A model of a general elastic curved rod

We indicate a new approach to the deformation of three‐dimensional curved rods with variable cross‐section. The model consists of a system of nine ordinary differential equations for which we prove existence and uniqueness via the coercivity of the association bilinear form. From the geometrical point of view, we are using the Darboux frame or a new local frame requiring just a C¹‐parameterization of the curve. Our model also describes the deformation occurring in the cross‐sections of the rod. Copyright © 2002 John Wiley & Sons, Ltd.     

Asymptotic behaviour of curved rods by the unfolding method

We consider in this work general curved rods with a circular cross‐section of radius δ. Our aim is to study the asymptotic behaviour of such rods as δ→0, in the framework of the linear elasticity according to the unfolding method. It consists in giving some decompositions of the displacements of such rods, and then in passing to the limit in a fixed domain. A first decomposition concerns the elementary displacements of a curved rod which characterize its translations and rotations, and the residual displacements related to the deformation of the cross‐section. The second decomposition concerns the displacements of the middle‐line of the rod. We prove that such a displacement can be written as the sum of an inextensional displacement and of an extensional one. An extensional displacement will modify the length of the middle‐line, while an inextensional displacement will not change this length in a first approximation. We show that the H¹‐norm of an inextensional displacement is of order 1, while that of an extensional displacement is in general, of order δ. A priori estimates are established and convergence results as δ→0, are given for the displacements. We give their unfolded limits, as well as the unfolded limits of the strain and stress tensors. To prove the convergence of the strain tensor, the introduction of elementary and residual displacements appears as essential. By passing to the limit as δ→0 in the linearized system of the elasticity, we obtain on the one hand, a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem coupling the limit of inextensional displacements and the limit of the angle of torsion. Copyright © 2004 John Wiley & Sons, Ltd.     

Derivation of the model of elastic curved rods from three-dimensional micropolar elasticity

In this paper we derive a model of curved elastic rods from the threedimensional linearized micropolar elasticity. Derivation is based on the asymptotic expansion method with respect to the thickness of the rod. The method is used without any a priori assumption on the scaling of the unknowns. The leading term, displacement and microrotation, is identified as the unique solution of a certain one-dimensional problem. Appropriate convergence results are proved.      

Asymptotic profile of solutions for some wave equations with very strong structural damping

We consider the Cauchy problem in R ⁿ for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L^1,1( R ⁿ) initial data by using a simple method introduced in by the first author. The obtained results will include regularity loss type estimates, which are essentially new in this kind of equation.     

Approximation of the vibration modes of a Timoshenko curved rod of arbitrary geometry

Rodolfo Rodríguez ^{The aim of this paper is to analyse a mixed finite-element methodfor computing the vibration modes of a Timoshenko curved rodwith arbitrary geometry. Optimal order error estimates are provedfor displacements, rotations and shear stresses of the vibrationmodes, as well as a double order of convergence for the vibrationfrequencies. These estimates are essentially independent ofthe thickness of the rod, which leads to the conclusion thatthe method is locking-free. Numerical tests are reported inorder to assess the performance of the method.}     

Asymptotic profile of solutions for strongly damped Klein‐Gordon equations

We consider the Cauchy problem in R ⁿ for strongly damped Klein‐Gordon equations. We derive asymptotic profiles of solutions with weighted L^1,1( R ⁿ) initial data by a simple method introduced by the second author. Furthermore, from the obtained asymptotic profile, we get the optimal decay order of the L²‐norm of solutions. The obtained results show that the wave effect will be relatively weak because of the mass term, especially in the low‐dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases to compare the difference.     

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.     

Asymptotic analysis for a dynamic piezoelectric shallow shell

The paper deals with the asymptotic formulation and justification of a mechanical model for a dynamic piezoelastic shallow shell in Cartesian coordinates. Starting from the three‐dimensional dynamic piezoelastic problem and by an asymptotic approach, the authors study the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. In order to obtain a nontrivial limit problem by asymptotic analysis, we need different scalings on the mass density. The authors show that the transverse mechanical displacement field coupled with the in‐plane components solves an problem with new piezoelectric characteristics and also investigate the very popular case of cubic crystals and show that, for two‐dimensional shallow shells, the coupling piezoelectric effect disappears. Copyright © 2017 John Wiley & Sons, Ltd.     

Asymptotic analysis for closed multiclass queueing networks in critical usage

We consider a class of closed multiclass queueing networks containing First-Come-First-Serve (FCFS) and Infinite Server (IS) stations. These networks have a productform solution for their equilibrium probabilities. We study these networks in an asymptotic regime for which the number of customers and the service rates at the FCFS stations go to infinity with the same order. We assume that the regime is in critical usage, whereby the utilizations of the FCFS servers slowly approach one. The asymptotic distribution of the normalized queue lengths is shown to be in many cases a truncated multivariate normal distribution. Traffic conditions for which the normalized queue lengths arealmost asymptotically independent are determined. Asymptotic expansions of utilizations and expected queue lengths are presented. We show through an example how to obtain asymptotic expansions of performance measures when the networks are in mixed usage and how to apply the results to networks with finite data.Supported partially by NSF grant NCR93-04601.     

Asymptotic analysis of a thin fluid layer-elastic plate interaction problem

We study the nonstationary flow of an incompressible fluid in a thin rectangle with an elastic plate as the upper part of the boundary. The flow is governed by a time-dependent pressure drop and an external force and it is modeled by Stokes equations. The dynamic of this fluid–structure interaction problem is studied in the limit when the thickness of the fluid domain tends to zero. Using the asymptotic techniques, we obtain for the effective plate displacement a sixth-order parabolic equation with a non standard boundary conditions. Results on existence, uniqueness and regularity of the solution are proved. The approximation is justified through a weak convergence theorem.     

Asymptotic profiles and asymptotic distributions of energy in elastic waves scattering

We compute the asymptotic wave profiles for the exterior problemin elasticity with homogeneous Neumann condition. We give preciseresults on the asymptotic distribution of the elastic energyin various subsets of space. These results are obtained viathe use of a recently developed extension of the scatteringtheory of C. Wilcox (Mabrouk & Helali, 2002).     

Asymptotic and numerical analysis of thermoelastic waves in an isotropic plate

^** Email: n.scott{at}uea.ac.uk This paper considers the propagation of a plane thermoelasticwave in an infinite homogeneous isotropic plate subjected toeither isothermal or thermally insulated traction-free boundaryconditions. The primary concerns are the derivation and numericalexamination of the dispersion relations along with the developmentof asymptotic models for the fundamental branches of the dispersionrelations. Thermal dissipation is also discussed.     

Asymptotic analysis for multi-objective sequential stochastic assignment problems

ABSTRACT

We provide an asymptotic analysis of multi-objective sequential stochastic assignment problems (MOSSAP). In MOSSAP, a fixed number of tasks arrive sequentially, with an n-dimensional value vector revealed upon arrival. Each task is assigned to one of a group of known workers immediately upon arrival, with the reward given by an n-dimensional product-form vector. The objective is to maximize each component of the expected reward vector. We provide expressions for the asymptotic expected reward per task for each component of the reward vector and compare the convergence rates for three classes of Pareto optimal policies.     

Asymptotic of solutions for second IBVP for hyperbolic systems in non-smooth domains

The purpose of this paper is to study the asymptotic behaviour of solutions of the second initial boundary value problem for hyperbolic systems in an infinite cylinder with the base containing conical points. Some new results on the asymptotics of solutions of elliptic problem depending on a parameter are also given.     

Asymptotic study of the elastic seadbed effects in ocean acoustics

A theoretical and asymptotic investigation of the Green' function for the system governing the propagation of time-harmonic acoustic waves in a horizontally stratified ocean with an elastic seabed is presented. Employing the surface Neumann-to-Dirichlet map for the elastic half space, we reduce the problem to an equivalent one in the layer, with a nonlocal boundarycondition at the fluid-bottom interface. The reduced problem is transformedby Hankel transform, to a non-selfadjoint boundary value problem for a second-order ordinary differential equation over the layer depth. The well posedness of this problem is investigated applying analytic Redholm theory for an equivalent Lippmann-Schwinger integral equation. An asymptotic expansionof the transformed nonlocal boundary condition is constructed in the case of a seabed with small shear modulus, and it is used to show that the Green function is a regular perturbation of that one in the case of a fluid bottom.     

Asymptotic distributions in the projection pursuit based canonical correlation analysis

In this paper, associations between two sets of random variables based on the projection pursuit (PP) method are studied. The asymptotic normal distributions of estimators of the PP based canonical correlations and weighting vectors are derived.     

Asymptotic analysis of integro-differential systems with an unstable spectral value of the integral operator’s kernel

A system of integro-differential equations with rapidly varying kernels, one of which has an unstable spectral value, is considered. An algorithm based on the method of normal forms is proposed for finding asymptotic solutions (of an arbitrary order). The contrast structures (internal transition layers) in solutions to the problem are investigated by analyzing the leading term of the asymptotic expansion. It is shown that the contrast structures are caused by the instability of the spectral value and by the presence of inhomogeneity. The role of the kernels of the integral operators in the formation of contrast structures is clarified.     

U(1)场中CP~1模型任意涡旋状孤立子解的存在性及渐近估计

对U(1)场中含有Chern-Simons项的CP~1模型用变分法证明了其拓扑孤立子解的存在性.进而,又用分析的方法讨论了该解的单调性和渐近性.     

2D Navier-Stokes方程的渐近吸引子

考虑了2D周期边界条件下Navier-Stokes方程渐近吸引子，即构造了一个有限维解序列，首先证明了该序列不会远离方程的整体吸引子，然后证明了它在长时间后无限趋于方程的整体吸引子，并给出了渐近吸引子的维数估计．