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.     

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.     

On a structural acoustic model which incorporates shear and thermal effects in the structural component

In this paper we consider a structural acoustic model which takes account of thermal effects over and above displacement, rotational inertia and shear effects in the flat flexible structural component of the model. Thus the structural medium is a Reissner-Mindlin plate into which an additional degree of freedom, viz. temperature variation in the plate, has been introduced and the constitutive equations for the structural acoustic model couple parabolic dynamics with hyperbolic dynamics. We show unique solvability of the mathematical model and investigate the effect of the presence of thermal effects on the mechanical dissipation devices needed to attain uniform stabilization of the two-dimensional model in which the structural component is a Timoshenko beam. It turns out that, as in linear structural acoustic models which use the Euler-Bernoulli equation or the Kirchoff equation to describe the deflections of the thermo-elastic structural medium, uniform stabilization of the energy associated with the model can be attained without introducing mechanical dissipation at the free edge of the beam. Open problems with regard to the stabilization of the three-dimensional model are outlined.     

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.     

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).     

A class of quasilinear degenerate elliptic problems

We establish the existence of solutions for a class of quasilinear degenerate elliptic equations. The equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain, and degeneracy only on the boundary. Equations of transonic gas dynamics, for example, satisfy this property in the region of subsonic flow and are degenerate across the sonic surface. We prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary. The maximal rate of blow-up at the degenerate boundary is bounded by the “degree of degeneracy” in the principal coefficients of the quasilinear elliptic operator. Our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation, the pressure-gradient equations of the compressible Euler equations, and the singular quasilinear anisotropic elliptic problems, and extend to the class of equations which satisfy the structure condition, such as the shallow water equation, compressible isentropic two-dimensional Euler equations, and general two-dimensional nonlinear wave equations. Our study provides a general framework to analyze degenerate elliptic problems arising in the self-similar reduction of a broad class of two-dimensional Cauchy problems.     

Low Mach number limit of viscous polytropic fluid flows

This paper studies the singular limit of the non-isentropic Navier-Stokes equations with zero thermal coefficient in a two-dimensional bounded domain as the Mach number goes to zero. A uniform existence result is obtained in a time interval independent of the Mach number, provided that the initial data satisfy the “bounded derivative conditions”, that is, the time derivatives up to order two are bounded initially, and Navier?s slip boundary condition is imposed.     

Bifurcation with a two-dimensional kernel

We present a theorem ensuring the existence of local solution branches for one-parameter bifurcation problems in which the linearization at the trivial solution possesses a two-dimensional kernel. In particular, we provide a straightforward “test” that is sufficient for the existence of local solution continua. We demonstrate our abstract theorem with several concrete examples for second-order systems of elliptic partial differential equations with symmetry.     

Transonic potential flows in a convergent-divergent approximate nozzle

In this paper we prove existence, uniqueness and regularity of certain perturbed (subsonic-supersonic) transonic potential flows in a two-dimensional Riemannian manifold with “convergent-divergent” metric, which is an approximate model of the de Laval nozzle in aerodynamics. The result indicates that transonic flows obtained by quasi-one-dimensional flow model in fluid dynamics are stable with respect to the perturbation of the velocity potential function at the entry (i.e., tangential velocity along the entry) of the nozzle. The proof is based upon linear theory of elliptic-hyperbolic mixed type equations in physical space and a nonlinear iteration method.     

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.     

Evidence for a Phase Transition in Three-Dimensional Lattice Models

It was recently discovered that an eigenvector structure of commutative families of layer-to-layer matrices in three-dimensional lattice models is described by a two-dimensional spin lattice generalizing the notion of one-dimensional spin chains. We conjecture the relations between the two-dimensional spin lattice in the thermodynamic limit and the phase structure of three-dimensional lattice models. We consider two simplest cases: the homogeneous spin lattice related to the Zamolodchikov–Bazhanov–Baxter model and a chess spin lattice related to the Stroganov–Mangazeev elliptic solution of the modified tetrahedron equation. Evidence for the phase transition is obtained in the second case.     

Uniform stability for the Timoshenko beam with tip load

In this paper we consider a hybrid elastic model consisting of a Timoshenko beam and a tip load at the free end of the beam. We show that uniform stabilization of the model which includes the rotary inertia of the tip load can be obtained when feedback boundary moment and force controls are applied at the point of contact between the beam and the tip load. However, in the presence of the load stabilization is “slower” and subject to a restriction on the boundary data at the free end of the beam.     

Competitive dynamics of e-commerce web sites

The characteristic of “rich gets richer” is much more conspicuous on the Internet, which is the important cause of “power-law” characteristic of the “winner-takes-all” markets. A competitive model of e-commerce web sites is presented in order to explore the effects of competition and the characteristic of “rich gets richer” on Internet economy, and also to investigate how the “winner-takes-all” phenomenon comes forth. Then these ordinary differential equations are qualitatively analyzed and numerically simulated to study the dynamics of competition. The effects of parameters on the dynamic behavior in the model are also discussed.     

Interval modeling of dynamics for multibody systems

Modeling of multibody systems is an important though demanding field of application for interval arithmetic. Interval modeling of dynamics is particularly challenging, not least because of the differential equations which have to be solved in the process. Most modeling tools transform these equations into a (non-autonomous) initial value problem, interval algorithms for solving of which are known. The challenge then consists in finding interfaces between these algorithms and the modeling tools. This includes choosing between “symbolic” and “numerical” modeling environments, transforming the usually non-autonomous resulting system into an autonomous one, ensuring conformity of the new interval version to the old numerical, etc. In this paper, we focus on modeling multibody systems’ dynamics with the interval extension of the “numerical” environment MOBILE, discuss the techniques which make the uniform treatment of interval and non-interval modeling easier, comment on the wrapping effect, and give reasons for our choice of MOBILE by comparing the results achieved with its help with those obtained by analogous symbolic tools.     

A metric approach to asymptotic analysis

We introduce a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. We relate this notion to a concept of “firm” asymptotic function. We use these notions to study boundedness properties which can be applied to continuity questions for some operations on sets and functions. Such questions arise in stability analysis of Hamilton-Jacobi equations. We present some other applications such as an extension of a theorem of Dieudonné and existence results in optimization and fixed point theory.     

Local radial point interpolation method for the fully developed magnetohydrodynamic flow

In this paper, a local radial point interpolation method (LRPIM) is presented to obtain the numerical solutions of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through a straight duct of rectangular section with arbitrary wall conductivity and orientation of applied magnetic field. Local weak forms are developed using weighted residual method locally for the governing equations of fully developed MHD flow. The shape functions from LRPIM possess the delta function property. Therefore, essential boundary conditions can be applied as easily as that in the finite-element method. The implementation procedure of LRPIM method is node based, and it doesn’t need any “mesh” or “element”. Computations have been carried out for different Hartmann numbers, wall conductivities and orientations of applied magnetic field.     

Continuous-time accelerated block successive overrelaxation methods for time-dependent Stokes equations

In order to solve the time-dependent Stokes equation, we follow the “Method of Lines” to obtain structured linear constant-coefficient differential–algebraic equations (DAEs). By taking advantage of the structure, we propose a class of waveform relaxation methods, called continuous-time accelerated block SOR (CABSOR) methods, for solving the obtained DAEs. The new methods are theoretically analyzed. The theory is applied to a two-dimensional time-dependent Stokes equation and verified by numerical experiments.     

On the stability of invariant sets of systems with impulse effect

A system of differential equations with impulse effect is considered. It is assumed that this system has an invariant set M$M$. By means of the direct Lyapunov method, the necessary and sufficient conditions of its uniform asymptotic stability are obtained. The conditions on the perturbations of right hand sides of differential equations and impulse effects, under which the uniform asymptotic stability of the invariant set M$M$ of the “nonperturbed” system implies the uniform asymptotic stability of the invariant set of the “perturbed” system, are obtained. The stability properties of invariant sets of periodic systems are also studied.     

Strongly Gorenstein projective, injective, and flat modules

In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three classes of modules give us a new characterization of the first modules, and confirm that there is an analogy between the notion of “Gorenstein projective, injective, and flat modules” and the notion of the usual “projective, injective, and flat modules”.     

The nonsteady-state planar oscillations of a plate and oscillator on a two-layer base

We consider a plane dynamic contact problem for an inhomogeneous base of the following form: a soft layer on a rigid layer of an elastic half-plane. The layer is represented by a Winkler model, corresponding to the long-wave asymptotics of the equations of elasticity theory. The problem is reduced to a system of integro-differential equations that is solved numerically. We present the results of the computations of dynamic characteristics describing the oscillations of a rigid body and an oscillator on this base.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 64–68.