

The questions of optimization in problems of oscillations in orthotropic shells of revolution of variable thickness are studied for the case when the thickness and radius of curvature of the shell generatrix are used as the controls. Restrictions are imposed on the principal oscillation eigenfrequency, thickness, internal volume and other parameters. It is shown that a solution of the problem exists and, that the problem can be approximated by a sequence of the finite-dimensional problems. Certain questions of the optimal control in the problem concerning the oscillations of plates of variable thickness with the thickness serving as the control, were studied in /1–4/.     

Solving some optimal control problems using the barrier penalty function method

In this paper we present a new approach to solve a two-level optimization problem arising from an approximation by means of the finite element method of optimal control problems governed by unilateral boundary-value problems. The problem considered is to find a minimum of a functional with respect to the control variablesu. The minimized functional depends on control variables and state variablesx. The latter are the optimal solution of an auxiliary quadratic programming problem, whose parameters depend onu.Our main idea is to replace this QP problem by its dual and then apply the barrier penalty method to this dual QP problem or to the primal one if it is in an appropriate form. As a result we obtain a problem approximating the original one. Its good property is the differentiable dependence of state variables with respect to the control variables. Furthermore, we propose a method for finding an approximate solution of a penalized lower-level problem if the optimal solution of the original QP problem is known. We apply the result obtained to some optimal shape design problems governed by the Dirichlet-Signorini boundary-value problem.This research was supported by the Academy of Finland and the Systems Research Institute of the Polish Academy of Sciences.     

Identification of a source factor in a control problem for the heat equation with a boundary memory term

We consider a material with thermal memory occupying a bounded region Ω with boundary Γ. The evolution of the temperature u(t,x) is described by an integrodifferential parabolic equation containing a heat source of the form f(t)z₀(x). We formulate an initial and boundary value control problem based on a feedback device located on Γ and prescribed by means of a quite general memory operator. Assuming both u and the source factor f are unknown, we study the corresponding inverse and control problem on account of an additional information. We prove a result of existence and uniqueness of the solution (u,f). Copyright © 2015 John Wiley & Sons, Ltd.     

Hall current effect on MHD mixed convection flow from an inclined continuously stretching surface with blowing/suction and internal heat generation/absorption

Hydromagnetic heat transfer by mixed convection along an inclined continuously stretching surface, with power-law variation in the surface temperature or heat flux, in the presence of Hall current and internal heat generation/absorption has been studied. The surface is considered to be permeable to allow fluid suction or blowing, and stretching with a surface velocity varied according to a power-law. Two cases of the temperature boundary conditions were considered at the surface. The governing equations have been transformed into non-similar partial differential equations which have been integrated by the forth-order Runge–Kutta method. The effect of Hall parameter, magnetic parameter, dimensionless blowing/suction parameter, space and temperature dependent internal heat generation/absorption parameters and buoyancy force parameters on the temperature, primary and secondary flow velocity have been studied parametrically. All parameters involved in the problem affect the flow and thermal distributions except the temperature-dependent internal heat generation/absorption in the case of prescribed heat flux (PHF). Numerical values of the local skin-friction and the local Nusselt numbers for various parametric conditions have been tabulated.     

Hamilton-Jacobi-Bellman inequality for the average control of piecewise deterministic Markov processes

ABSTRACT

The main goal of this paper is to study the infinite-horizon long run average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) with the control acting continuously on the jump intensity λ and on the transition measure Q of the process. We provide conditions for the existence of a solution to an integro-differential optimality inequality, the so called Hamilton-Jacobi-Bellman (HJB) equation, and for the existence of a deterministic stationary optimal policy. These results are obtained by using the so-called vanishing discount approach, under some continuity and compactness assumptions on the parameters of the problem, as well as some non-explosive conditions for the process.     

Estimations on Thermo-mechanical Dynamics of Vibration Elastomeric Isolators

This analysis deals with one of the basic problem category of vibratory systems, means the complete and complex characterization of elastic and viscous isolators behaviour under dynamic loads such as vibrations, seismic waves, shocks, etc. Usually, the dynamic characteristics of vibration isolators made by elastomeric materials are considered to have a constant shape for a certain practical case. It is ignored the thermal phenomenon inside the isolator block during the exploitation cycles and its influences on the proper characteristic parameters. This usual approximation leads to more or less significant differences between simulation and practical evolution of a vibration isolator subjected to the same dynamic load. Continuous changes of rigidity modulus and/or dissipative characteristics due to internal thermal effects imply aleatory evolution of the isolated system, unstable movements and resonance imminence danger. The partial results of this analysis dignify the linkage between thermal effects into the elastomeric isolator and its essential dynamic parameters. Using of these correlations frames the seismic shock and vibration protective devices designing and deployment areas. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of the temperatures and heat flows for a conical slide support taking account of frictional heating

We consider an axisymmetric problem of heat conduction taking account of frictional heating in a conetorus pair that models the functioning of a conical support. The bodies are pressed together and are rotating about a common axis. Heat is generated in the region of contact of the bodies due to frictional forces. Outside the region of contact there is heat exchange with the surrounding medium. The thermal contact between the two bodies is nonideal. The problem is reduced to a system of integral equations whose solution is constructed by the method of successive approximations. We give the results of numerical studies of the temperature distribution and heat flows from the geometric and thermophysical parameters of the body. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 19–27.     

Identification of Temperature Dependent Parameters in Radiative Heat Transfer

Laser-induced thermotherapy (LITT) is an established minimally invasive percutaneous technique of tumor ablation. Nevertheless, there is a need to predict the effect of laser applications and optimizing irradiation planning in LITT. Optical attributes (absorption, scattering) change due to thermal denaturation. The work presents the possibility to identify these temperature dependent parameters from given temperature measurements via an optimal control problem. The solvability of the optimal control problem is analyzed and results of successful implementations are shown. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)