arXiv subjects

Muthusamy Vanninathan

Publications and source records attributed to Muthusamy Vanninathan.

7 recordsLinked to original sources

arXiv2016

Homogenization of Stokes System using Bloch Waves

In this work, we study the Bloch wave homogenization for the Stokes system with periodic viscosity coefficient. In particular, we obtain the spectral interpretation of the homogenized tensor. The presence of the incompressibility constraint in the model raises new issues linking the homogenized tensor and the Bloch spectral data. The main difficulty is a lack of smoothness for the bottom of the Bloch spectrum, a phenomenon which is not present in the case of the elasticity system. This issue is solved in the present work, completing the homogenization process of the Stokes system via the Bloch wave method.

math.AP↗

arXiv2016

Inverse Diffusivity Problem via Homogenization Theory

Polarization tensor corresponding to near zero volume inhomogeneities was introduced in the pioneering work by Capdeboscq-Vogelius \cite{CV1,CV2}. A beautiful application of the polarization tensor to an inverse problem involving inhomogeneities was also given by them. In this article, we take an approach toward polarization tensor via homogenized tensor. Accordingly, we introduce polarization tensor corresponding to inhomogeneities with positive volume fraction.A relation between this tensor and the homogenized tensor is found. Next, we proceed to examine the sense in which this tensor is continuous as the volume fraction tends to zero. Our approach has its own advantages, as we will see. In particular, it provides another method to deduce optimal estimates on polarization tensors in any dimension from those on homogenized tensors, along with the information on underlying microstructures.

math.AP↗

arXiv2016

Bloch wave spectral analysis in the class of generalized Hashin-Shtrikman micro-structures

In this paper, we use spectral methods by introducing the Bloch waves to study the homogenization process in the non-periodic class of generalized Hashin-Shtrikman micro-structures \cite[page no. 281]{T}, which incorporates both translation and dilation with a family of scales, including one subclass of laminates. We establish the classical homogenization result with providing the spectral representation of the homogenized coefficients. It offers a new lead towards extending the Bloch spectral analysis in the non-periodic, non-commutative class of micro-structures.

math.AP↗

arXiv2016

Dispersion tensor and its unique minimizer in Hashin-Shtrikman micro-structures

In this paper, we introduce the macroscopic quantity, namely the dispersion tensor or the \textit{Burnett coefficient}s in the class of generalized Hashin-Shtrikman micro-structures \cite[page no. 281]{T}. In the case of two-phase materials associated with the periodic Hashin-Shtrikman structures, we settle the issue that the dispersion tensor has an unique minimizer, which is so called Apollonian-Hashin-Shtrikman micro-structure.

math.AP↗

arXiv2016

Bloch Wave Homogenization Relative to a Microstructure

In this work, we study the aspect of Bloch wave homogenization of the new notion of convergence of microstructures represented by matrices $B^ε$ related to the classical $H$-convergence of $A^ε$ introduced in \cite{TG-MV-1}. The new macro quantity $B^{#}$ appears to incorporate the interaction between the two microstructures $A^ε,B^ε$. Here we present its Bloch spectral representation along with the homogenization result.

math.AP↗

arXiv2015

Feedback stabilization of a simplified model of fluid-structure interaction on a tree

In this paper we study the dynamic feedback stability for a simplified model of fluid-structure interaction on a tree. We prove that, under some conditions, the energy of the solutions of the system decay exponentially to zero when the time tends to infinity. Our technique is based on a frequency domain method and a special analysis for the resolvent.

math.AP↗

arXiv2013

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 $\ep Y$-periodic conductivity. When the conductivity is bounded in $L^1$ and the constant of the Poincaré-Wirtinger weighted by the conductivity is very small with respect to $\ep^{-2}$, the first Bloch eigenvalue converges as $\ep\to 0$ 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 $L^1$-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 parameter around zero.

math.AP↗