Sound-hard Obstacles Research Articles

Identification of a non-convex obstacle for acoustical scattering

This paper deals with a two or three dimensional inverse scattering problem for identifying a sound hard obstacle in a homogeneous medium. We provide a reconstruction scheme for identifying the obstacle from far field patterns of scattered waves associated with incident waves in all directions when the obstacle is not necessarily convex.

Shape reconstruction of acoustic obstacles from the modulus of the far field pattern

We consider the inverse problem of time-harmonic acoustic wave scattering where the shape of an obstacle is reconstructed from a given incident field and the modulus of the far field pattern of the scattered field. Our approach is based on a pair of nonlinear and ill-posed integral equations to be solved for the shape of the unknown boundary. This approach is an extension of the method suggested by Kress and Rundell [5] for an inverse boundary value problem for the Laplace equation. Since the modulus of far field pattern is invariant under translations [4] we can reconstruct the shape of the obstacle but not the location.   &nbspThe numerical implementation of the method is described and it is illustrated by numerical examples that the method yields satisfactory reconstructions both for sound-soft and sound-hard obstacles, also in the case when the modulus is given in a limited aperture.

Scattering by a perfect conductor in a waveguide: energy-preserving schemes for integral equations

The scattering matrix for a perfectly conducting electrical cylinder (or a sound hard obstacle) in a waveguide is unitary. This is a well-known result which is a consequence of the conservation of power. When a numerical method is employed to approximate the reflection and transmission coefficients of the cylinder, an approximate scattering matrix can be constructed. An integral equation of the second kind for an unknown density can be solved, and the density can then be used for computing the entries of the approximate scattering matrix. We show that this approximate matrix is unitary for cylinders of symmetric cross-section, regardless of the order of the approximation. In the nonsymmetric case, the approximate scattering matrix still satisfies a conservation of energy condition, albeit in an unfamiliar form. As the order of the approximation is increased, conservation of energy is also satisfied in the more familiar form to machine accuracy.

BROADBAND MULTILEVEL FAST MULTIPOLE ALGORITHM FOR ACOUSTIC SCATTERING PROBLEMS

A broadband multilevel fast multipole algorithm (MLFMA) for the acoustic scattering from a sound-hard obstacle is presented. The formulation is based on the Burton–Miller boundary integral equation and Galerkin's method, avoiding any hypersingular integral operators. The resulting matrix equation has good iterative properties for all frequencies and avoids the interior resonance problem. The main novel feature is the use of a broadband MLFMA to accelerate the iterative generalized minimal residual (GMRES) solver. The algorithm is based on a combination of Rokhlin's translation formula for large division cubes and the spectral representation of the Green's function for cubes smaller than one half wavelength, thereby avoiding the sub-wavelength breakdown of the high-frequency MLFMA.

A hybrid method for sound-hard obstacle reconstruction

We are interested in solving the inverse problem of acoustic wave scattering to reconstruct the position and the shape of sound-hard obstacles from a given incident field and the corresponding far field pattern of the scattered field. The method we suggest is an extension of the hybrid method for the reconstruction of sound-soft cracks as presented in [R. Kress, P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems 21 (2005) 773–784] to the case of sound-hard obstacles. The designation of the method is justified by the fact that it can be interpreted as a hybrid between a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem and a decomposition method in the spirit of the potential method as described in [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Cannon, Hornung (Eds.), Inverse Problems, ISNM, vol. 77, 1986, pp. 93–102. Since the method does not require a forward solver for each Newton step its computational costs are reduced. By some numerical examples we illustrate the feasibility of the method.

Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave

We consider the two-dimensional inverse scattering problem of determining a sound-hard obstacle by the far field pattern. We establish the uniqueness within the class of polygonal domains by a single incoming plane wave.

The Herglotz wave function, the Vekua transform and the enclosure method

This paper gives applications of the enclosure method introduced by the author to typical inverse obstacle and crack scattering problems in two dimensions. Explicit extraction formulae of the convex hull of unknown polygonal sound-hard obstacles and piecewise linear cracks from the far field pattern of the scattered field at a fixed wave number and at most two incident directions are given. The main new points of this paper are: a combination of the enclosure method and the Herglotz wave function; explicit construction of the density in the Herglotz wave function by using the idea of the Vekua transform. By virtue of the construction, one can avoid any restriction on the wave number in the extraction formulae. An attempt for the case when the far field pattern is given on limited angles is also given.

Local absorbing boundaries of elliptical shape for scalar wave propagation in a half-plane

We have recently discussed the performance of local second-order two-dimensional absorbing boundary conditions of elliptical shape for scattering and radiation problems involving sound-hard obstacles embedded in a full-plane. In this article, using the method of images, we extend the applicability of elliptically shaped truncation boundaries to semi-infinite acoustic media. For problems in either the time- or the frequency-domains, involving near-surface structures of elongated cross-sections, we show that significant computational savings are attainable when compared against semi-circular truncation geometries.

Inverse scattering problems and the enclosure method

We show that the enclosure method introduced here is applicable to inverse scattering problems in two dimensions. Using the single set of Cauchy data of a solution of the Helmholtz equation, we give some formulae for extracting the value of the support functions of unknown sound-hard polygonal obstacles and piecewise linear cracks from the set.

A high-order algorithm for obstacle scattering in three dimensions

In this work we describe, implement and analyse in detail a high-order fully discrete spectral algorithm for solving the Helmholtz equation exterior to a bounded (sound-soft, sound-hard or absorbing) obstacle in three space dimensions, with Dirichlet, Neumann or Robin (impedance) boundary conditions. Our algorithm may be thought of as a discrete Galerkin method, but it is also equivalent to a Nyström method after a simple transformation. We test our algorithm with extensive computational experiments on a variety of three-dimensional smooth and non-smooth obstacles with conical singularities. Our tests include the computation of scattered and far fields induced by incident plane waves. Our method is shown to be very accurate for scattering from surfaces which are globally parameterised by spherical coordinates, and tests show that it performs very much better than several of the well-established fast algorithms for obstacle scattering on a range of such surfaces, even some which are non-smooth. Further, we prove superalgebraic convergence of the scattered and far fields obtained using our algorithm in the case of smooth scatterers.

Uniqueness in determining polygonal sound-hard obstacles

Uniqueness in determining polygonal sound-hard obstacles

Numerical method for the shape reconstruction of a hard target

A nonlinear optimization method was developed to solve the inverse problem of determining the shape of a hard target from the knowlegde of the far-field pattern of the acoustic scattering wave, it was achieved by solving independently an ill-posed linear system and a well-posed minimization problem. Such a separate numerical treatment for the illposedness and nonlinearity of the inverse problem makes the numerical implementation of the proposed method very easy and fast since there only involves the solution of a small scale minimization problem with one unknown function in the nonlinear optimization step for determining the shape of the sound-hard obstacle. Another particular feature of the method is that it can reproduce the shape of an unknown hard target efficiently from the knowledge of only one Fourier coefficient of the far-field pattern. Moreover, a two-step adaptive iteration algorithm was presented to implement numerically the nonlinear optimization scheme. Numerical experiments for several two-dimensional sound-hard scatterers having a variety of shapes provide an independent verification of the effectiveness and practicality of the inversion scheme.

Continuous Frechet differentiability with respect to a Lipschitz domain and a stability estimate for direct acoustic scattering problems

We consider direct acoustic scattering problems with either a sound-soft or sound-hard obstacle, or lossy boundary conditions, and establish continuous Frechet differentiability with respect to the shape of the scatterer of the scattered field and its corresponding far-field pattern. Our proof is based on the Implicit Function Theorem, and assumes that the boundary of the scatterer as well as the deformation are only Lipschitz continuous. From continuous Frechet differentiability, we deduce a stability estimate governing the variation of the far-field pattern with respect to the shape of the scatterer. We illustrate this estimate with numerical results obtained for a two-dimensional high-frequency acoustic scattering problem.

On the Far Field in Obstacle Scattering

We will establish far field identities for the scattering of time-harmonic acoustic waves from sound-soft and sound-hard obstacles and will illustrate two applications of these identities for the inverse obstacle scattering problem. First, we will provide an alternate proof for the Frechet differentiability of the far field pattern with respect to the boundary. Second, we will connect the identities with a new method for the approximate solution of the inverse obstacle scattering problem recently suggested by Potthast.

On a Concept of Uniqueness in Inverse Scattering for a Finite Number of Incident Waves

We introduce a concept of uniqueness in inverse scattering for a finite number of incident plane waves. Our results hold for inverse acoustic scattering from a sound-soft obstacle, a sound-hard obstacle, or an inhomogeneous medium. It is shown that for $\epsilon>0$ we can find an $N\in \nat$ such that a knowledge of the far field for N incident waves determines the support of the scatterer up to an error of size $\epsilon$ in the Hausdorff distance. In the limit $N\rightarrow \infty$ we obtain well-known uniqueness theorems which hold if the far field for all incident plane waves is known. Our results contain a new proof for the uniqueness of the support of an inhomogeneous medium in this case.

Scattering and inverse scattering of sound-hard obstacles via shape deformation

Direct and inverse scattering of plane acoustic waves from sound-hard obstacles are discussed. The direct problem is solved via the application of the Padé approximation. It is shown that this involves solving only certain algebraic recursion relations and requires neither Green's functions nor integral representations of the field. The shape of the scatterer is assumed to be a superposition of a deformation (allowed to be finite) over an underlying simple geometry. It is demonstrated that such a decomposition allows the scattered field to be obtained as solutions of classical Neumann problems in domains exterior to the underlying simple shape instead of the actual deformed contour. This introduces simplifications in the implementation of a Gauss - Newton type inversion procedure which was used in this study. Some inversions of two-dimensional scatterers are presented.

A Newton method for solving the inverse scattering problem for a sound-hard obstacle

We present a Newton method for solving the ill-posed and nonlinear inverse scattering problem. For its foundation we establish the Fréchet differentiability of the far-field operator with respect to the boundary and show that the Fréchet derivative can be obtained as the solution of a hypersingular integral equation. In order to derive the right-hand sides of this integral equation we use a quadrature method. We describe the numerical solution method and give some numerical results.

Karp’s theorem in acoustic scattering theory

Karp’s Theorem states that if the far field pattern corresponding to the scattering of a time harmonic plane acoustic wave by a sound-soft cylinder is of the form F 0 ( k ; θ − α ) {F_0}(k;\theta - \alpha ) where k k is the wave number, θ \theta the angle of observation and α \alpha the angle of incidence of the plane wave, then the cylinder must be circular. A new proof is given of this result and extended to the cases of scattering by a sound-hard obstacle and an inhomogeneous medium.

Surface fields on sound hard and sound soft obstacles with sharp and smooth boundaries

The computation of the surface fields generated on a sound hard or sound soft obstacle when irradiated by a plane harmonic wave is a challenging problem when the obstacle is of arbitrary shape and has corners. Not only is this computation the first step in determining scattered field, but it has other important practical applications such as the dynamic stress concentrations. The integral equation for the field on the surface of the obstacle is solved numerically using null field (T-matrix approach). It will be shown that for scatterers with corners, a piecewise basis must be chosen to represent the surface fields. Results will be presented for cylindrical scatterers with square, elliptical, and intersecting circular cross sections.