Unbounded Rough Surface Research Articles

The Nyström Method for Elastic Wave Scattering by Unbounded Rough Surfaces

The Nyström Method for Elastic Wave Scattering by Unbounded Rough Surfaces

A Priori Bounds for Elastic Scattering by Deterministic and Random Unbounded Rough Surfaces

A Priori Bounds for Elastic Scattering by Deterministic and Random Unbounded Rough Surfaces

The time‐domain scattering by the elastic shell in a two‐layered unbounded structure

The subject of this research is the time‐domain scattering problem for a three‐dimensional layered elastic shell submerged in a two‐layered fluid separated by an unbounded rough surface. The essence of the problem is to model the scattering interaction between the given elastic shell and some incident wave in a two‐layered medium. Using the exact transparent boundary condition (TBC), we reformulate the unbounded scattering problem into an equivalent initial‐boundary value problem. The well‐posedness is proved for the problem by the Laplace transform and variational method in the s‐domain. Moreover, we show that the reduced problem has a unique weak solution by the energy method. The priori estimates with explicit dependence on the time are derived for the acoustic pressure and the elastic displacement in the time domain.

Time‐domain PML analysis of a fluid–solid interaction problem above a rough surface

This paper is concerned with the time‐dependent acoustic–elastic interaction problem associated with a bounded elastic body immersed in a homogeneous air or fluid above an unbounded rough surface. The well‐posedness and stability of the problem are first established by using the Laplace transform and the energy method. A perfectly matched layer (PML) is then introduced to truncate the interaction problem above a finite layer containing the elastic body, leading to a PML problem in a finite strip domain. We further establish the existence, uniqueness, and stability estimate of solutions to the PML problem. Finally, we prove the exponential convergence of the PML problem in terms of the thickness and parameter of the PML, based on establishing an error estimate between the DtN operators of the original problem and the PML problem.

Recovering Unbounded Rough Surfaces with a Direct Imaging Method

In this paper, we consider the inverse acoustic scattering problem by an unbounded rough surface. A direct imaging method is proposed to reconstruct the rough surfaces from scattered-field data for incident plane waves and the performance analysis is also presented. The reconstruction method is very robust to noises of measured data and does’t need to know the type of the boundary conditions of the surfaces in advance. Finally, numerical examples are carried out to illustrate that our method is fast, accurate and stable even for the case of multiple-scale profiles.

Inverse obstacle scattering for Maxwell’s equations in an unbounded structure

This paper is concerned with the electromagnetic scattering of a point source by a perfectly electrically conducting obstacle which is embedded in a two-layered lossy medium separated by an unbounded rough surface. Given a dipole point source, the direct problem is to determine the electromagnetic wave field for the given obstacle and unbounded rough surface; the inverse problem is to reconstruct simultaneously the obstacle and unbounded rough surface from the reflected and transmitted fields measured on a plane surface which is above and below the unknown objects, respectively. For the direct problem, its well-posedness is established and a new boundary integral equation is proposed. The analysis is based on the exponential decay of the dyadic Green function for Maxwell’s equations in a lossy medium. For the inverse problem, the global uniqueness is proved and a local stability is discussed. A crucial step in the proof of the stability is to obtain the existence and characterization of the domain derivative of the electric field with respect to the shape of the obstacle and unbounded rough surface.

A non-iterative approach to inverse elastic scattering by unbounded rigid rough surfaces

This paper is concerned with the inverse time-harmonic elastic scattering problem of recovering unbounded rough surfaces in two dimensions. We assume that elastic plane waves with different directions are incident onto a rigid rough surface in a half plane. The elastic scattered field is measured within a finite distance above the rough surface. A sampling-type imaging algorithm is proposed to recover the unbounded rough surface from the scattered near-field data, which involves only inner products between the data. Numerical experiments are presented to show that the inversion scheme is not only efficient but also accurate and robust with respect to noise.

A Direct Imaging Method for Inverse Scattering by Unbounded Rough Surfaces

This paper is concerned with the inverse scattering problem by an unbounded rough surface. A direct imaging method is proposed to reconstruct the rough surface from the scattered near-field Cauchy ...

Uniqueness Results for Scattering and Inverse Scattering by Infinite Rough Surfaces with Tapered Wave Incidence

The scattering by a bounded or periodic rough surface is often modeled by assuming the plane wave incidence. However, when the rough surface is unbounded and nonperiodic, a tapered incident wave may be more appropriate to realize asymptotic truncation for the unbounded rough surface. Despite this extensive practical interest, little mathematical analysis of these problems has been carried out. This paper is concerned with the scattering and inverse scattering problem of the infinite impedance rough surfaces with tapered wave incidence. A boundary integral equation formulation is derived by using a half-plane impedance Green's function. It is shown that the integral equation is uniquely solvable in the space of bounded and continuous functions. Further, the impedance boundary value problem for the scattered field is proved to have a unique solution under certain constraints on the boundary impedance. It is also proved that a class of rough surfaces can be uniquely determined by the measurements of the scattered field generated by the tapered incident waves.

Analysis of Transient Acoustic-Elastic Interaction in an Unbounded Structure

Consider the wave propagation in a two-layered medium consisting of a homogeneous compressible air or fluid on top of a homogeneous isotropic elastic solid in three dimensions. The interface between the two layers is assumed to be an unbounded rough surface. This work presents the first mathematical analysis for the transient acoustic-elastic interaction problem in such an unbounded structure. Using an exact transparent boundary condition and suitable interface conditions, we study an initial boundary value problem for the coupling of the acoustic and elastic wave equations. The well-posedness and stability are established for the reduced problem. Moreover, a priori estimates with explicit dependence on the time are obtained for the acoustic pressure and the elastic displacement. The problem addressed is sufficiently general to have potential use in applications.

Imaging of Local Rough Surfaces by the Linear Sampling Method with Near-Field Data

Consider the scattering of a time-harmonic point source from an unbounded rough surface. The surface is assumed to be different from a plane surface over a finite interval, which leads to the situation that the model problem can be reduced to an equivalent boundary value problem with compactly supported boundary data. Well-posedness is thus proved by employing the variational method with classical Fredholm theory. By constructing a modified near-field equation, we then propose a novel sampling-type method for reconstructing the shape and location of the local rough surface, yielding a fast imaging algorithm. Numerical experiments are presented to illustrate the effectiveness of the method.

Multi-parameter identification and shape reconstruction for unbounded fractal rough surfaces with tapered wave incidence

In this paper, we study the inverse scattering of tapered waves from sound soft surfaces or horizontally polarized electromagnetic waves from perfectly conducting surfaces. By the related Fréchet derivative with respect to a given surface of the solution of direct scattering problems, an efficient algorithm is proposed to reconstruct the parameters of unbounded rough surfaces from a set of scattered field measurements. Tapered wave is introduced to realize the asymptotic truncation for unbounded fractal rough surface. In order to improve the results of reconstruction, multi-angle and multi-frequency incidence strategy has been used. The numerical results show that the method yields satisfactory reconstructions for some rough surface profiles.

Variational approach to scattering by unbounded rough surfaces with Neumann and generalized impedance boundary conditions

This paper is concerned with problems of scattering of time-harmonic electromagnetic and acoustic waves from an innite penetrable medium with a certain non-trapping and geometric conditions.

Boundary integral equation methods for the scattering problem by an unbounded sound soft rough surface with tapered wave incidence

In this paper, we consider the scattering problem of tapered acoustic wave by an unbounded sound soft surface. The scattering problem is modeled as a boundary value problem governed by the Helmholtz equation with Dirichlet boundary condition. Although the tapered wave is often introduced to realize asymptotic truncation for unbounded rough surface, the standard Helmholtz integral equations which derive for the scattering of plane waves by an arbitrary bounded obstacle are often used to generate benchmark numerical solutions. Different from the scattering of plane waves by an arbitrary bounded obstacle, we use the angular spectrum representation radiation condition to replace the Sommerfeld radiation condition, and derive a boundary integral equation for studying the scattering problem. Then we study the integral equation by the truncation method, whereby the integral equation posed on an unbounded region is approximated by an integral equation on a bounded region. Some properties of the integral equation in an energy space with weights are proved. Then the collocation method is used to solve the integral equation on a bounded region, and its convergence is also obtained.

A new spectral method for numerical solution of the unbounded rough surface scattering problem

A new spectral method is developed to solve the unbounded rough surface scattering problem. An unbounded rough surface is referred to as a non-local perturbation of an infinite plane surface such that the whole rough surface lies within a finite distance of the original plane. The method uses a transformed field expansion to reduce the boundary value problem with a complex scattering surface into a successive sequence of transmission problems of a planar surface. Hermite orthonormal basis functions are adopted to further simplify these problems to fully decoupled one-dimensional two-point boundary value problems, which are solved efficiently by the Legendre–Galerkin method. Numerical results indicate that the method is efficient, accurate, and well-suited for solving the scattering problem by unbounded rough surfaces.

Elastic scattering by unbounded rough surfaces: solvability in weighted Sobolev spaces

This paper is concerned with the variational approach in weighted Sobolev spaces to time-harmonic elastic wave scattering by one-dimensional unbounded rough surfaces. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the total elastic displacement satisfies either the Dirichlet or impedance boundary condition. We establish uniqueness and existence results at arbitrary frequency for both elastic plane wave and point source (spherical) wave incidence in the two-dimensional case. In particular, our approach covers the elastic scattering from periodic structures (diffraction gratings), and we prove quasiperiodicity of the scattered field whenever the incident field is quasiperiodic. Moreover, the diffraction grating problem is also uniquely solvable in the presented weighted Sobolev spaces for a broad class of non-quasiperiodic incident waves.

Analysis of the scattering by an unbounded rough surface

This paper is concerned with the mathematical analysis of the solution for the wave propagation from the scattering by an unbounded penetrable rough surface. Throughout, the wavenumber is assumed to have a nonzero imaginary part that accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Helmholtz equation with transparent boundary conditions proposed on plane surfaces confining the scattering surface. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. Furthermore, the scattering problem is investigated for the case when the scattering profile is a sufficiently small and smooth deformation of a plane surface. Under this assumption, the problem is equivalently formulated into a set of two‐point boundary value problems in the frequency domain, and the analytical solution, in the form of an infinite series, is deduced by using a boundary perturbation technique combined with the transformed field expansion approach. Copyright © 2012 John Wiley & Sons, Ltd.

Elastic Scattering by Unbounded Rough Surfaces

We consider the two-dimensional time-harmonic elastic wave scattering problem for an unbounded rough surface, due to an inhomogeneous source term whose support lies within a finite distance above the surface. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the elastic displacement vanishes. We propose an upward propagating radiation condition (angular spectrum representation) for solutions of the Navier equation in the upper half-space above the rough surface, and we establish an equivalent variational formulation. Existence and uniqueness of solutions at arbitrary frequency is proved by applying a priori estimates for the Navier equation and perturbation arguments for semi-Fredholm operators.

Electromagnetic Scattering by Unbounded Rough Surfaces

This paper is concerned with the analysis of electromagnetic wave scattering in inhomogeneous medium with infinite rough surfaces. Consider a time-harmonic electromagnetic field generated by either a magnetic dipole or an electric dipole incident on an infinite rough surface. The dielectric permittivity is assumed to have a positive imaginary part which accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Maxwell equations, with transparent boundary conditions proposed on plane surfaces with inhomogeneity in between. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. The perfectly matched layer (PML) method is investigated to truncate the unbounded rough surface electromagnetic scattering problem in the direction away from the rough surfaces. It is shown that the truncated PML problem attains a unique solution. An explicit error estimate is given between the solution of the scattering problem and that of the truncated PML problem. The error estimate implies that the PML solution converges exponentially to the scattering solution by increasing either the PML medium parameter or the PML layer thickness. The convergence result is expected to be useful for determining the PML medium parameter in the computational electromagnetic scattering problem.

Variational Approach in Weighted Sobolev Spaces to Scattering by Unbounded Rough Surfaces

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.