Time discrete wave equations: Boundary observability and control

In this paper we study the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. We prove that the projection of the solution in an appropriate filtered space is exactly controllable with uniformly bounded cost with respect to the time-step. In this way, the well-known exact-controllability property of the wave equation can be reproduced as the limit, as the time step h → 0, of the controllability of projections of the time-discrete one. By duality these results are equivalent to deriving uniform observability estimates (with respect to h → 0) within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. The later is established by means of a time-discrete version of the classical multiplier technique. The optimality of the order of the filtering parameter is also established, although a careful analysis of the expected velocity of propagation of time-discrete waves indicates that its actual value could be improved

Homogenization of fiber reinforced brittle materials: The extremal cases

We analyze the behavior of a fragile material reinforced by a reticulated elastic unbreakable structure in the case of antiplane shear. The microscopic geometry of this material is described by means of two small parameters: the period ε of t he grid and the ratio δ between the thickness of the fibers and the period ε. We show that the asymptotic behavior as ε → 0+ and δ → 0+ depends dramatically on the relative size of these parameters. Indeed, in the two cases considered, i.e., ε &≪ δ and ε 3≫ δ, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively. © 2009 Society for Industrial and Applied Mathematics

Simulation of DC dual-laterolog measurements in complex formations: A Fourier-series approach with nonorthogonal coordinates and self-adapting finite elements

Dual laterolog (DLL) makes use of a galvanic conduction principle to focus electrical currents into rock formations, thereby minimizing shoulder and borehole effects in the measurement of formation resistivity. The tool includes two separate focusing systems: deep-sensing (LLd) and shallow-sensing modes (LLs). Laterolog current-focusing systems were designed for operation primarily in vertical boreholes penetrating horizontal layers; only recently their design has been revised for operation in deviated wells in the presence of electrical anisotropy. We simulated three-dimensional (3D) DLL measurements in dipping, invaded, and electrically anisotropic formations and appraised the corresponding effects on apparent resistivity logs. Simulations were performed by combining the use of a Fourier-series expansion in a nonorthogonal system of coordinates with an existing 2D goal-oriented, higher-order, and self-adaptive finite-element method. This numerical algorithm yields accurate solutions in limited CPU time because only a few Fourier modes are needed to simulate practical applications. For the calculation of focused currents, we introduced an embedded postprocessing method that incorporates a synthetic focusing principle to compute current intensities at each iterative step of optimal mesh refinements. Our numerical method accurately simulates 3D DLL measurements in rock formations that exhibit extreme contrasts of electrical resistivity. Simulations indicate that LLs resistivity logs are more sensitive to both invaded and anisotropic layers than LLd resistivity logs. In deviated wells, shoulder-bed effects on apparent resistivity logs increase with an increase of dip angle, and are emphasized across thin conductive layers. Electrical anisotropy effects on apparent resistivity logs increase substantially with dip angle. © 2009 Society of Exploration Geophysicists. All rights reserved

On the Gittins index in the M/G/1 queue

For an M/G/1 queue with the objective of minimizing the mean number of jobs in the system, the Gittins index rule is known to be optimal among the set of non-anticipating policies. We develop properties of the Gittins index. For a single-class queue it is known that when the service time distribution is of type Decreasing Hazard Rate (New Better than Used in Expectation), the Foreground-Background (First-Come-First-Served) discipline is optimal. By utilizing the Gittins index approach, we show that in fact, Foreground-Background and First-Come-First-Served are optimal if and only if the service time distribution is of type Decreasing Hazard Rate and New Better than Used in Expectation, respectively. For the multi-class case, where jobs of different classes have different service distributions, we obtain new results that characterize the optimal policy under various assumptions on the service time distributions. We also investigate distributions whose hazard rate and mean residual lifetime are not monotonic. © Springer Science+Business Media, LLC 2009

Energy-norm-based and goal-oriented automatic hp adaptivity for electromagnetics: Application to waveguide Discontinuities

The finite-element method (FEM) enables the use of adapted meshes. The simultaneous combination of h (local variations in element size) and p (local variations in the polynomial order of approximation) refinements, i.e., hp-adaptivity, is the most powerful and flexible type of adaptivity. In this paper, two versions of a fully automatic hp-adaptive FEM for electromagnetics are presented. The first version is based on minimizing the energy-norm of the error. The second, namely the goal oriented strategy, is based on minimizing the error of a given (user-prescribed) quantity of interest. The adaptive strategy delivers exponential convergence rates for the error, even in the presence of singularities. The hp adaptivity is presented in the context of 2-D analysis of H -plane rectangular waveguide discontinuities. Stabilized variational formulations and H(curl) FEM discretizations in terms of quadrilaterals of variable order of approximation supporting anisotropy and hanging nodes are used. Comparison of energy-norm and goal-oriented hp-adaptive strategies in the context of waveguiding problems is provided. Specifically, the scattering parameters of the discontinuity are used as goal

Sensitivity study of borehole-to-surface and crosswell electromagnetic measurements acquired with energized steel casing to water displacement in hydrocarbon-bearing layers

We study the theoretical response of electromagnetically energized steel casing in the presence of subsurface variations of electrical resistivity. Casing is energized with a finite-size solenoid antenna located along the axis of the borehole. Measurements consist of the azimuthal component of the electric field acquired either on the surface or in a separate well in the same hydrocarbon field. We assume two-dimensional (2D) axisymmetric variations of subsurface electrical resistivity and casing excitation. Simulations of electromagnetic (EM) fields excited by energized steel casing are performed with a goal-oriented hp-adaptive finite-element method that automatically generates a sequence of optimal grids delivering exponential convergence rates in terms of the EM fields at the receiver antennas against CPU time. This advanced finite-element method enables accu-rate modeling of problems with high conductivity contrasts inlarge computational domains. Numerical simulations quantify the measurement sensitivity to variations of frequency, distance from casing to receivers, resistivity of the target oil-bearing layer, and pistonlike radial invasion of water within a target layer initially saturated with oil. When receivers are placed in a nearby well, numerical results indicate that measurements exhibit the largest sensitivity to the target (oil-saturated) layer when the transmitter or receiver antenna is located just above the target layer, and another antenna is located below the target layer. A frequency range from 5-30 Hz provides optimal results for the detection of oil-bearing layers and estimation of radial extent of water invasion. Large horizontal distances (up to 1500 m) between transmitter and receivers and a background material with resistivity above 50 Ωm also enhance the measurement sensitivity to radial variations of water invasion. This sensitivity can be as large as 15%-20% of the measured electric field

Leaky Cell Model of Hard Spheres

We study packings of hard spheres on lattices. The partition function, and therefore the pressure, may be written solely in terms of the accessible free volume, i.e., the volume of space that a sphere can explore without touching another sphere. We compute these free volumes using a leaky cell model, in which the accessible space accounts for the possibility that spheres may escape from the local cage of lattice neighbors. We describe how elementary geometry may be used to calculate the free volume exactly for this leaky cell model in two- and three-dimensional lattice packings and compare the results to the well-known Carnahan–Starling and Percus–Yevick liquid models. We provide formulas for the free volumes of various lattices and use the common tangent construction to identify several phase transitions between them in the leaky cell regime, indicating the possibility of coexistence in crystalline materials