Modeling junctions of plates and beams by means of self-adjoint extensions

On the basis of an asymptotic analysis of elliptic problems on thin domains and their junctions, a model of a mixed boundary value problem for a second-order scalar differential equation on the union of 3D thin beams and a plate is constructed. One end of each beam is attached to the plate, and on the other end, the Dirichlet conditions are imposed; on the remaining part of the joint boundary, the Neumann boundary conditions are set. An asymptotic expansion of the solution to such a problem has certain distinguishing features; namely, the expansion coefficients turn out to be rational functions of the large parameter |lnh| (where h ∈ (0, 1] is a small geometric parameter), and the solution to the limit problem in the longitudinal section of the plate has logarithmic singularities at the junction points with the beams. Thus, the classical settings of boundary value problems are inadequate to describe the asymptotics, and the technique of self-adjoint extensions and function spaces with separated asymptotics must be used.     

The flexural rigidity of a thin plate reinforced with periodic systems of separated rods

A two-dimensional model of the flexure of a thin plate, reinforced with periodic families of separated thin rods, symmetrical about the middle plane, is constructed. Since the rods only interact through the pliable matrix material, the algorithm for constructing the asymptotics is essentially different from the classical procedure in the theory of composite plates and leads to new results. Explicit formulae are obtained for the coefficients of the fourth order differential equation which arises.     

Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension three

Homogenized laws for sequences of high-contrast two-phase non-symmetric conductivities perturbed by a parameter $h$ are derived in two and three dimensions. The parameter $h$ characterizes the antisymmetric part of the conductivity for an idealized model of a conductor in the presence of a magnetic field. In dimension two an extension of the Dykhne transformation to non-periodic high conductivities permits to prove that the homogenized conductivity depends on $h$ through some homogenized matrix-valued function obtained in the absence of a magnetic field. This result is improved in the periodic framework thanks to an alternative approach, and illustrated by a cross-like thin structure. Using other tools, a fiber-reinforced medium in dimension three provides a quite different homogenized conductivity.     

Minimal bases and maximal nonbases in additive number theory

An asymptotic basis $\text{A}$ of order h is minimal if no proper subset of $\text{A}$ is an asymptotic basis of order h. Examples are constructed of minimal asymptotic bases, and also of an asymptotic basis of order two no subset of which is minimal.If $\text{B}$ is a set of nonnegative integers which is not a basis (resp. asymptotic basis) of order h, but such that every proper superset of $\text{B}$ is a basis (resp. asymptotic basis) of order h, then $\text{B}$ is a maximal nonbasis (resp. maximal asymptotic nonbasis) of order h. Examples of such sets are constructed, and it is proved that every set not a basis of order h is a subset of a maximal nonbasis of order h.     

The time-dependent von K��rm��n plate equation as a limit of 3d nonlinear elasticity

The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness h of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of h, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von Kármán plate equation.     

On the derivation of an asymptotically correct shear correction factor for the Reissner-Mindlin plate model

In the framework of thin linear elastic plates it is known that the solutions of both the three-dimensional problem and the Reissner-Mindlin plate model can be developed into asymptotic expansions. By comparing the particular asymptotic expansions with respect to the half-thickness ɛ of the plate in the case of periodic boundary conditions on the lateral side, the shear correction factor in the Reissner-Mindlin plate model can be determined in such a way that this model approximates the three-dimensional solution with one order of the plate thickness better than the classical Kirchhoff model. This fails for hard clamped lateral boundary conditions so that the Reissner-Mindlin model is in this case asymptotically as good as the Kirchhoff model.     

Asymptotics for polynomials orthogonal over the unit disk with respect to a positive polynomial weight

We derive asymptotics for polynomials orthogonal over the complex unit disk with respect to a weight of the form ²|h(z)|, with h(z) a polynomial without zeros in |z|<1. The behavior of the polynomials is established at every point of the complex plane. The proofs are based on adapting to the unit disk a technique of J. Szabados for the asymptotic analysis of polynomials orthogonal over the unit circle with respect to the same type of weight.     

Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

On the bending of plates with transverse shear deformation and mixed periodic boundary conditions

The method of matched asymptotic expansions is used to find a homogenized problem whose solution is an approximation to the solution of a mixed periodic boundary value problem in the theory of bending of thin elastic plates. A critical size for the fixed parts of the boundary is found such that the boundary condition of the homogenized problem is an intermediate case between that for the clamped edge plate and that for the free boundary plate.     

Homogenization of a Thick Plate Model with Inclusions or Openings Periodically Distributed in the Thickness

AMS (MOS): 65 M-N

Using Hencky-Mindlin thermoelastic linear model for plates with a large number of small identical inclusions or openings in their thickness, we obtain equilibrium equations in which coefficients depend on x and periodically on x/ε where ε is the diameter of the cell of the periodic structure.

Formal asymptotic expansions give “homogenized” equations with coefficients independant of ε corresponding to an equivalent homogeneous plate.

Solution of these homogenized equations is proved to be (in a weak sense) the limit, when ε tends to zero, of original equations solution.

Moreover this result leads to a method giving explicit computation of thermal and elastic coefficients of the equivalent homogeneous plate in the case of right cylindrical openings. Numerical results for different shapes of openings are given.     

Uniform Estimates of Remainders in Asymptotic Expansions of Solutions to the Problem on Eigenoscillations of a Piezoelectric Plate

A method for obtaining estimates of asymptotic remainders is presented. The constants in estimates are independent of the number of the eigenvalue, as well as of the small parameter h, the thickness of the plate. Owing to an information about connections between frequencies of eigenoscillations of the three-dimensional plates and its two-dimensional model obtained under various restrictions to h, it is possible to divide the asymptotics in collective and individual ones. Only in the case of the individual asymptotics, i.e., under rigid restrictions on h, it is possible to construct asymptotic expansions for the corresponding eigenvectors. We consider arbitrarily anizotropic composed cylindrical plates in whcih piezoeffects can dominate along longitudinal directions, as well as along transverse directions. The connectedness of elastic and electric fields Implies the appearance of a nontrivial dissipative components of the operator of the problem under consideration, but its spectrum remains real and positive. Bibliography: 43 titles.     

Asymptotics for steady‐state voltage potentials in a bidimensional highly contrasted medium with thin layer

We study the behaviour of steady‐state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively, α) surrounded by a thin membrane of thickness h and of complex permittivity α (respectively, 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady‐state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter α is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady‐state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so‐called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and α with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviours of the potentials. Copyright © 2007 John Wiley & Sons, Ltd.     

Étude d'un modèle thermo-chimique de formation d'un matériau composite

We consider a composite material constituted of carbon or glass fibres included in a resin which becomes solid when it is heated up (reaction of reticulation). The mathematical modelling of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period $?>0$. First we prove the existence and uniqueness of a solution by using Schauder's fixed point theorem. Then, by using an asymptotic expansion, we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero and we obtain an error estimate in a case of weak non-linearity. Finally we solve numerically the homogenized problem. To cite this article: S. Meliani et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Convolutions of semi-classical spectral distributions and periodic-orbit theory

We study the convolution of semi-classical spectral distributions associated to h-pseudodifferential operators on Rn. Under standard assumptions the micro-support of this object can be characterized via families of periodic orbits correlated simultaneously by energy and periods. When all the orbits are non-degenerate the convolution admits, as h tends to 0, an explicit asymptotic expansion in term of the respective dynamical systems. In this setting, this result validates the theory of orbits pairs used by physicists in quantum chaos. Some new contributions, related to the crossing of the period functions, are also analyzed.     

Asymptotic approximation of eigenelements of the Dirichlet problem for the Laplacian in a domain with shoots

We study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a two‐dimensional bounded domain with thin shoots, depending on a small parameter ε. Under the assumption that the width of the shoots goes to zero, as ε tends to zero, we construct the limit (homogenized) problem and prove the convergence of the eigenvalues and eigenfunctions to the eigenvalues and eigenfunctions of the limit problem, respectively. Under the additional assumption that the shoots, in a fixed vicinity of the basis, are straight and periodic, and their width and the distance between the neighboring shoots are of order ε, we construct the two‐term asymptotics of the eigenvalues of the problem, as ε→0. Copyright © 2009 John Wiley & Sons, Ltd.     

On multiscale homogenization problems in boundary layer theory

This paper is concerned with the homogenization of the equations describing a magnetohydrodynamic boundary layer flow past a flat plate, the flow being subjected to velocities caused by injection and suction. The fluid is assumed incompressible, viscous and electrically conducting with a magnetic field applied transversally to the direction of the flow. The velocities of injection and suction and the applied magnetic field are represented by rapidly oscillating functions according to several scales. We derive the homogenized equations, prove convergence results and establish error estimates in a weighted Sobolev norm and in C ⁰-norm. We also examine the asymptotic behavior of the solutions of the equations governing a boundary layer flow past a rough plate with a locally periodic oscillating structure.     

A bias corrected nonparametric regression estimator

In this article, we propose a new method of bias reduction in nonparametric regression estimation. The proposed new estimator has asymptotic bias order h⁴, where h is a smoothing parameter, in contrast to the usual bias order h² for the local linear regression. In addition, the proposed estimator has the same order of the asymptotic variance as the local linear regression. Our proposed method is closely related to the bias reduction method for kernel density estimation proposed by Chung and Lindsay (2011). However, our method is not a direct extension of their density estimate, but a totally new one based on the bias cancelation result of their proof.     

Inverse heat conduction problem in a thin circular plate and its thermal deflection

An inverse problem of transient heat conduction in a thin finite circular plate with the given temperature distribution on the interior surface of a thin circular plate being a function of both time and position has been solved with the help of integral transform technique and also determine the thermal deflection on the outer curved surface of a thin circular plate defined as 0 ? r ? a, 0 ? z ? h. The results, obtained in the series form in terms of Bessel’s functions, are illustrated numerically.     

Calculation of effective mechanical properties of textiles by asymptotic approach

We consider plates with 2-D periodic rod or fabric structure, used as geotextiles or textiles. The period of structure as well as the hight of a plate are much smaller compared to its depth and width. This makes a direct numerical computation of boundary value or contact elasticity problem too expensive. Three small parameters are introduced for the asymptotic analysis: the first one connected with the period of structure, the second one with the thickness of fibers or beams inside the periodicity cell and the last one – with the hight of a plate. The overcoming to the limit with respect to period of structure provides equivalent homogenized plate of the finite hight. Calculation of its outer-plane stiffness is a new aspect of this work. The next overcoming to the limit with respect to the hight reduces the 3-D problem to the homogenized equations, fourth order PDEs. The effective elasticity moduli and outer-plane stiffness can be obtained numerically solving cell experiments. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vorticity measure of type-II superconducting thin films

In this paper, we study the Ginzburg-Landau system of variable thickness superconducting thin films, placed in an applied magnetic field h _ex , when h _ex is of the order of the “first critical field”, i.e. of the order of |ln?ε|. We examine the asymptotic behavior of the “vorticity-measures” associated to the vortices of the solution and describe the distribution of vortices.