Universal bounds on the electrical and elastic response of two-phase bodies and their application to bounding the volume fraction from boundary measurements

Universal bounds on the electrical and elastic response of two-phase (and multiphase) ellipsoidal or parallelopipedic bodies have been obtained by Nemat-Nasser and Hori. Here we show how their bounds can be improved and extended to bodies of arbitrary shape. Although our analysis is for two-phase bodies with isotropic phases it can easily be extended to multiphase bodies with anisotropic constituents. Our two-phase bounds can be used in an inverse fashion to bound the volume fractions occupied by the phases, and for electrical conductivity reduce to those of Capdeboscq and Vogelius when the volume fraction is asymptotically small. Other volume fraction bounds derived here utilize information obtained from thermal, magnetic, dielectric or elastic responses. One bound on the volume fraction can be obtained by simply immersing the body in a water filled cylinder with a piston at one end and measuring the change in water pressure when the piston is displaced by a known small amount. This bound may be particularly effective for estimating the volume of cavities in a body. We also obtain new bounds utilizing just one pair of (voltage, flux) electrical measurements at the boundary of the body.Comment: 5 figures, 27 page

UNLV Authors Honored for Publications in 2009

Lists UNLV authors and their publications who were honored at the 2010 UNLV authors reception. Includes cover art

Strongly Elliptic Systems and Boundary Integral Equations

Partial differential equations provide mathematical models of many important problems in the physical sciences and engineering. This book treats one class of such equations, concentrating on methods involving the use of surface potentials. It provides the first detailed exposition of the mathematical theory of boundary integral equations of the first kind on non-smooth domains. Included are chapters on three specific examples: the Laplace equation, the Helmholtz equation and the equations of linear elasticity. The book is designed to provide an ideal preparation for studying the modern research literature on boundary element methods. Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. Cambridge University Press 978-0-521-66332-8 -Strongly Elliptic Systems and Boundary Integral Equations William McLean Frontmatte

Stability of Traveling Waves in Thin Liquid Films Driven by Gravity and Surfactant

A thin layer of fluid flowing down a solid planar surface has a free surface height described by a nonlinear PDE derived via the lubrication approximation from the Navier Stokes equations. For thin films, surface tension plays an important role both in providing a significant driving force and in smoothing the free surface. Surfactant molecules on the free surface tend to reduce surface tension, setting up gradients that modify the shape of the free surface. In earlier work [12, 13J a traveling wave was found in which the free surface undergoes three sharp transitions, or internal layers, and the surfactant is distributed over a bounded region. This triple-step traveling wave satisfies a system of PDE, a hyperbolic conservation law for the free surface height, and a degenerate parabolic equation describing the surfactant distribution. As such, the traveling wave is overcornpressive. An examination of the linearized equations indicates the direction and growth rates of one-dimensional waves generated by small perturbations in various parts of the wave. Numerical simulations of the nonlinear equations offer further evidence of stability to one-dimensional perturbations

Collective motion of living organisms: the Vicsek model

The purpose of this work is to study the Vicsek model for self-driven articles, in particular its time-continuous version. We will study its mean-field limit, as well as the large-scale behaviour of the model. In particular, we find that the equilibrium distributions change according to whether the density is above or below a given threshold. Below this value, the only equilibrium distribution is isotropic in velocity direction and is stable; moreover, the convergence to this equilibrium is exponentially fast. When the density is above the threshold, we have a second class of anisotropic equilibria, formed by Von-Mises-Fischer distributions of arbitrary orientation. In this case, the isotropic equilibria become unstable and there is exponentially fast convergence to the anisotropic ones

Physical-Mathematical modeling and numerical simulations of stress-strain state in seismic and volcanic regions

The strain-stress state generated by faulting or cracking and influenced by the strong heterogeneity of the internal earth structure precedes and accompanies volcanic and seismic activity. Particularly, volcanic eruptions are the culmination of long and complex geophysical processes and physical processes which involve the generation of magmas in the mantle or in the lower crust, its ascent to shallower levels, its storage and differentiation in shallow crustal chambers, and, finally, its eruption at the Earth’s surface. Instead, earthquakes are a frictional stick-slip instability arising along pre-existing faults within the brittle crust of the Earth. Long-term tectonic plate motion causes stress to accumulate around faults until the frictional strength of the fault is exceeded. The study of these processes has been traditionally carried out through different geological disciplines, such as petrology, structural geology, geochemistry or sedimentology. Nevertheless, during the last two decades, the development of physical of earth as well as the introduction of new powerful numerical techniques has progressively converted geophysics into a multidisciplinary science. Nowadays, scientists with very different background and expertises such as geologist, physicists, chemists, mathematicians and engineers work on geophysics. As any multidisciplinary field, it has been largely benefited from these collaborations. The different ways and procedures to face the study of volcanic and seismic phenomena do not exclude each other and should be regarded as complementary. Nowadays, numerical modeling in volcanology covers different pre-eruptive, eruptive and post-eruptive aspects of the general volcanic phenomena. Among these aspects, the pre-eruptive process, linked to the continuous monitoring, is of special interest because it contributes to evaluate the volcanic risk and it is crucial for hazard assessment, eruption prediction and risk mitigation at volcanic unrest. large faults. The knowledge of the actual activity state of these sites is not only an academic topic but it has crucial importance in terms of public security and eruption and earthquake forecast. However, numerical simulation of volcanic and seismic processes have been traditionally developed introducing several simplifications: homogeneous half-space, flat topography and elastic rheology. These simplified assumptions disregards effects caused by topography, presence of medium heterogeneity and anelastic rheology, while they could play an important role in Moreover, frictional sliding of a earthquake generates seismic waves that travel through the earth, causing major damage in places nearby to the modeling procedure This thesis presents mathematical modeling and numerical simulations of volcanic and seismic processes. The subject of major interest has been concerned on the developing of mathematical formulations to describe seismic and volcanic process. The interpretation of geophysical parameters requires numerical models and algorithms to define the optimal source parameters which justify observed variations. In this work we use the finite element method that allows the definition of real topography into the computational domain, medium heterogeneity inferred from seismic tomography study and the use of complex rheologies. Numerical forward method have been applied to obtain solutions of ground deformation expected during volcanic unrest and post-seismic phases, and an automated procedure for geodetic data inversion was proposed for evaluating slip distribution along surface rupture