A rectangular element for the Reissner–Mindlin plate

We investigate a new rectangular element for the Reissner–Mindlin model based on the primitive variable system. Nonconforming rotated Q₁ element is used to approximate the transverse displacement, and the biquadratic element is used for the rotation. A convergent error estimate is obtained, which is independent of the thickness of the plate. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 184–193, 2000     

Dynamic boundary stabilization of a Reissner–Mindlin plate with Timoshenko beam

This paper is concerned with well‐posedness results for a mathematical model for the transversal vibrations of a two‐dimensional hybrid elastic structure consisting of a rectangular Reissner–Mindlin plate with a Timoshenko beam attached to its free edge. The model incorporates linear dynamic feedback controls along the interface between the plate and the beam. Classical semigroup methods are employed to show the unique solvability of the coupled initial‐boundary‐value problem. We also show that the energy associated with the system exhibits the property of strong stability. Copyright © 2004 John Wiley & Sons, Ltd.     

Spectral convergence for vibrating systems containing a part with negligible mass

We consider a set of Neumann (mixed, respectively) eigenvalue problems for the Laplace operator. Each problem is posed in a bounded domain Ω_R of ?ⁿ, with n=2,3, which contains a fixed bounded domain B where the density takes the value 1 and 0 outside. Ω_R has a diameter depending on a parameter R, with R?1, diam(Ω_R) →∞ as R→∞ and the union of these sets is the whole space ?ⁿ (the half space {x∈?ⁿ/x_n<0}, respectively). Depending on the dimension of the space n, and on the boundary conditions, we describe the asymptotic behaviour of the eigenelements as R→∞. We apply these asymptotics in order to derive important spectral properties for vibrating systems with concentrated masses. Copyright © 2005 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.     

Analysis of a 3‐node mixed‐shear‐projected triangular element method for Reissner–Mindlin plates

This article analyses an existing 3‐node hybrid triangular element, called MiSP3, for Reissner–Mindlin plates which behaves robustly in numerical benchmark tests (Ayad, Dhatt, and Batoz, Int J Numer Method Eng 42 (1998), 1149–1179). Based on Hellinger‐Reissner variational principle and the mixed shear interpolation/projection technique of MITC family, the MiSP3 element uses continuous piecewise linear polynomials for the approximations of displacements and a piecewise‐independent equilibrium mode for the approximations of bending moments/shear stresses. Due to local elimination of the parameters of moments/stresses, the element is almost of the same computational cost as the conforming linear triangular displacement element. We derive uniform stability and convergence results with respect to the plate thickness. The main tools of our analysis are the self‐equilibrium relation of the moments/stresses approximations, the properties of the mixed shear interpolation and the discrete Helmholtz decomposition of the shear stress approximation. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 241–258, 2017     

A-priori and a-posteriori error analysis for a family of Reissner–Mindlin plate elements

A family of plate elements introduced by Falk and Tu [25] is considered. A new stability and a-priori error analysis is given. In addition, an a-posteriori error estimate is proved. The analysis is confirmed by numerical benchmark computations. AMS subject classification (2000)  65F20     

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number of ?‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ? → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ? → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.     

Interior estimates for a low order finite element method for the Reissner–Mindlin plate model

Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner–Mindlin plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the global approximability of the finite element method. As an application, we show that for the soft simply supported plate, the Arnold–Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal order convergence cannot hold globally due to the boundary layer. Numerical results are given which support our conclusion. This revised version was published online in June 2006 with corrections to the Cover Date.     

On stability of hyperbolic thermoelastic Reissner–Mindlin–Timoshenko plates

In the present article, we consider a thermoelastic plate of Reissner–Mindlin–Timoshenko type with the hyperbolic heat conduction arising from Cattaneo's law. In the absence of any additional mechanical dissipations, the system is often not even strongly stable unless restricted to the rotationally symmetric case, and so on. We present a well‐posedness result for the linear problem under general mixed boundary conditions for the elastic and thermal parts. For the case of a clamped, thermally isolated plate, we show an exponential energy decay rate under a full damping for all elastic variables. Restricting the problem to the rotationally symmetric case, we further prove that a single frictional damping merely for the bending component is sufficient for exponential stability. To this end, we construct a Lyapunov functional incorporating the Bogovski? operator for irrotational vector fields, which we discuss in the appendix. Copyright © 2014 John Wiley & Sons, Ltd.     

An efficient numerical method for the resolution of the Kirchhoff‐Love dynamic plate equation

We solve numerically the Kirchhoff‐Love dynamic plate equation for an anisotropic heterogeneous material using a spectral method. A mixed velocity‐moment formulation is proposed for the space approximation allowing the use of classical Lagrange finite elements. The benefit of using high order elements is shown through a numerical dispersion analysis. The system resulting from this spatial discretization is solved analytically. Hence this method is particularly efficient for long duration experiments. This time evolution method is compared with explicit and implicit finite differences schemes in terms of accuracy and computation time. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A global attractor for a fluid–plate interaction model accounting only for longitudinal deformations of the plate

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier–Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in‐plane motions on a flexible flat part of the boundary. The main novelty of the model is the assumption that the transversal displacements of the plate are negligible relative to in‐plane displacements. These kinds of models arise in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. Under some conditions this attractor is an exponentially attracting single point. We also show that the corresponding linearized system generates an exponentially stable C₀‐semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results mean that dissipation of the energy in the fluid because of viscosity is sufficient to stabilize the system. Copyright © 2011 John Wiley & Sons, Ltd.     

On a generalization of the Schauder and Krasnosel'skii fixed points theorems on Dunford–Pettis spaces and applications

In this paper, we give a generalization of the Schauder and Krasnosel'skii fixed point theorems in Dunford–Pettis spaces. Both of these theorems can be used to resolve some open problems posed by Jeribi (Nonlinear Anal.: Real World Appl. 2002; 3 :85–105); and Latrach (J. Math. Phys. 1996; 37 :1336–1348). Further, we applied our work to prove some existence results for a source problem with general boundary conditions in L₁ spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

On the Leibniz Cohomology of Vector Fields

Gelfand and Fuks have studied the cohomology of the Lie algebra of vector fields on a manifold. In this article, we generalize their main tools to compute the Leibniz cohomology, by extending the two spectral sequences associated to the diagonal and the order filtration. In particular, we determine some new generators for the diagonal Leibniz cohomology of the Lie algebra of vector fields on the circle.     

On the Spectrum of the Orr–Sommerfeld Equation on the Semiaxis

The Orr–Sommerfeld equation is a spectral problem which is known to play an important role in hydrodynamic stability. For an appropriate operator theoretical realization of the equation, we will determine the essential spectrum, and calculate an enclosure of the set of all eigenvalues by elementary analytical means.     

On a compound Poisson risk model with dependence and in the presence of a constant dividend barrier

In this paper, we consider a classical risk process with dependence and in the presence of a constant dividend barrier. The dependence structure between the claim amounts and the interclaim times is introduced through a Farlie–Gumbel–Morgenstern copula. We analyze the expectation of the discounted penalty function and the expectation of the present value of the distributed dividends. For each function, an integro‐differential equation with boundary conditions is derived, and the solution is provided. Finally, we find an explicit solution for each function when the claim amounts are exponentially distributed. We illustrate the impact of the dependence on these two quantities. Copyright © 2012 John Wiley & Sons, Ltd.     

Spectral gaps for the Dirichlet‐Laplacian in a 3‐D waveguide periodically perturbed by a family of concentrated masses

We consider a spectral problem for the Laplace operator in a periodic waveguide perturbed by a family of “heavy concentrated masses”; namely, Π contains small regions of high density, which are periodically distributed along the z axis. Each domain has a diameter and the density takes the value in and 1 outside; m and ε are positive parameters, , . Considering a Dirichlet boundary condition, we study the band‐gap structure of the essential spectrum of the corresponding operator as . We provide information on the width of the first bands and find asymptotic formulas for the localization of the possible gaps.     

The construction of plane elastomechanics and Mindlin plate elements of B-spline wavelet on the interval

Based on two-dimensional tensor product B-spline wavelet on the interval (BSWI), a class of C0 type plate elements is constructed to solve plane elastomechanics and moderately thick plate problems. Instead of traditional polynomial interpolation, the scaling functions of two-dimensional tensor product BSWI are employed to form the shape functions and construct BSWI elements. Unlike the process of direct wavelets adding in the previous work, the elemental displacement field represented by the coefficients of wavelets expansions is transformed into edges and internal modes via the constructed transformation matrix in this paper. The method combines the versatility of the conventional finite element method (FEM) with the accuracy of B-spline functions approximation and various basis functions for structural analysis. Some numerical examples are studied to demonstrate the proposed method and the numerical results presented are in good agreement with the closed-form or traditional FEM solutions.     

On the global existence and small dispersion limit for a class of complex Ginzburg–Landau equations

In this paper we consider a class of complex Ginzburg–Landau equations. We obtain sufficient conditions for the existence and uniqueness of global solutions for the initial‐value problem in d‐dimensional torus ??^d, and that solutions are initially approximated by solutions of the corresponding small dispersion limit equation for a period of time that goes to infinity as dispersive coefficient goes to zero. Copyright © 2008 John Wiley & Sons, Ltd.     

On the stability of a quadratic Jensen type functional equation

In this paper we obtain the general solution of the quadratic Jensen type functional equation

and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and G vruta.