Stabilization of the wave equation on 1-D networks

In this paper we study the stabilization of the wa ve equation on general 1-d networks. For that, we transfer known observability results in the context of control problems of conservative systems (see [R. Dáger and E. Zuazua, Wave Propagation, Observation, and Control in 1-d Flexible Multi-structures, Math. Appl. 50, Springer-Verlag, Berlin, 2006]) into a weighted observability estimate for dissipative systems. Then we use an interpolation inequality similar to the one proved in [P. Bégout and F. Soria, J. Differential Equations, 240 (2007), pp. 324-356] to obtain the explicit decay estimates of the energy for smooth initial data. The obtained decay rate depends on the geometric and topological properties of the network. We also give some examples of particular networks in which our results apply, yielding different decay rates. © 2009 Society for Industrial and Applied Mathematics

Effect of vibro stone column installation on the performance of reinforced soil

Empirical design methods for stone column foundations are often on single stone columns or as a homogeneous medium of soil/column. These methods underestimate the capacity of the composite system because they do not take into account the increased confining stress acting on the stone column or the increased stiffness of the soil. This study used Plaxis 2D to study the effect of the installation method on the confining pressure and soil stiffness around a single column by assuming the installation of the column could be modelled as an expanding cavity followed by consolidation of the surrounding soil. The mean stress and stiffness generated during installation between two, adjacent columns was used in Plaxis 3D to compare the settlement of circular foundations on estuarine deposits reinforced by stone columns at a site in Santa Barbara, California. Good agreement was found between the predicted and actual settlement of the trial foundations on three column arrangements. The predictions gave a better estimate of the settlement compared to those using a unit cell or homogeneous medium showing that improvements to the soil should be taken into account when assessing stone column performance

Mathematical analysis of plasmonic nanoparticles: the scalar case

Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is threefold: (i) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (ii) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle. Optimal bounds on the enhancement factors are also derived; (iii) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation