Thermoelastic System Research Articles

On the long-time behavior of a nonlinear damped porous thermoelastic system with second sound

In this paper, we consider a porous thermoelastic system with a micro-heat dissipation and a nonlinear frictional damping. We establish an explicit and general decay rate result, using some properties of the convex functions and the multiplier method. Our result is obtained without imposing any restrictive growth assumption on the damping term.

Uniform Stability of Second Sound Thermoelasticity with Distributed Delay

In this paper we consider a thermoelastic system of second sound with internal delay. Under suitable assumption on the weight of the delay, we prove, using the energy method, that the damping effect through heat conduction given by Cattaneo’s law is still strong enough to uniformly stabilize the system even in the presence of time delay.

Pullback attractors for a class of non-autonomous thermoelastic plate systems

In this article we study the asymptotic behavior of solutions, in the sense of pullback attractors, of the evolution system $\begin{cases}u_{tt} +Δ^2 u+a(t)Δθ=f(t,u),&t>τ,\ x∈Ω,\\θ_t-κΔ θ-a(t)Δ u_t=0,&t>τ,\ x∈Ω,\end{cases}$ subject to boundary conditions $u=Δ u=θ=0,\ t>τ,\ x∈\partial Ω,$ where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N≥ 2$, which boundary $\partialΩ$ is assumed to be a $\mathcal{C}^4$-hypersurface, $κ>0$ is constant, $a$ is an Hölder continuous function and $f$ is a dissipative nonlinearity locally Lipschitz in the second variable. Using the theory of uniform sectorial operators, in the sense of P. Sobolevskiǐ ([23]), we give a partial description of the fractional power spaces scale for the thermoelastic plate operator and we show the local and global well-posedness of this non-autonomous problem. Furthermore we prove existence and uniform boundedness of pullback attractors.

Direct imaging of the SSD and USB memory drives heating by thermo-elastic optical indicator microscopy

The visualization of mass storage controllers and flash memory chips in the USB2.0, USB3.0 and solid state device (SSD) devices was performed using a thermo-elastic optical indicator microscopy system. Heat distributions corresponding to the various operation/loading parameters such as writing block sizes were analyzed, which was showed device behavior on heating. Direct measurements of the temperature by a thermocouple were implemented in order to investigate the operation thermal states of the devices under test (DUTs) in the idle regime. Heat temperature differences dependence on loading block size (0.25kB – 64MB) was found for all DUTs. In order to explain the heat variation dependence on block size a series of measurements of the DUTs transfer rates were performed. The data transfer rate is drastically changed depending on the transferred block size (8B – 128MB), (0.2kB/s – 7MB/s), (0.3kB/s – 21MB/s), (2kB/s – 92MB/s) and (3.7kB/s – 222MB/s) for USB2.0, USB3.0, HDD, and SSD, respectively.

EXPONENTIAL DECAY OF THERMO-ELASTIC BRESSE SYSTEM WITH DISTRIBUTED DELAY TERM

The paper considered here is one-dimensional linear thermo-elastic Bresse system with a distributed delay term in the first equation. We prove the well-posedness and exponential stability result, this later will be shown without the usual assumption on the wave speeds. To achieve our goals, we make use of the semi-group method.

Decay rates for delayed abstract thermoelastic systems with Cattaneo law

In this paper we consider a thermoelastic type system with Cattaneo’s law and internal time delay. Under a suitable assumption on the weight of delay, we prove that the exponential stability of this system is reduced to an observability estimate for the corresponding uncontrolled system. The proof of the main results uses the methodology introduced by Haraux (Port Math 46:245–258, 1989) and generalized by Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001). Illustrating example is given.

Stabilisation of a thermo-elastic system under nonuniformly bounded disturbances in the boundary input

ABSTRACTWe consider the exponential stabilisation for a thermo-elastic system of type II with nonuniformly bounded external disturbances in the boundary control. We begin with finding feedback controls for the system without disturbances by Lyapunov function approach. Then, based on the resulting feedback control, we give new dynamic feedback control laws that make the system exponentially stable by tracking disturbances exponentially through a time-varying high gain observer. We apply the linear operator semi-group theory to prove the well-posedness of the resulting closed system. Finally, we prove that the solution of the closed-loop system can approach zero polynomially or exponentially by the Lyapunov function approach.

A generalized finite element method for linear thermoelasticity

We propose and analyze a generalized finite element method designed for linear quasistatic thermoelastic systems with spatial multiscale coefficients. The method is based on the local orthogonal decomposition technique introduced by Målqvist and Peterseim (Math. Comp. 83 (2014) 2583–2603). We prove convergence of optimal order, independent of the derivatives of the coefficients, in the spatial H1-norm. The theoretical results are confirmed by numerical examples.

Well-posedness and a general decay for a nonlinear damped porous thermoelastic system with second sound

Abstract In this paper, we consider a one-dimensional porous thermoelastic system with second sound and nonlinear feedback. We show the well-posedness, using the semigroup theory, and establish an explicit and general decay rate result, using some properties of convex functions and the multiplier method. Our result is obtained without imposing any restrictive growth assumption on the damping term.

Energy decay rate of transmission problem between thermoelasticity of type I and type II

In this paper, the energy decay rate of a 1-d mixed type I and type II thermoelastic system is considered. The system consists of two kinds of thermoelastic components. One is the classical thermoelasticity (so-called type I), another one is nonclassical thermoelasticity without dissipation (named type II). These two components are coupled at the interface satisfying certain transmission condition. We prove that the system is lack of uniform exponential decay rate and further obtain the sharp polynomial decay rate by resolvent estimates together with the diagonalization argument in linear algebra. Moreover, we present some numerical simulations to support these theoretical results.

On the numerical solution of a singular second-order thermoelastic system

In this article, we study an initial boundary value problem for a coupled system of thermoelasticity. Some existence and uniqueness results are given. The homotopy analysis method is employed to ob...

Recovery of a space-dependent vector source in anisotropic thermoelastic systems

We investigate the theoretical and numerical determination of a space-dependent vector source (load) in an anisotropic thermoelastic system of type-III from the knowledge of an additional final time measurement. The uniqueness of a solution to this inverse source problem is proved for various assumptions made on the convolution kernel. A convergent and stable iterative algorithm is proposed for the recovery of the unknown vector source in the linear case and, at the same time, a stopping criterion is also given. Three numerical experiments are considered to validate the properties of the proposed iterative procedure and the regularizing/stabilizing character of the corresponding stopping criterion. The numerical experiments carried out showed that it exists a certain limitation of the method with respect to the recovery of non-symmetric sources.

Non-Homogeneous Thermoelastic Timoshenko Systems

The well-established Timoshenko system is characterized by a particular relation between shear stress and bending moment from its constitutive equations. Accordingly, a (thermal) dissipation added on the bending moment produces exponential stability if and only if the so called “equal wave speeds” condition is satisfied. This remarkable property extends to the case of non-homogeneous coefficients. In this paper, we consider a non-homogeneous thermoelastic system with dissipation restricted to the shear stress. To this new problem, by means of a delicate control observability analysis, we prove that a local version of the equal wave speeds condition is sufficient for the exponential stability of the system. Otherwise, we study the polynomial stability of the system with decay rate depending on the regularity of initial data.

Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback

In this paper, we study the well-posedness and exponential decay for the porous thermoelastic system with the heat conduction given by Cattaneo’s law and a time-varying delay term, the coefficient of which is not necessarily positive. Using the semigroup arguments and variable norm technique of Kato, we first prove that the system is well-posed under a certain condition on the weight of the delay term, the weight of the elastic damping term and the speed of the delay function. By introducing a suitable energy and an appropriate Lyapunov functional, we then establish an exponential decay rate result.

Sensitivity analysis in thermoelastic problems using the complex finite element method

ABSTRACTThis article presents the complex finite element method (ZFEM) for the sensitivity analysis of thermoelastic systems. ZFEM, based on the complex Taylor series approach, performs finite element procedures using complex variables such that the response variables (temperature, stress) and their sensitivities with respect to an input parameter of interest (shape, mechanical and thermal properties, loading) are obtained simultaneously. ZFEM offers significant advantages over alternative sensitivity analyses that require direct derivations of the sensitivity formulae, multiple runs, and/or remeshing. To verify the numerical implementation, a hollow cylinder with convective boundary conditions on the inside and outside surface was considered. First-order derivatives of the stress fields were compared with exact solutions to demonstrate the accuracy of ZFEM sensitivities. The results indicate that the ZFEM-based derivatives are of high accuracy, thereby showing its applicability in the design and analysis of thermoelastic problems.

A general decay result for a multi-dimensional weakly damped thermoelastic system with second sound

In this article we consider an n-dimensional system of thermoelasticity with second sound in the presence of a weak frictional damping. We establish an explicit and general decay rate result, using some properties of convex functions. Our result is obtained without imposing any restrictive growth assumption on the frictional damping term.

Динамическая устойчивость нагретых геометрически нерегулярных пластин на основе модели Рейснера

На основе континуальной модели геометрически нерегулярной пластинки в рамках модели типа Рейснера решается задача динамической устойчивости нагретой ребристой пластинки под действием периодических по временной координате тангенциальных усилий. Тангенциальные усилия в уравнениях динамической устойчивости нагретой пластины конкретизируются на основании решения неоднородной краевой задачи безмоментной термоупругости в перемещениях. Система сингулярных уравнений динамической устойчивости записана в функции прогиба и дополнительных функциях, характеризующих закон изменения касательных напряжений в вертикальных плоскостях по переменным $x$ и $y$. Решение сводится к уравнению Матье, параметры которого представлены в терминах классической теории пластин и содержат поправки от температуры, поперечных сдвигов и подкрепляющих ребер. Определяются первые три области динамической устойчивости термоупругой системы. Проводится количественный анализ влияния температуры, деформации сдвига в вертикальных плоскостях и относительной высоты ребер на конфигурацию областей динамической устойчивости.

Stability for thermoelastic plates with two temperatures

We investigate the well-posedness, the exponential stability, or the lack thereof, of thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial boundary value problems for different boundary conditions deal with systems of partial differential equations involving Schrodinger like equations, hyperbolic and elliptic equations, which have a different character compared to the classical one with the usual single temperature. Depending on the model -with Fourier or with Cattaneo type heat conduction -we obtain exponential resp. non-exponential stability, thus providing another examples where the change from Fourier's to Cattaneo's law leads to a loss of exponential stability.

Time Optimal Control of a Thermoelastic System

This paper considers the numerical approximation for the time optimal control problem of a thermoelastic system with some control and state constraints. By the Galerkin finite element method (FEM), the original problem is projected into a semidiscrete optimal control problem governed by a system of ordinary differential equations. Then the optimal time and control parameterization method is applied to reduce the original system to an optimal parameter selection problem, in which both the optimal time and control are taken as decision variables to be optimized. This problem can be solved as a nonlinear optimization problem by a hybrid algorithm consisting of chaotic particle swarm optimization (CPSO) and sequential quadratic programming (SQP) algorithm. The numerical simulations demonstrate the effectiveness of the proposed numerical approximation method.

Spectral analysis of thermoelastic systems under nonclassical thermal models

ABSTRACTWe study some spectral properties of the solutions to generalized thermoelastic systems under Lord–Shulman, Green–Lindsay, and Green–Naghdi of type-II models. First, we prove that the linear operator of each model has compact resolvent and generates a C0−semigroup in an appropriate Hilbert space. We also show that there is a sequence of generalized eigenfunctions of the linear operator that forms a Riesz basis. By a detailed spectral analysis, we obtain the expressions of the spectrum and we deduce that the spectrum-determined growth condition holds. Therefore, if the imaginary axis is not an asymptote of the spectrum, we prove that the energy of each model decays exponentially to a rate determined explicitly by the physical parameters. Finally, some simulations are given for each model to support our results.