Time-harmonic Plane Wave Research Articles

Inverse obstacle scattering for elastic waves in three dimensions

Consider an exterior problem of the three-dimensional elastic wave equation, which models the scattering of a time-harmonic plane wave by a rigid obstacle. The scattering problem is reformulated into a boundary value problem by introducing a transparent boundary condition. Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle's surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, we consider both the direct and inverse problems. The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles with embedded objects

This paper considers the inverse problem of scattering of time-harmonic acoustic and electromagnetic plane waves by a bounded, inhomogeneous, penetrable obstacle with embedded objects inside. A new method is proposed to prove that the inhomogeneous penetrable obstacle can be uniquely determined from the far-field pattern at a fixed frequency, disregarding its contents. Our method is based on constructing a well-posed interior transmission problem in a small domain associated with the Helmholtz or modified Helmholtz equation and the Maxwell or modified Maxwell equations. A key role is played by the smallness of the domain which ensures that the lowest transmission eigenvalue is large so that a given wave number k is not an eigenvalue of the interior transmission problem. Another ingredient in our proofs is a priori estimates of solutions to the transmission scattering problems with data in Lp (1<p<2), which are established in this paper by using the integral equation method. A main feature of the new method is that it can deal with the acoustic and electromagnetic cases in a unified way and can be easily applied to deal with inverse scattering by unbounded rough interfaces.

English

We consider the inverse scattering of time-harmonic plane waves to reconstruct the shape of a sound-soft crack from a knowledge of the given incident field and the phaseless data, and we check the invariance of far field data with respect to translation of the crack. We present a numerical method that is based on a system of nonlinear and ill-posed integral equations, and our scheme is easy and simple to implement. The numerical implementation is described and numerical examples are presented to illustrate the feasibility of the proposed method.

A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers

Consider the scattering of a time-harmonic plane wave by heterogeneous media consisting of linear or nonlinear point scatterers and extended obstacles. A generalized Foldy–Lax formulation is developed to take fully into account of the multiple scattering by the complex media. A new imaging function is proposed and an FFT-based direct imaging method is developed for the inverse obstacle scattering problem, which is to reconstruct the shape of the extended obstacles. The novel idea is to utilize the nonlinear point scatterers to excite high harmonic generation so that enhanced imaging resolution can be achieved. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Convergence of the PML solution for elastic wave scattering by biperiodic structures

This paper is concerned with the analysis of elastic wave scattering of a time-harmonic plane wave by a biperiodic rigid surface, where the wave propagation is governed by the three-dimensional Navier equation. An exact transparent boundary condition is developed to reduce the scattering problem equivalently into a boundary value problem in a bounded domain. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by developing a PML equivalent transparent boundary condition. The proofs rely on a careful study of the error between the two transparent boundary operators. The work significantly extend the results from the one-dimensional periodic structures to the two-dimensional biperiodic structures. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.

Modelling wave-induced sea ice break-up in the marginal ice zone.

A model of ice floe break-up under ocean wave forcing in the marginal ice zone (MIZ) is proposed to investigate how floe size distribution (FSD) evolves under repeated wave break-up events. A three-dimensional linear model of ocean wave scattering by a finite array of compliant circular ice floes is coupled to a flexural failure model, which breaks a floe into two floes provided the two-dimensional stress field satisfies a break-up criterion. A closed-feedback loop algorithm is devised, which (i) solves the wave-scattering problem for a given FSD under time-harmonic plane wave forcing, (ii) computes the stress field in all the floes, (iii) fractures the floes satisfying the break-up criterion, and (iv) generates an updated FSD, initializing the geometry for the next iteration of the loop. The FSD after 50 break-up events is unimodal and near normal, or bimodal, suggesting waves alone do not govern the power law observed in some field studies. Multiple scattering is found to enhance break-up for long waves and thin ice, but to reduce break-up for short waves and thick ice. A break-up front marches forward in the latter regime, as wave-induced fracture weakens the ice cover, allowing waves to travel deeper into the MIZ.

An adaptive finite element PML method for the elastic wave scattering problem in periodic structures

An adaptive finite element method is presented for the elastic scattering of a time-harmonic plane wave by a periodic surface. First, the unbounded physical domain is truncated into a bounded computational domain by introducing the perfectly matched layer (PML) technique. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by developing an equivalent transparent boundary condition. Second, an a posteriori error estimate is deduced for the discrete problem and is used to determine the finite elements for refinements and to determine the PML parameters. Numerical experiments are included to demonstrate the competitive behavior of the proposed adaptive method.

Relativistic Aspects of Plane Wave Scattering by a Perfectly Conducting Half-Plane With Uniform Velocity Along an Arbitrary Direction

The scattering of time-harmonic plane waves by a perfectly electrically conducting (PEC) half-plane (H-P) in relativistic uniform motion is discussed. The problem is formulated using the special theory of relativity by means of the Lorentz transformation applied to the classic Sommerfeld solution. Exact fields are obtained for the 3-D vector problem of scattering of an obliquely incident and arbitrarily polarized plane wave by a PEC H-P in relativistic translational motion. The total fields are then presented as the relativistic analog of their motionless counterparts and an association with the uniform asymptotic theory of diffraction framework is inferred. In addition to recovering known features such as the relativistic Doppler effect and shadow boundary shifts, it is herein depicted a polarization coupling effect due to motion that can give rise to a 3-D scattered field with all components present even for a linearly polarized incident wave. Validating results and illustrative examples are also presented.

Asymptotic Analysis of Plane Wave Scattering by a Fast Moving PEC Wedge

This contribution is concerned with the exact and asymptotic scattering of an oblique incident time-harmonic electromagnetic plane wave from a fast moving perfectly electric conducting wedge. By utilizing the Lorentz transformation and applying Maxwelln’s boundary conditions in the (scatterer) co-moving frame, an exact solution for the total fields is obtained in both the co-moving (scatterer) and the laboratory (incident field) frames. The fields are evaluated asymptotically in the high frequency regime in which the scattered field is presented as a sum of three wave types: the direct (reflected) wave, a shadowing wave, and a diffraction wave. Novel relativistic wave phenomena that are associated with the scatterer dynamics are explored.

Inverse electromagnetic diffraction by biperiodic dielectric gratings

Consider the incidence of a time-harmonic electromagnetic plane wave onto a biperiodic dielectric grating, where the surface is assumed to be a small and smooth perturbation of a plane. The diffraction is modeled as a transmission problem for Maxwell’s equations in three dimensions. This paper concerns the inverse diffraction problem which is to reconstruct the grating surface from either the diffracted field or the transmitted field. A novel approach is developed to solve the challenging nonlinear and ill-posed inverse problem. The method requires only a single incident field and is realized via the fast Fourier transform. Numerical results show that it is simple, fast, and stable to reconstruct biperiodic dielectric grating surfaces with super-resolved resolution.

Dynamic fracture behavior of a nanocrack in a piezoelectric plane

AbstractScattering of time harmonic plane waves by a finite blunt nano‐crack in a homogeneous transversely isotropic piezoelectric plane under plane strain conditions is studied. The mechanical model combines: (a) classical elastodynamic theory for the bulk piezoelectric solid, where the total wave comprises both incident and scattered wave field; (b) non‐classical boundary conditions and localized constitutive equation for the interface between the crack and piezoelectric matrix within the frame of the Gurtin‐Murdoch surface elasticity theory. The computational approach uses a non‐hypersingular traction based boundary integral equation method (BIEM) basing on the fundamental solution derived in closed form by Radon transforms. Furthermore, the parametric study is presented which demonstrates the sensitivity of the generalized stress concentration fields near the crack‐tip to the key parameters as the size of the crack, surface properties, coupled nature of the piezoelectricity phenomenon, type and properties of the incident wave such as frequency, wave length and wave propagation direction, dynamic interaction between the crack, incident wave and coupled electro‐elastic continuum.

Recovering an electromagnetic obstacle by a few phaseless backscattering measurements

We consider the electromagnetic scattering from a convex polyhedral PEC or PMC obstacle due to a time-harmonic incident plane wave. It is shown that the modulus of the far-field pattern in the backscattering aperture possesses a certain local maximum behavior. Using the local maximum indicating phenomena, one can determine the exterior unit normal directions, as well as the face areas, of the front faces of the obstacle. Then we propose a recovery scheme of reconstructing the obstacle by phaseless backscattering measurements. This work significantly extends our recent study in Li and Liu (2014 preprint) from two dimensions and acoustic scattering to the more challenging three dimensions and electromagnetic scattering.

Waves in Nonlocal Elastic Solid with Voids

In this paper, the governing relations and equations are derived for nonlocal elastic solid with voids. The propagation of time harmonic plane waves is investigated in an infinite nonlocal elastic solid material with voids. It has been found that three basic waves consisting of two sets of coupled longitudinal waves and one independent transverse wave may travel with distinct speeds. The sets of coupled waves are found to be dispersive, attenuating and influenced by the presence of voids and nonlocality parameters in the medium. The transverse wave is dispersive but non-attenuating, influenced by the nonlocality and independent of void parameters. Furthermore, the transverse wave is found to face critical frequency, while the coupled waves may face critical frequencies conditionally. Beyond each critical frequency, the respective wave is no more a propagating wave. Reflection phenomenon of an incident coupled longitudinal waves from stress-free boundary surface of a nonlocal elastic solid half-space with voids has also been studied. Using appropriate boundary conditions, the formulae for various reflection coefficients and their respective energy ratios are presented. For a particular model, the effects of non-locality and dissipation parameter ( $\tau $ ) have been depicted on phase speeds and attenuation coefficients of propagating waves. The effect of nonlocality on reflection coefficients has also been observed and shown graphically.

Physical (Geometrical) Analysis of Polarization Phenomena Arising in the Diffraction by Scatterers in Relativistic Uniform Motion

The scattering of time-harmonic plane waves by a perfectly electrically conducting half-plane in relativistic uniform motion has been previously shown to give rise to polarization effects when the line of motion is along an arbitrary direction. A physical interpretation for the observation of these polarization effects is herein presented, based on an analysis of the geometric properties of the scatterer under Lorentz transformations.

Numerically stable method for wave-field calculation and band-gap estimation in layered phononic crystals

High-frequency wave propagation in layered waveguides composed of periodic unit-cells is studied. Each unit-cell consists of a certain number of various elastic layers. Wave motion in the considered elastic structures (phononic crystals) is characterized by frequency bands, known as forbidden zones or band-gaps, where the full reflection of time-harmonic plane waves incident from an external half-space is observed. Using the transfer matrix method, the mathematical model of elastic wave propagation in anisotropic layered phononic crystals between two half-spaces is developed. A stable numerical algorithm for calculation of wave fields in the cells of phononic crystals is proposed. The amplitude coefficients of transmitted longitudinal and shear body waves are expressed in terms of transfer-matrix eigenvalues. A classification of band-gaps and pass-bands in layered anisotropic phononic crystals is proposed. It is based on the analysis of Bloch wavenumbers (expressed via the eigenvalues of the unit-cell transfer matrix) and the derived semi-analytical asymptotic expression of the transmission coefficients. Each Bloch wave has attenuation within the band-gaps of first kind. Waves propagating without attenuation are not excited within the band-gaps of second kind due to their specific polarization and boundary conditions at the interface between the multi-layered structure and the external half-space. By changing the parameters of the incident field, for instance, angles of incidence, band-gaps are converted into low transmission pass-bands where transmission coefficients are rather small. Therefore, a low transmission pass-band is similar to the traditional band-gap from an engineering point of view. The numerical results for energy transmission coefficient and Bloch waves are given at different angles of incidence; they demonstrate the formation of band-gaps in anisotropic phononic crystals.

A fast hybrid Galerkin method for high-frequency acoustic scattering

We consider the scattering of a time-harmonic acoustic incident plane wave by a smooth convex object. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. Based on the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary. To achieve a good approximation at high frequencies with a relative low number of degrees of freedom, we propose a novel Galerkin boundary element method with a hybrid approximation space, consisting of the products of plane wave basis functions with piecewise polynomials supported on several overlapping meshes: a polynomial grading on the illuminated side and a geometric grading on the shadow side. Using the asymptotic expansions of the solution, we prove that, as , the number of degree of freedom is able to decrease only very modestly to maintain a fixed absolute error bound ( is a typical behavior). Numerical experiments also show that the method achieves a better accuracy as , for a fixed number of degrees.

Inverse obstacle scattering for elastic waves

Consider the scattering of a time-harmonic plane wave by a rigid obstacle which is embedded in an open space filled with a homogeneous and isotropic elastic medium. An exact transparent boundary condition is introduced to reduce the scattering problem into a boundary value problem in a bounded domain. Given the incident field, the direct problem is to determine the displacement of the wave field from the known obstacle; the inverse problem is to determine the obstacle’s surface from the measurement of the displacement on an artificial boundary enclosing the obstacle. In this paper, we consider both the direct and inverse problems. The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is derived for the displacement with respect to the variation of the surface. A continuation method with respect to the frequency is developed for the inverse problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

The direct scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder

In this paper we consider the direct scattering problem of obliquely incident time-harmonic electromagnetic plane waves by an infinitely long dielectric cylinder. We assume that the cylinder and the outer medium are homogeneous and isotropic. From the symmetry of the problem, Maxwell's equations are reduced to a system of two 2D Helmholtz equations in the cylinder and two 2D Helmholtz equations in the exterior domain where the fields are coupled on the boundary. We prove uniqueness and existence of this differential system by formulating an equivalent system of integral equations using the direct method. We transform this system into a Fredholm type system of boundary integral equations in a Sobolev space setting. To handle the hypersingular operators we take advantage of Maue's formula. Applying a collocation method we derive an efficient numerical scheme and provide accurate numerical results using as test cases transmission problems corresponding to analytic fields derived from fundamental solutions.

Mathematical basis for a mixed inverse scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R2 as cross section. To this end, we solve a mixed scattering problem for the Helmholtz equation in R2 where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we set suitable boundary conditions on Γ and ∂D (∂D∈C2). The boundary integral method is employed to study the direct scattering problem. The mathematical basis is given to reconstruct the shape of the crack and the obstacle by using the linear sampling method. The numerical examples are given to show the viability of the method.

A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems

The present work is concerned with the shape reconstruction problem of isotropic elastic inclusions from far-field data obtained by the scattering of a finite number of time-harmonic incident plane waves. This paper aims at completing the theoretical framework which is necessary for the application of geometric optimization tools to the inverse transmission problem in elastodynamics. The forward problem is reduced to systems of boundary integral equations following the direct and indirect methods initially developed for solving acoustic transmission problems. We establish the Fréchet differentiability of the boundary to far-field operator and give a characterization of the first Fréchet derivative and its adjoint operator. Using these results we propose an inverse scattering algorithm based on the iteratively regularized Gauß–Newton method and show numerical experiments in the special case of star-shaped obstacles.