Sound-hard Obstacles Research Articles

Reconstructing the shape and material parameters of dissipative obstacles using an impedance model

In inverse scattering problems, a model that allows for the simultaneous recovery of both the domain shape and an impedance boundary condition covers a wide range of problems with impenetrable domains, including recovering the shape of sound-hard and sound-soft obstacles and obstacles with thin coatings. This work develops an optimization framework for recovering the shape and material parameters of a penetrable, dissipative obstacle in the multifrequency setting, using a constrained class of curvature-dependent impedance function models proposed by Antoine et al (2001 Asymptotic Anal. 26 257–83). We find that in certain regimes this constrained model improves the robustness of the recovery problem, compared to more general models, and provides meaningfully better obstacle recovery than simpler models. We explore the effectiveness of the model for varying levels of dissipation, for noise-corrupted data, and for limited aperture data in the numerical examples.

Computation of scattering poles using boundary integrals

Scattering resonances have important applications in mathematics, physics and engineering. They can be viewed as the poles of the meromorphic extension of the scattering operator. In this paper, we consider the computation of the scattering poles for sound soft obstacles. The scattering problem is formulated using boundary integrals, which is then discretized by the Nyström method. The discrete scattering poles are computed using the contour integral method. The proposed method is highly accurate, free of spurious modes, and can be extended to treat other obstacles, e.g., sound hard obstacles. Numerical examples are presented to validate the effectiveness and accuracy. The current paper is among the very few computational studies of scattering poles for obstacles.

A novel quantitative inverse scattering scheme using interior resonant modes

This paper is devoted to a novel quantitative imaging scheme of identifying impenetrable obstacles in time-harmonic acoustic scattering from the associated far-field data. The proposed method consists of two phases. In the first phase, we determine the interior eigenvalues of the underlying unknown obstacle from the far-field data via the indicating behavior of the linear sampling method. Then we further determine the associated interior eigenfunctions by solving a constrained optimization problem, again only involving the far-field data. In the second phase, we propose a novel iteration scheme of Newton’s type to identify the boundary surface of the obstacle. By using the interior eigenfunctions determined in the first phase, we can avoid computing any direct scattering problem at each Newton’s iteration. The proposed method is particularly valuable for recovering a sound-hard obstacle, where the Newton’s formula involves the geometric quantities of the unknown boundary surface in a natural way. We provide rigorous theoretical justifications of the proposed method. Numerical experiments in both 2D and 3D are conducted, which confirm the promising features of the proposed imaging scheme. In particular, it can produce quantitative reconstructions of high accuracy in a very efficient manner.

An iterative scheme for imaging acoustic obstacle from phaseless total-field data

<p style='text-indent:20px;'>In this paper, we consider the inverse problem of determining the location and the shape of a sound-soft or sound-hard obstacle from the modulus of the total-field collected on a measured curve for an incident point source. We propose a system of nonlinear integral equations based iterative scheme to reconstruct both the location and the shape of the obstacle. Several validating numerical examples are provided to illustrate the effectiveness and robustness of the proposed inversion algorithm.</p>

The interior transmission eigenvalue problem for elastic waves in media with obstacles

In this paper, we investigate the interior transmission eigenvalue problem for elastic waves propagating outside a sound-soft or a sound-hard obstacle surrounded by an anisotropic layer. This study is motivated by the inverse problem of identifying an object embedded in an inhomogeneous media in the presence of elastic waves. Our analysis of this non-selfadjoint eigenvalue problem relies on the weak formulation of involved boundary value problems and some fundamental tools in functional analysis.

On controllability methods for the Helmholtz equation

When the Helmholtz equation is discretized by standard finite difference or finite element methods, the resulting linear system is highly indefinite and thus notoriously difficult to solve, in fact increasingly so at higher frequency. The exact controllability approach (Bristeau et al., 1994) instead reformulates the problem in the time domain and seeks the time-harmonic solution of the corresponding wave equation. By iteratively reducing the mismatch between the solution at initial time and after one period, the controllability method greatly speeds up the convergence to the time-harmonic asymptotic limit. Moreover, each conjugate gradient iteration solely relies on standard numerical algorithms, which are inherently parallel and robust against higher frequencies. The original energy functional used to penalize the departure from periodicity is strictly convex only for sound-soft scattering problems. To extend the controllability approach to general boundary-value problems governed by the Helmholtz equation, new penalty functionals are proposed, which are numerically efficient. Numerical experiments for wave scattering from sound-soft and sound-hard obstacles, inclusions, but also for wave propagation in closed wave guides illustrate the usefulness of the resulting controllability methods.

Generalized impedance boundary conditions and shape derivatives for 3D Helmholtz problems

This paper is concerned with the shape sensitivity analysis of the solution to the Helmholtz transmission problem for three-dimensional sound-soft or sound-hard obstacles coated by a thin layer. This problem can be asymptotically approached by exterior problems with an improved condition on the exterior boundary of the coated obstacle, called generalized impedance boundary condition (GIBC). Using a series expansion of the Laplacian operator in the neighborhood of the exterior boundary, we retrieve the first-order GIBCs characterizing the presence of an interior thin layer with a constant thickness. The first shape derivative of the solution to the original Helmholtz transmission problem solves a new thin layer transmission problem with non-vanishing jumps across the exterior and the interior boundary of the thin layer. We show that we can interchange the first-order differentiation with respect to the shape of the exterior boundary and the asymptotic approximation of the solution. Numerical experiments are presented to highlight the various theoretical results.

Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects

We present a high-order accurate boundary-based solver for three-dimensional (3D) frequency-domain scattering from a doubly-periodic grating of smooth axisymmetric sound-hard or transmission obstacles. We build the one-obstacle solution operator using separation into P azimuthal modes via the FFT, the method of fundamental solutions (with N proxy points lying on a curve), and dense direct least-squares solves; the effort is O(N3P) with a small constant. Periodizing then combines fast multipole summation of nearest neighbors with an auxiliary global Helmholtz basis expansion to represent the distant contributions, and enforcing quasiperiodicity and radiation conditions on the unit cell walls. Eliminating the auxiliary coefficients, and preconditioning with the one-obstacle solution operator, leaves a well-conditioned square linear system that is solved iteratively. The solution time per incident wave is then O(NP) at fixed frequency. Our scheme avoids singular quadratures, periodic Green's functions, and lattice sums, and its convergence rate is unaffected by resonances within obstacles. We include numerical examples such as scattering from a grating of period 13λ×13λ comprising highly-resonant sound-hard “cups” each needing NP=64800 surface unknowns, to 10-digit accuracy, in half an hour on a desktop.

Defect Detection from Multi-frequency Limited Data via Topological Sensitivity

In this work we investigate the reconstruction of sound-hard obstacles buried in a bounded material medium by a non-iterative method based on the computation of topological derivatives. The main purpose is to suitably combine multi-frequency data in a fairly demanding measurement configuration: a very reduced number of aligned emitters and receivers are considered. The performance of the algorithm is illustrated for both single and multiple obstacle reconstructions.

Regularized Transformation-Optics Cloaking for the Helmholtz Equation: From Partial Cloak to Full Cloak

We develop a very general theory on the regularized approximate invisibility cloaking for the wave scattering governed by the Helmholtz equation in any space dimensions \({N \geq 2}\) via the approach of transformation optics. There are four major ingredients in our proposed theory: (1) The non-singular cloaking medium is obtained by the push-forwarding construction through a transformation that blows up a subset \({K_\varepsilon}\) in the virtual space, where \({\varepsilon \ll 1}\) is an asymptotic regularization parameter. \({K_\varepsilon}\) will degenerate to K 0 as \({\varepsilon \rightarrow +0}\), and in our theory K 0 could be any convex compact set in \({\mathbb{R}^N}\), or any set whose boundary consists of Lipschitz hypersurfaces, or a finite combination of those sets. (2) A general lossy layer with the material parameters satisfying certain compatibility integral conditions is employed right between the cloaked and cloaking regions. (3) The contents being cloaked could also be extremely general, possibly including, at the same time, generic mediums and, sound-soft, sound-hard and impedance-type obstacles, as well as some sources or sinks. (4) In order to achieve a cloaking device of compact size, particularly for the case when \({K_\varepsilon}\) is not “uniformly small”, an assembly-by-components, the (ABC) geometry is developed for both the virtual and physical spaces and the blow-up construction is based on concatenating different components.Within the proposed framework, we show that the scattered wave field \({u_\varepsilon}\) corresponding to a cloaking problem will converge to u 0 as \({\varepsilon \rightarrow +0}\), with u 0 being the scattered wave field corresponding to a sound-hard K 0. The convergence result is used to theoretically justify the approximate full and partial invisibility cloaks, depending on the geometry of K 0. On the other hand, the convergence results are conducted in a much more general setting than what is needed for the invisibility cloaking, so they are of significant mathematical interest for their own sake. As for applications, we construct three types of full and partial cloaks. Some numerical experiments are also conducted to illustrate our theoretical results.

The factorization method for scatterers with different physical properties

The scattering of time-harmonic acoustic plane waves by a mixed type scatterer is considered. Such a scatterer is given as the union ofseveral components with different physical properties. Some of them are impenetrable obstacles with Dirichlet or impedance boundary conditions,while the others are penetrable inhomogeneous media with compact support.This paper is concerned with modifications of the factorization method for the following two basic cases,one is the scattering by a priori separated sound-soft and sound-hard obstacles,the other one is the scattering by a scatterer with impenetrable (Dirichlet) and penetrable components. The other cases can be dealt with similarly.Finally, some numerical experiments are presented to demonstrate the feasibility and effectiveness of the modified factorization methods.

Reconstruction of Neumann eigenvalues and support of sound hard obstacles

Recent study in inverse scattering theory shows that Dirichlet and transmission eigenvalues for sound soft obstacles and inhomogeneous non-absorbing media, respectively, can be reconstructed from scattering data. The purpose of the current paper is to show that Neumann eigenvalues can be estimated from scattering data and to provide a thorough numerical study. To reconstruct an eigenvalue, one usually chooses a point inside the obstacle and solves some linear ill-posed integral equations for wavenumbers in an interval containing the eigenvalue. For a specific eigenvalue, it is noted that there are points inside the obstacle which cannot be used. In addition, we present some numerical examples relating to the behavior of the solutions of the integral equations for points outside the obstacle. Finally, an eigenvalue method is employed to reconstruct the support of the obstacle.

The factorization method for inverse acoustic scattering in a layered medium

In this paper, we consider a problem of inverse acoustic scattering by an impenetrable obstacle embedded in a layered medium. We will show that the factorization method can be applied to recover the embedded obstacle; that is, the equation is solvable if and only if the sampling point z is in the interior of the unknown obstacle. Here, is a self-adjoint operator related to the far field operator and ϕz is the far field pattern of the Green function with respect to the problem of scattering by the background medium for point z. The validity of the factorization method is proven with the help of a mixed reciprocity principle and an application of the scattering operator. Due to the established mixed reciprocity principle, knowledge of the Green function for the background medium is no longer required, which makes the method attractive from the computational point of view. The paper is only concerned with sound-soft obstacles, but the analysis can be easily extended for sound-hard obstacles, or obstacles with separated sound-soft and sound-hard parts. Finally, we provide an explicit example for a radially symmetric case and present some numerical examples.

Solving an inverse obstacle problem for the wave equation by using the boundary control method

We introduced in Oksanen (2011 arXiv:1106.3204) a method to locate discontinuities of a wave speed in dimension 2 from acoustic boundary measurements modeled by the hyperbolic Neumann-to-Dirichlet operator. Here we extend the method for sound hard obstacles in arbitrary dimension. We present numerical experiments with simulated noisy data suggesting that the method is robust against measurement noise.

Stability for the acoustic scattering problem for sound-hard scatterers

We study the stability for the direct acoustic scattering problem with sound-hard scatterers with minimal regularity assumptions on the scatterers. The main tool we use for this purpose is the convergence in the sense of Mosco. We obtain uniform decay estimates for scattered fields and we investigate how a sound-hard screen may be approximated by thin sound-hard obstacles.

An integral equation approach to acoustic cloaking

The paper presents an original boundary integral equation (BIE) formulation for the analysis of the acoustic cloaking of a scatterer. The advantage of such an approach is the lower computational burden, especially when the analysis of a large portion of the hosting domain is required. The partial differential equation governing the propagation inside the cloak is recast in the form of non-homogeneous wave equation, with field sources depending on the mechanical properties of the cloak. The boundary integral formulation is derived using the standard procedure. The boundary element method (BEM) is used to derive the matrix transfer function of the cloak. The latter is applied to the incident field at the cloak's outer boundary to obtain the total field at arbitrary locations in the host. The formulation is applied to the simple case of a radially symmetric cloak embedding a circular obstacle. Numerical results are presented for sound-hard and sound-soft obstacles, including a study of the cloaking efficiency as a function of the frequency.

The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: II. Obstacles with a dissipative boundary or finite refractive index and back-scattering data

In this paper, a wave is generated by the initial data whose support is localized at the outside of unknown obstacles and observed in a limited time on a known closed surface or the same position as the support of the initial data. The observed data in the latter process are nothing but the back-scattering data. Two types of obstacles are considered. One type is of obstacles with a dissipative boundary condition, which is a generalization of the sound-hard obstacles; another type is of obstacles with a finite refractive index, the so-called transparent obstacles. For each type of obstacle, two formulas that yield explicitly the distance from the support of the initial data to unknown obstacles are given.

Inverse obstacle scattering problems with a single incident wave and the logarithmic differential of the indicator function in the enclosure method

This paper gives a remark on the enclosure method by considering inverse obstacle scattering problems with a single incident wave whose governing equation is given by the Helmholtz equation in two dimensions. It is concerned with the indicator function in the enclosure method. The previous indicator function is essentially real-valued since only its absolute value is used. In this paper, another method for the use of the indicator function is introduced. The method employs the logarithmic differential with respect to the independent variable of the indicator function and yields directly the coordinates of the vertices of the convex hull of unknown polygonal sound-hard obstacles or thin ones. The convergence rate of the obtained formulae is better than that of the previous indicator function. Some other applications of this method are also given.

A note on the enclosure method for an inverse obstacle scattering problem with a single point source

This paper gives a note on an application of the enclosure method to an inverse obstacle scattering problem governed by the Helmholtz equation in two dimensions. It is shown that one can uniquely determine the convex hull of an unknown sound-hard polygonal obstacle from the trace of the total wave that was exerted by a single point source onto a known circle surrounding the obstacle provided the source is sufficiently far from the obstacle. The result contains a formula that extracts the value of the support function of the obstacle at a generic direction. Some other applications to thin obstacles, obstacles in a layered medium and the far-field equation in the linear sampling method are also included.

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

We consider the inverse acoustic scattering problem of determining a sound-hard obstacle by far-field measurements. It is proved that a polyhedral scatterer in , consisting of finitely many solid polyhedra, is uniquely determined by a single incoming plane wave.