M. Vanninathan

A spectral problem arising in fluid-solid structures

In this paper, we study a simplified model that describes the eigenfrequencies and eigenmotions of a periodic tube bundle immersed in an incompressible, perfect fluid. This model involves Laplace equations and a nonlocal boundary condition. The eigenvalues of the problem appear in this nonlocal condition. In practice we are interested in the case where one has very many tubes. Our goal in this article is to study the asymptotic behaviour of the spectrum of these problems as the number of tubes goes to infinity. We do this in terms of the convergance of the spectral families associated with these problems.

On Burnett coefficients in periodic media

The aim of this work is to demonstrate a curious property of general periodic structures. It is well known that the corresponding homogenized matrix is positive definite. We calculate here the next order Burnett coefficients associated with such structures. We prove that these coefficients form a tensor which is negative semidefinite. We also provide some examples showing degeneracy in multidimension.

On Burnett coefficients in periodic media in low contrast regime

In this work, we consider low contrast periodic media and we study the dependence of the effective or homogenized tensor and the dispersion tensor in terms of the microstructure. We treat both one-dimensional structures and some laminated structures in higher dimension. Interesting properties of the sign of these coefficients are found. Surprisingly, these depend on the microstructure only through the local proportion parameter, and in some cases, they do not depend on the microstructure at all.

Limits of the resonance spectrum of tube arrays immersed in a fluid

The resonance frequency spectrum of an elastic tube array placed in a still fluid is spread over a relatively wide interval. This interval depends on the added mass matrix of the bundle. This matrix reflects the mechanical interaction of tubes through the fluid. When the bundle is practically infinite (i.e. with a large number of tubes), the added mass matrix can be expressed by the Bloch wave” decomposition, valid for infinite periodic structures. That allows to show that the resonance spectrum of the fluid-bundle system is continuous and therefore spread over an interval. It then becomes easy to compute the limits of this interval. Both cases of incompressible and compressible fluids are investigated.

Limiting behaviour of a spectral problem in fluid-solid structures

Conca, c., J. Planchard and M. Vanninathan, Limiting behaviour of a spectral problem in fluid-solid structures, Asymptotic Analysis 6 (1993) 365-389 The aim of this paper is the asymptotic analysis of a spectral problem which involves Helmholtz equation coupled with a nonlocal Neumann boundary condition on the boundary of a periodic perforated domain of [J;l2. This eigenvalue problem represents the vibrations Ceigenfrequencies and eigenmotions) of a tube-bundle immersed in a perfect compressible fluid. Our analysis of convergence shows that the vibrations of this fluid-solid structure with a large number of tubes are close to the spectrum of an unbounded operator in a Hilbert space. Using the method of Bloch and exploiting the periodic structure of the problem, we derive the spectral family of this limit operator and we prove that its spectrum can be completely determined by only computing local eigenvalue problems in the basic cell representing the periodic structure in the problem.

Burnett coefficients and laminates

The object of discussion of this article is the fourth-order tensor d introduced as a set of macro coefficients associated with fine periodic structures. Focus of attention is its variation on laminated microstructures. Complete bounds are obtained on its quartic form along with the corresponding optimal structures. Differences with corresponding results for the homogenized matrix are pointed out. Using Blossoming Principle, it is shown that d is not negative in the sense of Legendre–Hadamard, even though its quartic form is negative.

Null-controllability for a Coupled Heat-Finite-dimensional Beam System

A model representing a coupling between a heat conducting medium and a finite-dimensional approximation of a beam equation is considered. We establish a Carleman inequality for this model. Next we deduce a nullcontrollability result with an internal control in the conducting medium and there is no control in the structure equation.

First Bloch eigenvalue in high contrast media

This paper deals with the asymptotic behavior of the first Bloch eigenvalue in a heterogeneous medium with a high contrast ɛY-periodic conductivity. When the conductivity is bounded in L1 and the constant of the Poincare-Wirtinger weighted by the conductivity is very small with respect to ɛ−2, the first Bloch eigenvalue converges as ɛ → 0 to a limit which preserves the second-order expansion with respect to the Bloch parameter. In dimension two the expansion of the limit can be improved until the fourth-order under the same hypotheses. On the contrary, in dimension three a fibers reinforced medium combined with a L1-unbounded conductivity leads us to a discontinuity of the limit first Bloch eigenvalue as the Bloch parameter tends to zero but remains not orthogonal to the direction of the fibers. Therefore, the high contrast conductivity of the microstructure induces an anomalous effect, since for a given low-contrast conductivity the first Bloch eigenvalue is known to be analytic with respect to the Bloch...

Higher Order Macro Coefficients in Periodic Homogenization

A first set of macro coefficients known as the homogenized coefficients appear in the homogenization of PDE on periodic structures. If energy is increased or scale is decreased, these coefficients do not provide adequate approximation. Using Bloch decomposition, it is first realized that the above coefficients correspond to the lowest energy and the largest scale. This naturally paves the way to introduce other sets of macro coefficients corresponding to higher energies and lower scales which yield better approximation. The next task is to compare their properties with those of the homogenized coefficients. This article reviews these developments along with some new results yet to be published.

Un problème de fréquences propres en couplage fluide-structure

The aim of this paper is the determination of the eigenfrequencies of elastic tube-bundles immersed in a perfect fluid. Two computational methods in the case of tube bundles with periodic structure are described. These methods are based on homogenization techniques and pseudo-periodic functions.