On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam

This paper is concerned with a model which describes the interaction of sound and elastic waves in a structural acoustic chamber in which one “wall” is flexible and flat. The model is new in the sense that the composite dynamics of the three-dimensional structure is described by the linearized equations for a gas defined on the interior of the chamber and the Reissner-Mindlin plate equations on the two-dimensional flat wall of the chamber, while, if a two-dimensional acoustic chamber is considered, the Timoshenko beam equations describe the deflections of the one-dimensional “wall.” With a view to achieving uniform stabilization of the structure linear feedback boundary damping is incorporated in the model, viz. in the wave equation for the gas and in the system of equations for the vibrations of the elastic medium. We present the uniform stability result for the case of a two-dimensional chamber and outline the method for the three-dimensional model which shows strong resemblance with the system of dynamic plane elasticity.     

Boundary stabilization of a 3-dimensional structural acoustic model

The main result of this paper provides uniform decay rates obtained for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall. The uniform stabilization is achieved by introducing a nonlinear dissipation acting via boundary forces applied at the edge of the plate and viscous or boundary damping applied to the wave equation. The results obtained in this paper extend, to the non-analytic, hyperbolic-like setting, the results obtained previously in the literature for acoustic problems modeled by structurally damped plates (governed by analytic semigroups). As a bypass product, we also obtain optimal uniform decay rates for the Euler Bernoulli plate equations with nonlinear boundary dissipation acting via shear forces only and without (i) any geometric conditions imposed on the domain ,(ii) any growth conditions at the origin imposed on the nonlinear function. This is in contrast with the results obtained previously in the literature ([22] and references therein).     

Strong stabilization of a structural acoustic model,which incorporates shear and thermal effects in the structural component

In this paper we consider the question of stabilization of a linear three‐dimensional structural acoustic model, which incorporates displacement, rotational inertia, shear and thermal effects in the flat flexible structural component of the model. We show strong stabilization of the coupled model without incorporating viscous or boundary damping in the equations for the gas dynamics and without imposing geometric conditions. It turns out that damping is needed in the interior of the plate. Our main tool is an abstract resolvent criterion due to Y. Tomilov. Copyright © 2009 John Wiley & Sons, Ltd.     

On the stabilizability of a structural acoustic model which incorporates shear effects in the structural component

In this paper we consider a linear three-dimensional structural acoustic model which takes account of displacement, rotational inertia and shear effects in the flat flexible structural component of the model. Thus the deflections of the structural component of the structure are governed by the Reissner–Mindlin plate equations. We show strong stabilization of the coupled model without incorporating viscous or boundary damping in the equations for the gas dynamics and without imposing geometric conditions. It turns out that damping is needed in the interior of the plate, to which end Kelvin–Voigt damping is introduced in the plate equations. As our main tool we use a resolvent criterion for strong stability due to Tomilov.     

Stability analysis of coupled structural acoustics PDE models under thermal effects and with no additional dissipation

In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber Ω. This acoustic wave equation is coupled on a boundary interface Γ₀ to a two dimensional system of thermoelasticity: this thermoelastic PDE is composed in part of a structural beam or plate equation, which governs the vibrations of flexible wall portion Γ₀ of the chamber Ω. Moreover, this elastic dynamics is coupled to a heat equation which also evolves on Γ₀, and which imparts a thermal damping onto the entire structural acoustic system. As we said, the interaction between the wave and thermoelastic PDE components takes place on the boundary interface Γ₀, and involves coupling boundary terms which are above the level of finite energy. We analyze the stability properties of this coupled structural acoustics PDE model, in the absence of any additive feedback dissipation on the hard walls Γ₁ of the boundary . Under a certain geometric assumption on Γ₁, an assumption which has appeared in the literature in connection with structural acoustic flow, and which allows for the invocation of a recently derived microlocal boundary trace estimate, we show that classical solutions of this thermally damped structural acoustics PDE decay uniformly to zero, with a rational rate of decay.     

Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

We consider coupled PDE systems comprising of a hyperbolic and a parabolic-like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A specific prototype consists of a wave equation defined on a three-dimensional bounded domain Ω coupled with a thermoelastic plate equation defined on Γ ₀—a flat surface of the boundary \partial Ω . Thus, the coupling between the wave and the plate takes place on the interface Γ ₀. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the ``minimal amount' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to Γ <sub>0</sub>, suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity is accounted for in the plate model; (ii) the plate model does not account for any type of mechanical damping, including the structural damping most often considered in the literature; (iii) there is no mechanical damping placed on the interface Γ <sub>0</sub>; (iv) the boundary damping is nonlinear without a prescribed growth rate at the origin; (v) the undamped portions of the boundary \partial Ω are subject to Neumann (rather than Dirichlet) boundary conditions, which is a recognized difficulty in the context of stabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold. The main mathematical challenge is to show how the thermal energy is propagated onto the hyperbolic component of the structure. This is achieved by using a recently developed sharp theory of boundary traces corresponding to wave and plate equations, along with the analytic estimates recently established for the co-continuous semigroup associated with thermal plates subject to free boundary conditions. These trace inequalities along with the analyticity of the thermoelastic plate component allow one to establish appropriate inverse/ recovery type estimates which are critical for uniform stabilization. Our main result provides ``optimal' uniform decay rates for the energy function corresponding to the full structure. These rates are described by a suitable nonlinear ordinary differential equation, whose coefficients depend on the growth of the nonlinear dissipation at the origin. \par Accepted 12 May 2000. Online publication 6 October 2000.     

Boundary stabilization of structural acoustic interactions with interface on a Reissner–Mindlin plate

Boundary stabilization of a structural acoustic model comprised of a wave and a Reissner–Mindlin plate is addressed. Both the components of the dynamics are subject to localized nonlinear boundary damping: the acoustic dissipative feedback is restricted to the flexible boundary and only a portion of the rigid wall; the plate is damped only on a segment of its edge.Derivation of stabilization/observability inequalities for a coupled system requires weighted energy multipliers dependent on the geometry of the domain, and special microlocal trace estimates for the Reissner–Mindlin plate. The behavior of the energy at infinity can be quantified by a solution to an explicitly constructed nonlinear ODE. The nonlinearities in the feedbacks may include sub- and superlinear growth at infinity, in which case the decay scheme presents a trade-off between the regularity of trajectories and attainable uniform dissipation rates of the finite energy.     

Thermal Effects in a Two-Dimensional Hybrid Elastic Structure

In this paper we consider a two-dimensional hybrid thermo-elastic structure consisting of a thermo-elastic plate which has a beam attached to its free end. We show that the initial-boundary-value problem for the interactive system of partial differential equations which take account of the mechanical strains/stresses and the thermal stresses in the plate and the beam, can be associated with a uniformly bounded evolution operator. It turns out that the interplay of parabolic dynamics due to the thermal effects in the hybrid structure and the hyperbolic dynamics associated with the elasticity of the structure yields analyticity for the entire system. This result yields solvability for the problem under optimal initial freedom of the displacement, velocity, and temperature in the plate and the beam, while uniform stability is readily available.     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

A rate of rational decay is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity Ω and (ii) a structurally damped elastic equation, to describe time‐evolving displacements along the flexible portion Γ₀ of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter η∈[0,1]. The coupling between these two distinct dynamics occurs across the boundary interface Γ₀. Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin–Voight damping is present in fourth‐order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of . By way of deriving these stability results, necessary a priori inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay. Copyright © 2016 John Wiley & Sons, Ltd.     

Control-theoretic properties of structural acoustic models with thermal effects,I. Singular estimates

We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic equations with Neumann boundary data.     

Uniform stabilization of a nonlinear structural acoustic model with a Timoshenko beam interface

This paper is concerned with a nonlinear model which describes the interaction of sound and elastic waves in a two‐dimensional acoustic chamber in which one flat ‘wall’, the interface, is flexible. The composite dynamics of the structural acoustic model is described by the linearized equations for a gas defined on the interior of the chamber and the nonlinear Timoshenko beam equations on the interface. Uniform stability of the energy associated with the interactive system of partial differential equations is achieved by incorporating a nonlinear feedback boundary damping scheme in the equations for the gas and the beam. Copyright © 2006 John Wiley & Sons, Ltd.     

Uniform stability in structural acoustic models with flexible curved walls

The aim of this paper is twofold. First, we develop an explicit extension of the Kirchhoff model for thin shells, based on the model developed by Michel Delfour and Jean-Paul Zolésio. This model relies heavily on the oriented distance function which describes the geometry. Once this model is established, we investigate the uniform stability of a structural acoustic model with structural damping. The result no longer requires that the active wall be a plate. It can be virtually any shell, provided that the shell is thin enough to accommodate the curvatures.     

多孔介质平板通道发展传热中非局部热平衡时的温度分布特征

研究了多孔介质平板通道中,Darcy流体发展传热强迫对流非局部热平衡下,固相骨架和孔隙流体的温度分布特征．考虑流体流动方向的热传导以及固相和流相相互作用的粘性耗散,根据非局部热平衡的两能量方程模型,得到了常壁温度时多孔介质固相骨架温度和孔隙流体温度的解析解．证明了当两相间的热交换系数趋于无穷大时,两能量方程的温度解趋于局部热平衡时一能量方程的温度解．针对不同的无量纲参数,给出了固相和流相的温度分布状态,通过参数研究,揭示了非局部热平衡强迫对流时温度对无量纲参数的依赖关系．     

Thermoacoustic response of a simply supported isotropic rectangular plate in graded thermal environments

A theoretical model is developed to investigate the thermoacoustic response of a simply supported plate subjected to combined thermal and acoustic excitations, with two typical graded thermal environments considered. The thermoacoustic governing equation derived by incorporating the thermal moments, membrane forces and acoustic loads into the plate vibration equation is solved using the modal decomposition approach. In combination with the thermal boundary conditions, the Fourier heat conduction equation is solved for the graded temperature distribution in the plate. Fluid-structure coupling between acoustic excitation and the plate is ensured by adopting the velocity continuity condition at the fluid–plate interface. With focus placed on the effect of graded thermal environments on plate vibroacoustic response, numerical investigations reveal the necessity for considering thermal moments in theoretical modeling, particularly when graded thermal environments are of common concern for aircraft structures.     

Inverse problem for structural acoustic interaction

We consider an inverse problem of determining a source term for a structural acoustic partial differential equation (PDE) model that is comprised of a two- or a three-dimensional interior acoustic wave equation coupled to an elastic plate equation. The coupling takes place across a boundary interface. For this PDE system, we obtain uniqueness and stability estimates for the source term from a single measurement of boundary values of the “structure” (acceleration of the elastic plate). The proof of uniqueness is based on a Carleman estimate (first version) of the wave problem within the chamber. The proof of stability relies on three main points: (i) a more refined Carleman estimate (second version) and its resulting implication, a continuous observability-type estimate; (ii) a compactness/uniqueness argument; (iii) an operator theoretic approach for obtaining the needed regularity in terms of the initial conditions.     

Generalized thermoelasticity of beams under partial thermal shock

In this paper, the generalized thermoelastic response of a beam subjected to a partial lateral thermal shock is analysed. The beam is made of homogeneous and isotropic material and is assumed to follow the Hooke law for its constitutive material. The displacement gradient is small and the linear form of strain-displacement relations is used for the beam. The equations of motion and the boundary conditions of the beam are derived based on Hamilton’s principle. According to the first and second laws of thermodynamics, a non-Fourier constitutive equation is employed to derive the energy equation of the beam. The non-Fourier effects lead to the constitutive equation of the hyperbolic type and thus the thermal and mechanical waves can be observed. The propagation of waves in the beam are simulated by finite element model and the wave reflections for different types of boundary conditions are studied. The relaxation time is considered as a significant parameter and results show that energy absorption of the structure and the wave propagation speed depend upon this parameter.     

Uniform stabilization of a quasilinear plate model in hyperbolic thermoelasticity

We study dynamic elastic deformations of a quasilinear plate model of Timoshenko's type under thermal effects, which are modeled by Cattaneo's law. We prove uniform exponential stabilization of the total energy as time approaches infinity. We show global wellposedeness of the model and build a convenient Lyapunov function, which allow us to conclude the main result of this work. Copyright © 2015 John Wiley & Sons, Ltd.     

Perturbation analysis of radiative effect on free convection flows in porous medium in the presence of pressure work and viscous dissipation

The effect of thermal radiation with a regular three-parameter perturbation analysis has been studied for the effects in some free convection flows of Newtonian fluid-saturated porous medium. The effects of the thermal radiation, permeability of the porous medium, pressure stress work and viscous dissipation on the flows and temperature fields have been included in the analysis. Four different vertical flows have been analyzed, those adjacent to an isothermal surface, uniform heat flux surface, a plane plume and flow generated from a horizontal line energy source, and, a vertical adiabatic surface. Rosseland approximation is used to describe the radiative heat flux in the energy equation. The numerical results of the perturbation analysis for four conditions are solved numerically by the fourth-order Runge–Kutta integration scheme. Numerical values of the main physical quantities are the skin friction and a heat transfer and total heat and mass convected downstream are presented in a tabular form with the parameters characterizing the radiation, permeability of the porous medium, pressure stress work and viscous dissipation. The obtained results are compared and a representative set is displayed graphically to illustrate the influences of the radiation, permeability of the porous medium, pressure stress work and viscous dissipation on the velocity and the temperature profiles.     

Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks

We consider a dynamic linear shallow shell model, subject to nonlinear dissipation active on a portion of its boundary in physical boundary conditions. Our main result is a uniform stabilization theorem which states a uniform decay rate of the resulting solutions. Mathematically, the motion of a shell is described by a system of two coupled partial differential equations, both of hyperbolic type: (i) an elastic wave in the 2-d in-plane displacement, and (ii) a Kirchhoff plate in the scalar normal displacement. These PDEs are defined on a 2-d Riemann manifold. Solution of the uniform stabilization problem for the shell model combines a Riemann geometric approach with microlocal analysis techniques. The former provides an intrinsic, coordinate-free model, as well as a preliminary observability-type inequality. The latter yield sharp trace estimates for the elastic wave—critical for the very solution of the stabilization problem—as well as sharp trace estimates for the Kirchhoff plate—which permit the elimination of geometrical conditions on the controlled portion of the boundary.