Approximation of a circular cylindrical shell by Clough-Johnson flat plate finite elements

Summary In this paper, we study the approximation of a right circular cylindrical shell by a nonconforming method using Clough-Johnson flat plate finite elements. Compatibility conditions which have to be satisfied by the degrees of freedom at every node of the triangulation are given. Then, we prove that convergence is insured for shallow shells when using particular families of triangulations which are practically easy to implement. Finally, we propose a new approximation method by flat plate finite elements which assures the convergence for any kind of circular cylindrical shell.     

A-posteriori modeling error estimation for hierarchic plate models

Summary. Hierarchic plate models are dimensional reductions obtained by semidiscretization of the three-dimensional plate problem in the transverse direction and energy projection. We derive computable a-posteriori estimators for the modeling error thus incurred under the assumption that the resulting two-dimensional plate models are solved exactly. The estimators are valid for homogeneous, monoclinic materials, for plates with unsmooth midsurfaces and for a wide class of variational edge conditions. Computable a-posteriori bounds on the effectivity indices are also derived and sufficient conditions for the asymptotic (i.e. as the plate thickness tends to zero) and the spectral (i.e. as the order of the plate model tends to infinity) exactness of the estimators are given. Numerical examples are presented. Received August 16, 1994 / Revised version received March 20, 1995     

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.     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω. Research supported by the RIP program of Oberwolfach Institut and by the Tunisian Ministry for Scientific Research and Technology (MRST) under Grant 02/UR/15-01. Research supported by the RIP program of Oberwolfach Institut. (Received: September 17, 2003; revised: February 26, 2004)     

Diffraction of transverse horizontal waves in Fibonacci piezoelectric superlattices

The transfer matrix and global matrix techniques are used in the analysis of diffraction of transverse horizontal waves in piezoelectric superlattices. The system is composed of a set of different piezoelectric layers with quasiperiodic structure according to the Fibonacci sequence. The diffraction spectrum of different configurations of piezoelectric Fibonacci structures is studied. Numerical results for the diffraction spectrum for the piezoelectric Fibonacci superlattice are obtained and a comparison between the results of the transfer and global matrix methods is given, exhibiting the advantage of the latter.     

Approximation of general arch problems by straight beam elements

Summary In this paper, we approximate the solution of a problem of a general arch by a nonconforming method using straight beam elements and taking into account numerical integration. Compatibility conditions which have to be satisfied at the mesh points are given. These conditions ensure for this method the same order of convergence as usual conforming finite element methods.     

Nonlinear vibrations of offshore structures under random wave excitation

An account is given of some mathematical methods which have been used to analyse the response of offshore structures to random wave excitation. The analysis of nonlinear phenomena and the assessment of related nongaussian probability distributions are emphasized. The following problems are analysed in some depth: probability distributions for Morison-type wave loading; response near resonance of nonlinearly damped systems to random excitation; parametric resonance and instability of nonlinearly damped systems with randomly fluctuating restoring force coefficient. Solutions to each of these problem areas are illustrated by practical examples.     

Sur la convergence des méthodes d'éléments finis nonconformes pour des problèmes linéaires de coques minces

Résumé Dans cet article, nous étudions la convergence des méthodes d'éléments finis nonconformes pour l'approximation des problèmes de coques minces générales. Nous établissons des conditions suffisantes de convergence pour une large classe d'éléments finis, puis nous donnons des estimations de l'erreur sur les déplacements et les contraintes.

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On the convergence of nonconforming finite element method for linear thin shell problems

Summary In this paper, we study the convergence of nonconforming finite element method for the approximation of general thin shell problems. We give sufficient conditions of convergence for a large class of finite elements, then we estimate the error on the displacements and on the stresses.

     

Finite homogeneous deformations of symmetrically loaded compressible membranes

The equilibrium problem of nonlinear, isotropic and hyperelastic square membranes, stretched by a double symmetric system of dead loads, is investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions in addition to the expected symmetric solutions. For compressible materials, the mathematical condition allowing the computation of these asymmetric solutions is given. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a compressible Mooney–Rivlin material and a broad numerical analysis is performed. The qualitatively more interesting branches of asymmetric equilibrium are shown and the influence of the material parameters is discussed. Finally, using the energy criterion, some stability considerations are made.     

Asymptotic behavior of eigenfunctions and eigenfrequencies of oscillation boundary value problems of the linear theory of elastic mixtures

The asymptotic behavior of eigenoscillation and eigen-vector-function is studied for the internal boundary value problems of oscillation of the linear theory of a mixture of two isotropic elastic media.     

Shock structure in viscoelasticity of relaxation type

We are concerned with the shock structure in viscoelasticity of rate type. The existence of a shock profile, which in the vanishing relaxation limit becomes an elastic shock, is demonstrated by an explicit construction of the solution of viscoelastic equations provided that the equilibrium stress–strain relation is governed by a parabolic law. For a more general stress–strain law which does not differ substantially from a parabolic one, the solution is obtained from a perturbation expansion.     

An isoperimetric estimate and W^{1,p}-quasiconvexity in nonlinear elasticity

A class of stored energy densities that includes functions of the form with , g and h convex and smooth, and is considered. The main result shows that for each such W in this class there is a such that, if a 3 by 3 matrix satisfies , then W is -quasiconvex at on the restricted set of deformations that satisfy condition (INV) and a.e. (and hence that are one-to-one a.e.). Condition (INV) is (essentially) the requirement that be monotone in the sense of Lebesgue and that holes created in one part of the material not be filled by material from other parts. The key ingredient in the proof is an isoperimetric estimate that bounds the integral of the difference of the Jacobians of and by the -norm of the difference of their gradients. These results have application to the determination of lower bounds on critical cavitation loads in elastic solids. Received January 5, 1998 / Accepted March 13, 1998     

Some exact velocity profiles for granular flow in converging hoppers

Gravity flow of granular materials through hoppers occurs in many industrial processes. For an ideal cohesionless granular material, which satisfies the Coulomb-Mohr yield condition, the number of known analytical solutions is limited. However, for the special case of the angle of internal friction equal to ninety degrees, there exist exact parametric solutions for the governing coupled ordinary differential equations for both two-dimensional wedges and three-dimensional cones, both of which involve two arbitrary constants of integration. These solutions are the only known analytical solutions of this generality. Here, we utilize the double-shearing theory of granular materials to determine the velocity field corresponding to these exact parametric solutions for the two problems of gravity flow through converging wedge and conical hoppers. An independent numerical solution for other angles of internal friction is shown to coincide with the analytical solution.Received: July 24, 2002     

On a Stochastic Plate Equation

In this paper we discuss an initial—boundary value problem for an elastic plate driven by a space-time white noise. The existence and uniqueness of a weak solution is established. We use a specialized PDE method based upon the results for the deterministic equation. Accepted 2 February 2001. Online publication 4 May 2001.     

A classification of thin shell theories

This paper gives a modern mathematical analysis of the relationships between several, different linear shell theories. It also discusses the asymptotic role played by membrane theory. It presents theorems on the existence and uniqueness of solutions of membrane equations depending on the concavity of the surface.     

Locking and robustness in the finite element method for circular arch problem

Summary. In this paper we discuss locking and robustness of the finite element method for a model circular arch problem. It is shown that in the primal variable (i.e., the standard displacement formulation), the p-version is free from locking and uniformly robust with order and hence exhibits optimal rate of convergence. On the other hand, the h-version shows locking of order , and is uniformly robust with order for which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience. If mixed method is used, both the h-version and the p-version are free from locking. Furthermore, the mixed method even converges uniformly with an optimal rate for the stress. Received June 5, 1992 / Revised version received May 17, 1994     

Ergodicity of stochastic plates

Summary We prove the stabilization of plates subject to a stochastic evolution of Ginzburg Landa type. We study the asymptotic behaviour of the rate of stabilization when the intensity of the noise vanishes. An approximate simulated annealing property is also given.This article was processed by the author using the Springer-Verlag T_EX QPMZGHB macro package 1991.     

An eigenvalue problem for a hemivariational inequality involving a nonlinear compact operator

The present paper deals with an eigenvalue problem arising in hemivariational inequalities involving a nonlinear compact operator. This problem has been studied concerning the existence of its solution by applying a critical point approach suitable for nonconvex, nonsmooth energy functions. The method is based on Ekeland's variational principle and on the results of Chang and Szulkin.     

A posteriori error estimates for the Morley plate bending element

A local a posteriori error indicator for the well known Morley element for the Kirchhoff plate bending problem is presented. The error indicator is proven to be both reliable and efficient. The technique applied is general and it is shown to have also other applications.