Distribution of Resonances and Decay Rate of the Local Energy for the Elastic Wave Equation

We study resonances (scattering poles) associated to the elasticity operator in the exterior of an arbitrary obstacle with Neumann or Dirichlet boundary conditions. We prove that there exists an exponentially small neighborhood of the real axis free of resonances. Consequently we prove that for regular data, the energy for the elastic wave equation decays at least as fast as the inverse of the logarithm of time. According to Stefanov–Vodev ([SV1, SV2]), our results are optimal in the case of a Neumann boundary condition, even when the obstacle is a ball of ℝ³. The main difference between our case and the case of the scalar Laplacian (see Burq [Bu]) is the phenomenon of Rayleigh surface waves, which are connected to the failure of the Lopatinskii condition. Received: 22 February 2000 / Accepted: 28 June 2000     

Yield drag in a two-dimensional foam flow around a circular obstacle: Effect of liquid fraction

We study the two-dimensional flow of foams around a circular obstacle within a long channel. In experiments, we confine the foam between liquid and glass surfaces. In simulations, we use a deterministic software, the Surface Evolver, for bubble details and a stochastic one, the extended Potts model, for statistics. We adopt a coherent definition of liquid fraction for all studied systems. We vary it in both experiments and simulations, and determine the yield drag of the foam, that is, the force exerted on the obstacle by the foam flowing at very low velocity. We find that the yield drag is linear over a large range of the ratio of obstacle to bubble size, and is independent of the channel width over a large range. Decreasing the liquid fraction, however, strongly increases the yield drag; we discuss and interpret this dependence.     

Polymer chain elasticity in the presence of a topological obstacle

The theory of high elasticity of polymer networks is developed under the assumption that each subchain is situated near a topological obstacle having the form of an infinite straight line, which cannot be crossed by the subchain. The statistics of a polymer chain in the presence of such an obstacle is considered using the well-known results of Edwards, Prager and Frisch, and Saito and Chen. It is shown that qualitative features of the stress-strain curves under uniaxial extension-compression obtained using the present approach correspond to those known from the experiments. However, quantitatively the deviations from the classical high elasticity theory appear to be too small.     

Nonequilibrium-driven motion in actin networks: comet tails and moving beads

We present 3D dynamic Monte-Carlo simulations of the growth of an actin network close to an obstacle coated with Wiskott-Aldrich syndrome protein (WASP), an inducer of actin branching. Our simulations incorporate both elasticity and relaxation of the actin tail, thus allowing for local network compression. Whilst steady state motility derives mainly from polymerization at the leading edge, nonthermal stored elastic energy and retrograde flow are observed in a thin slab of material close to the obstacle. We observe a crossover from steady to hopping bead motion as the branching rate is decreased.     

Neumann resonances in linear elasticity for an arbitrary body

We study resonances (scattering poles) associated to the elasticity operator in the exterior of an arbitrary obstacle inR ³ with Neumann boundary conditions. We prove that there exists a sequence of resonances tending rapidly to the real axis.Partly supported by BSF under grant MM 401.     

Discrete rearranging disordered patterns, part II: 2D plasticity, elasticity and flow of a foam

The plastic flow of a foam results from bubble rearrangements. We study their occurrence in experiments where a foam is forced to flow in 2D: around an obstacle; through a narrow hole; or sheared between rotating disks. We describe their orientation and frequency using a topological matrix defined in the companion paper (F. Graner, B. Dollet, C. Raufaste, and P. Marmottant, this issue, 25 (2008) DOI 10.1140/epje/i2007-10298-8), which links them with continuous plasticity at large scale. We then suggest a phenomenological equation to predict the plastic strain rate: its orientation is determined from the foam's local elastic strain; and its rate is determined from the foam's local elongation rate. We obtain a good agreement with statistical measurements. This enables us to describe the foam as a continuous medium with fluid, elastic and plastic properties. We derive its constitutive equation, then test several of its terms and predictions.     

Factorization method for inverse obstacle scattering problem in three-dimensional planar acoustic waveguides

In this paper, we consider the inverse scattering problem of reconstructing a bounded obstacle in a three-dimensional planar waveguide from the scattered near-field data measured on a finite cylindrical surface containing the obstacle and corresponding to infinitely many incident point sources also placed on the measurement surface. The obstacle is allowed to be an impenetrable scatterer or a penetrable scatterer. We establish the validity of the factorization method with the nearfield data to characterize the obstacle in the planar waveguide by constructing an outgoing-to-incoming operator which is an integral operator defined on the measurement surface with the kernel given in terms of an infinite series.     

On Uniqueness in the General Inverse Transmisson Problem

In this paper we demonstrate uniqueness of a transparent obstacle, of coefficients of rather general boundary transmission condition, and of a potential coefficient inside obstacle from partial Dirichlet-to Neumann map or from complete scattering data at fixed frequency. The proposed transmission problem includes in particular the isotropic elliptic equation with discontinuous conductivity coefficient. Uniqueness results are shown to be optimal. Hence the considered form can be viewed as a canonical form of isotropic elliptic transmission problems. Proofs use singular solutions of elliptic equations and complex geometrical optics. Determining an obstacle and boundary conditions (i.e. reflecting and transmitting properties of its boundary and interior) is of interest for acoustical and electromagnetic inverse scattering, for modeling fluid/structure interaction, and for defects detection.     

Anomalous elasticity of an ordered lamellar liquid foam

We investigate experimentally the linear viscoelastic properties of a lamellar liquid foam as a function of the cell size and spatial organisation. The system consists of multilamellar vesicles generated by a simple shear flow on a lyotropic lamellar phase. The vesicles can be prepared either in an amorphous or a spatially ordered state. Their size is easily tunable in the range R = 0.5-15 μm. Whereas the shear modulus of the amorphous lamellar foam is alike that of usual liquid foams or concentrated emulsions and scales linearly with 1/R, the elastic modulus of the ordered foam is almost independent of the cell size. This result --probably the first describing the elasticity of an ordered foam-like system-- remains unexplained. Received 7 August 2000     

Calculation of phonon life time in amorphous solids

We present the calculation of phonon life time at low frequencies in an amorphous solid, which is assumed to be characterized by an elasticity that exhibits spatial fluctuation. Thermodynamic Green’s function method is used to compute phonon self-energy, and an iterative method is devised to obtain an improvement upon the first order perturbation calculation. The elasticity correlation is taken to be an exponentially falling function of distance. We obtain an inverse life time that varies as the fourth power of phonon frequency for small values of the latter, and whose frequency-dependence becomes weaker and weaker as the frequency increases.     

Two-dimensional foam rheology with viscous drag

We formulate and apply a continuum model that incorporates elasticity, yield stress, plasticity, and viscous drag. It is motivated by the two-dimensional foam rheology experiments of Debregeas et al. [Phys. Rev. Lett. 87, 178305 (2001)10.1103/PhysRevLett.87.178305] and Wang et al. [Phys. Rev. E 73, 031401 (2006)10.1103/PhysRevE.73.031401], and is successful in exhibiting their principal features, which are an exponentially decaying velocity profile and strain localization. Transient effects are also identified.     

Elasticity reconstruction for ultrasound elastography using a radial compression: an inverse approach

To reduce the inherent mechanical artifacts in the strain images, many groups have investigated solutions to the inverse problem in elastography. However, in prostate elastography or intravascular elastography where the compression direction is radial, the inverse problem has not been studied thoroughly. In this paper, an iterative approach is proposed to reconstruct tissue elasticity for ultrasound elastography using a radial compression. The method is based upon the stress-strain relations in the polar coordinates. Computer simulations in an intravascular model are performed to illustrate the feasibility of this method in reducing the mechanical artifacts of the strain images. The reconstructed elasticity error and the contrast-transfer efficiency (CTE) as a function of the iteration number show that the inverse approach converges with a few iterations.     

A method to solve an acoustic inverse scattering problem involving smart obstacles

A time-harmonic acoustic inverse scattering problem involving smart obstacles is formulated and a method to solve it is proposed. A smart obstacle is an obstacle that, when hit by an incoming acoustic wave, tries to pursue a given goal circulating a suitable pressure current on its boundary. A pressure current is a quantity whose physical dimension is pressure divided by time. The goals pursued by the smart obstacles that we have considered are the following ones: to be undetectable or to appear with a shape and/or acoustic boundary impedance different from its actual ones eventually in a location in space different from the actual location. The following time-harmonic inverse scattering problem is considered: from the knowledge of several far fields generated by the smart obstacle when hit by known time-harmonic waves, the knowledge of the goal pursued by the smart obstacle and of its acoustic boundary impedance reconstruct the boundary of the obstacle. A method to solve this inverse problem that generalizes the so-called Herglotz function method is proposed. Some numerical experiments that validate the method proposed are presented. The website http://www.econ.univpm.it/recchioni/w13 contains some auxiliary material that helps the understanding of the current paper.     

Relaxation time of the topological T1 process in a two-dimensional foam

The elementary topological T1 process in a two-dimensional foam corresponds to the flip of one film with respect to the geometrical constraints, and is a process by which the structure of an out-of-equilibrium foam evolves. We study both experimentally and theoretically the T1 dynamics in a dry two-dimensional foam. The dynamics is controlled by the surface viscoelastic properties of the films (surface shear plus dilatational viscosity, mu_{s}+kappa, and Gibbs elasticity ), and is independent of the shear viscosity of the bulk liquid. Moreover, the dynamics of the T1 process provides a tool for measuring the surface rheological properties: we obtained =32+/-8 mN/m and mu_{s}+kappa=1.3+/-0.7 mPa.m.s for sodium dodecyl sulfate, and =65+/-12 mN/m and mu_{s}+kappa=31+/-12 mPa.m.s for bovine serum albumin, in good agreement with literature values.     

Reduced-order modeling of time-dependent reflectance profiles from purely scattering media

Due to the widespread existence and importance of foam, inverse techniques for characterizing industrial foams are of interest. An essential element in an inverse method used to characterize a foam layer is a model of the time-dependent reflectance of a laser pulse. Monte Carlo methods may be used to accurately model reflectance, but these methods are computationally expensive. Computationally efficient methods based on the diffusion approximation have been developed, but this approach is not sufficiently accurate in many cases of interest. Therefore, a computationally efficient and robust method is desirable. This paper presents a computationally efficient method for modeling the time-dependent reflectance of a laser pulse from a non-absorbing, scattering plane layer that is based on reduced-order modeling techniques. The accuracy of the proposed method is demonstrated by comparing reflectance profiles for randomly selected foam layer properties with corresponding profiles that were generated from Monte Carlo simulations.     

Non-ambiguous recovery of Biot poroelastic parameters of cellular panels using ultrasonicwaves

The inverse problem of the recovery of the poroelastic parameters of open-cell soft plastic foam panels is solved by employing transmitted ultrasonic waves (USW) and the Biot-Johnson-Koplik-Champoux-Allard (BJKCA) model. It is shown by constructing the objective functional given by the total square of the difference between predictions from the BJKCA interaction model and experimental data obtained with transmitted USW that the inverse problem is ill-posed, since the functional exhibits several local minima and maxima. In order to solve this problem, which is beyond the capability of most off-the-shelf iterative nonlinear least squares optimization algorithms (such as the Levenberg Marquadt or Nelder-Mead simplex methods), simple strategies are developed. The recovered acoustic parameters are compared with those obtained using simpler interaction models and a method employing asymptotic phase velocity of the transmitted USW. The retrieved elastic moduli are validated by solving an inverse vibration spectroscopy problem with data obtained from beam-like specimens cut from the panels using an equivalent solid elastodynamic model as estimator. The phase velocities are reconstructed using computed, measured resonance frequencies and a time-frequency decomposition of transient waves induced in the beam specimen. These confirm that the elastic parameters recovered using vibration are valid over the frequency range ofstudy.     

Room evacuation in the presence of an obstacle

The investigation of human behaviour while trying to escape from a room under panic is an important issue in complex systems research. Several authors have called attention to the fact that placing an obstacle near the exit improves the evacuation time of the room (Helbing et al. (2000, 2005) [2] and [8], Hughes (2003) [6], Johansson and Helbing (2005) [16], Piccoli and Tosin (2009) [5]). We studied this effect in the context of the “social force model” (Helbing et al. (2000) [2]). We show that placing an obstacle does not guarantee, by itself, better chances of survival for all pedestrians. The way they choose to avoid the obstacle is critical for their own performance. We found not only that the faster they try to escape, the slower they get out (“faster is slower” effect), but also, the short cut they might take in order to get to the exit will probably do no better (“clever is not always better” effect).     

A SIMPLIFIED METHOD TO DESIGN SUSPENDED CABS FOR COUNTERBALANCE TRUCKS

A “low-frequency” suspension system, placed between the driving cab and chassis of an existing fork lift truck was designed. The aim of this project was to develop a design procedure which is easy to implement and suitable for all types of fork lift trucks. It was also to show how the use of numerical simulation could be helpful to optimize the efficiency of such suspension systems. The cab specifications were: (1) to achieve a vertical vibration attenuation of at least 50% when this truck is tested under severe but realistic conditions, (2) to operate with no specific adjustment for drivers weighing between 60 and 100 kg, (3) to be efficient with a reasonable dynamic stroke (about 3 cm maximum). The suspended cab was modelled using ADAMS software. In the simplified method, the input acceleration signals (at the four fixing points of the cab) were not computed from a vehicle model (chassis and wheels) but directly measured under various driving conditions (passage of two or four wheels over an obstacle with a loaded or unloaded fork lift truck). This model allowed evaluation of the theoretical attenuation, obtained below the driver's seat along the three axes, in comparison with an infinitely rigid suspension. The attenuation ratio was calculated for several values of the characteristics of the suspension components (stiffness and damping). Similarly, for every design value tested, the design constraints were evaluated and at the end of this parametrical study, optimal suspension components were found. Finally, the suspended cab was built according to the results of the parametrical study and measurements subsequently confirmed that the attenuation of vertical accelerations was more than 50%.     

Identifying the elastic properties of an inhomogeneously thick layer

We consider the problem of reconstructing inhomogeneously thick elasticity moduli of an isotropic layer in analyzing steady-state vibration. The problem reduces to a one-dimensional coefficient of an inverse problem whose solution is constructed with the use of iteration schemes for integral Fredholm equations of the first and second kind. We consider model examples of reconstructing the layer characteristics and discuss various aspects of numerical realization.