Francesco Zirilli

Boundary layers and non-linear vibrations in an axially moving beam

Abstract The non-linear oscillations of a one-dimensional axially moving beam with vanishing flexural stiffness and weak non-linearities are analysed. The solution of the initial-boundary value problem for the partial differential equation that describes the motion of the beam when two parameters related to the flexural stiffness and the non-linear terms vanish is expanded into a perturbative double series. Two singular perturbation effects due to the small flexural stiffness and to the weak non-linear terms arise: (i) a boundary layer effect when the flexural stiffness vanishes, (ii) a secular effect. Some tests are performed to compare the “first order” perturbative solution with an approximate solution obtained by a finite difference scheme. The effect of the oscillation amplitude combined with the presence of small bending stiffness and axial transport velocity is investigated enlighting some interesting aspects of axially moving systems. The value of the perturbative series as a computational tool is shown.

A fast phase unwrapping algorithm for SAR interferometry

Phase unwrapping is the key problem in building the elevation map of a scene from interferometric synthetic aperture radar (SAR) system data. Phase unwrapping consists in the reconstruction of the phase difference of the radiation received by two SAR systems as a function of the azimuth and slant range coordinates. The data available to reconstruct the phase difference are a measure of the difference module 2/spl pi/. The authors propose a phase unwrapping method that makes use of the equivalent, in a discrete space, of the irrotational property of a gradient vector field. This property is used first to locate the areas where the discrete vector field estimated from the available data must be corrected, and then, with the knowledge of some a priori information, to perform the correction needed to obtain a useful estimate of the discrete gradient of the phase difference function, from which the phase difference function is reconstructed. The use of the fast Fourier transform makes it possible to have a fast algorithm, that is to process an image of N pixel in O(NlogN) elementary operations. Tests of the method proposed here on real and simulated data are presented.

The fusion of different resolution SAR images

We explore the possibility of using different data sets relative to the same scene to obtain a better knowledge of the scene than the one obtained using only one data set. In particular we concentrate on the fusion of two different spatial resolution images, although the method we propose can be regarded as a method of more general interest. The fused image has the least mean square deviation from the finer resolution image, subject to the constraints imposed by the knowledge of the coarser resolution image. Fusion is obtained by solving a constrained quadratic highly parallelizable minimization problem. Explicit formulas for the solution of the minimization problem are given. The number of elementary operations required is proportional to the number of pixels of the finer resolution image. We test our method on a class of simulated images that reproduce some features of synthetic aperture radar (SAR) images. As an example we consider the problem of detection of point scatterers in a uniform background. The results obtained show that the information from the coarser resolution image can significantly improve the quality of the reconstructed scene obtained from the finer resolution image.

The use of the Pontryagin maximum principle in a furtivity problem in time-dependent acoustic obstacle scattering

Abstract In this paper we consider a furtivity problem in the context of time-dependent three-dimensional acoustic obstacle scattering. The scattering problem for a ‘passive’ obstacle is the following: an incoming acoustic wavepacket is scattered by a bounded simply connected obstacle with locally Lipschitz boundary having a known boundary acoustic impedance. The scattered wave is the solution of an exterior problem for the wave equation. To make the obstacle furtive we leave ‘passive’ obstacles and we consider ‘active’ obstacles, that is obstacles that, when hit by the incoming wavepacket, react with a pressure current circulating on their boundary. The furtivity problem consists of making the acoustic field scattered by the obstacle ‘as small as possible’ by choosing a control function, that is a pressure current on the boundary of the obstacle, in the function space of the admissible controls. It consists of finding the control function that minimizes a cost functional that will be made precise later. This furtivity problem is of great relevance in many applications. The mathematical model for this furtivity problem is a control problem for the wave equation. In the boundary condition for the wave equation on the boundary of the obstacle we introduce a control function, the so-called pressure current. The cost functional depends on the control function, and on the scattered acoustic field. Note that the scattered field depends on the control function via the boundary conditions. Using the Pontryagin maximum principle we show that, for a suitable choice of the cost functional, the first-order optimality conditions for the furtivity problem considered can be formulated as an exterior problem defined outside the obstacle for a system of two coupled wave equations. This is the main purpose of the paper. Moreover, to solve this exterior problem numerically we develop a highly parallelizable method based on a ‘perturbative series’ of the type proposed in 1. This method obtains the time-dependent scattered field and the control function as superpositions of time harmonic functions. The space-dependent parts of each time harmonic component of the scattered field and of the control function are obtained by solving an exterior boundary value problem for two coupled Helmholtz equations. The mathematical model and the numerical method proposed are validated by studying some test problems numerically. The results obtained with a parallel implementation of the numerical method proposed on the test problems are shown and discussed from the numerical and the physical point of view. The quantitative character of the results obtained is established. Animations (audio, video) relative to the numerical experiments can be found at stacks.iop.org/WRM/11/549. MThis article features online multimedia enhancements

Three-dimensional inverse obstacle scattering for time harmonic acoustic waves: a numerical method

A numerical method for a three-dimensional inverse acoustic scattering problem is considered. From the knowledge of several far fields patterns of the Helmholtz equation a closed surface

An inverse problem for the three dimensional vector Helmholtz equation for a perfectly conducting obstacle

\partial D

A Quadratically Convergent Method for Linear Programming

representing the boundary of an unknown obstacle D is reconstructed. The obstacle D is supposed to be acoustically soft or acoustically hard or characterized by a given acoustic impedance.

A masking problem in time dependent acoustic obstacle scattering

Abstract A numerical algorithm for a three-dimensional inverse electromagnetic scattering problem is considered. For time-harmonic waves the Maxwells equations are reduced to the vector Helmholtz equation. From the knowledge of several far fields generated by an obstacle D when hit by incoming linearly polarized plane waves the boundary ∂D of the obstacle is reconstructed. The obstacle D is supposed to be bounded, connected, with smooth boundary and perfectly conducting. The reconstruction procedure proposed here generalizes the “Herglotz function method” introduced by Colton and Monk [1] for the acoustic problem and is effective in the so-called resonance region.

A new formalism for wave scattering from a bounded obstacle

Abstract A new method to solve linear programming problems is introduced. This method follows a path defined by a system of o.d.e., and for nondegenerate problem is quadratically convergent.

The time harmonic electromagnetic field in a disturbed half‐space: An existence theorem and a computational method

A masking problem in time dependent three dimensional acoustic obstacle scattering is considered. The masking problem consists in making masked a bounded scatterer characterized by an acoustic boundary impedance and immersed in a homogeneous isotropic medium that, when hit by an incident acoustic field, generates a scattered acoustic field. The precise definition of the masking problem is given later. This problem has been formulated as an optimal control problem for the wave equation. The corresponding first order optimality condition is derived and solved with a highly parallelizable numerical method. Some numerical experience on a test problem is shown.