Time-harmonic Waves Research Articles

Plane harmonic waves in a thermoelastic solid with double porosity

Propagation of time harmonic plane waves in an infinite thermoelastic solid medium with double porosity is studied in this paper. It is found that there may exist five basic waves consisting of four sets of coupled longitudinal waves and an independent transverse wave traveling at different speeds. Each set of coupled longitudinal waves is found to be dispersive, attenuating and depends upon the presence of both types of voids and thermal properties in the medium. The lone transverse wave is found to be non-dispersive and non-attenuating, and does not depend on the presence of voids and thermal properties, and travels with the speed of the shear wave of classical elasticity. Two sets of the coupled longitudinal waves face critical frequencies, below which these waves do not propagate in the medium. These critical frequencies depend upon the presence of voids and thermal parameters of the medium. The reflection phenomenon of a set of incident coupled longitudinal wave from the stress-free and thermally insulated boundary surface of a half-space has been studied. For a particular model, the phase speeds, attenuation coefficients, amplitude ratios and energy ratios have been computed numerically, presented graphically and discussed.

A boundary knot method for 3D time harmonic elastic wave problems

This paper makes the first attempt to apply the boundary knot method (BKM) to 3D time harmonic elastic wave problems. Unlike the famous method of fundamental solutions (MFS), the BKM employs non-singular general solutions as the basis functions to approximate the solutions. Therefore, the BKM overcomes the auxiliary boundary issue of the MFS, and inherits the advantages of the MFS, i.e., mathematically simple, easy-to-program, meshless, integration-free and accurate. Particularly, the non-singular general solutions of 3D time harmonic elastic wave problems for the BKM are firstly derived in this study. Two examples are carried out to verify the accuracy and convergence of the BKM.

Natural Domain Decomposition Algorithms for the Solution of Time-Harmonic Elastic Waves

We study for the first time Schwarz domain decomposition methods for the solution of the Navier equations modeling the propagation of elastic waves. These equations in the time-harmonic regime are difficult to solve by iterative methods, even more so than the Helmholtz equation. We first prove that the classical Schwarz method is not convergent when applied to the Navier equations and can thus not be used as an iterative solver, only as a preconditioner for a Krylov method. We then introduce more natural transmission conditions between the subdomains and show that if the overlap is not too small, this new Schwarz method is convergent. We illustrate our results with numerical experiments, both for situations covered by our technical two subdomain analysis and situations that go far beyond, including many subdomains, cross points, heterogeneous materials in a transmission problem, and Krylov acceleration. Our numerical results show that the Schwarz method with adapted transmission conditions leads systematically to a better solver for the Navier equations than the classical Schwarz method.

Inverse Scattering by a Random Periodic Structure

This paper develops an efficient numerical method for the inverse scattering problem of a time-harmonic plane wave incident on a perfectly reflecting random periodic structure. The method is based on a novel combination of the Monte Carlo technique for sampling the probability space, a continuation method with respect to the wavenumber, and the Karhunen--Loève expansion of the random structure, which reconstructs key statistical properties of the profile for the unknown random periodic structure from boundary measurements of the scattered fields away from the structure. Numerical results are presented to demonstrate the reliability and efficiency of the proposed method.

An Adaptive Finite Element DtN Method for the Open Cavity Scattering Problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper concerns the numerical solutions of the open cavity scattering problems in both transverse magnetic and transverse electric polarizations. Based on the Dirichlet-to-Neumann (DtN) map for each polarization, a transparent boundary condition is imposed to reduce the scattering problem equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed. The estimate consists of the finite element approximation error and the truncation error of the DtN operator, which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented for both polarizations to illustrate the competitive behavior of the adaptive method.

Concentration of dynamic stresses in an elastic space with twoperiodic array of elliptical cracks

Normal incidence of the plane time-harmonic longitudinal wave on double-periodic array of coplanar elliptical cracks, which are located in 3D infinite elastic space is considered. Corresponding symmetric wave scattering problem is reduced to a boundary integral equation for the displacement jump across the crack surfaces in a unit cell by means of periodic Green’s function, which is presented in the form of Fourier integrals. A regularization technique for this Green’s function involving special lattice sums in closed forms is adopted, which allows its effective calculation in a wide range of wave numbers. The boundary integral equation is correctly solved by using the mapping method. The frequency dependencies of mode-I stress intensity factor in the vicinity of the crack front points for periodic distances in the system of elliptical cracks are revealed.

Determining a Multi-layered Fluid-solid Medium from the Acoustic Measurements

Consider the inverse problem of recovering a multi-layered fluid-solid medium from many acoustic measurements corresponding to time-harmonic acoustic plane waves. We prove that both the supports of the embedded solid obstacle and the surrounding layered fluid medium can be uniquely identified by means of acoustic far-field pattern for all incident wave fields at a fixed frequency. Our proof is based on the constructions of some well-posed partial differential equation systems in sufficiently small domains combined with the a priori estimates for the solutions of the forward scattering problem.

Relation between crack density and acoustic nonlinearity in thermally damaged sandstone

This paper investigates thermal damage (microcracking) in sandstone using ultrasonic methods. Previous investigations have demonstrated that microcracks and lightly compressed interfaces in solids can generate strong higher harmonic signals in a propagating time-harmonic wave. Distributed cracks in rock will generate such nonlinearity and thus measurements of the acoustic nonlinearity can be a useful experimental means to quantify microcracking damage. In fact, experimental results have shown that the acoustic nonlinearity increases with the progress of microcracking in different rocks and concrete. This research attempts to experimentally determine the relation between the crack density and the acoustic nonlinearity in thermally damaged sandstone. The dynamic Young's modulus and longitudinal wave velocity are measured in sandstone specimens that are exposed to high temperatures, from which the crack density is calculated using a micromechanics model. A recently proposed standing wave based ultrasonic technique is employed to measure the acoustic nonlinearity parameters in these specimens. It is shown that the relation between the crack density and the acoustic nonlinearity are approximately linear. This result indicates that the acoustic nonlinearity measurement can be useful for evaluating thermal damage in rock materials.

Stress intensity factors of a Griffith crack in a porous medium subjected to a time-harmonic stress wave

In many practices, cracks are filled with fluid and their background materials can be modelled within the framework of poroelasticity. Understanding the crack-tip mechanical behaviors of such a cracked medium subjected to dynamic loads is of importance for prediction of fracture failure. In this paper, based on Biot’s theory of poroelastodynamics, solutions of mode-I and mode-II dynamic stress intensity factors (SIFs) of a Griffith crack are derived for the scattering problem of oblique incidence of in-plane time-harmonic P-SV waves. The crack surfaces are assumed to be permeable, and the crack is saturated by fluid. With the aid of Fourier integral transforms, the problem is reduced to a system of coupled Fredholm integral equations with special emphasis placed on finding the near-field solution. Impacts of incident wave type, material permeability and incident angle on the frequency-dependent behaviors of the SIFs have been shown graphically. It is found that the induced fluid pressure inside the crack can significantly change the effective normal stress so that the magnitude of mode-I SIF is much lower than that of a dry impermeable crack. Furthermore, the diffusion-type fluid flow (scattered slow P wave) can greatly lower the magnitude of mode-I SIF at low frequencies (much smaller than the resonance frequency). The characteristic frequency at which the curves of SIFs decline strongest occurs when the characteristic diffusion length is of the same order of the crack size. The fluid flow can also lower mode-II SIF at low frequencies and the effects are strong in the low permeability case. Specifically, in the case of scattering of SV waves, it is shown that the maximum drop of mode-II SIF in peak with respect to that of the corresponding dry crack in an elastic medium is about 10%. The obtained results reveal significant influences of the presence of pore fluid upon the SIFs, both magnitudes and frequency-dependent trends. Such information is useful in predicting the fracture strength of saturated porous materials subjected to oscillating loads.

Discrete scattering by a pair of parallel defects.

Scattering of a time-harmonic anti-plane shear wave due to either a pair of crack tips or a pair of rigid constraint tips on square lattice is considered. The two problems correspond to the so-called zero-offset case of scattering due to a pair of identical Sommerfeld screens. The peculiar structural symmetry allows the reduction of coupled equations to two scalar Wiener-Hopf equations and a total of four geometrically reduced problems on lattice half-plane. Exact solution of each problem for incidence from the bulk lattice, as well as from an associated lattice waveguide, is constructed. A suitable superposition of the four expressions is used to construct the solution of the main problem. The discrete paradigm involving the wave mode incident from the waveguide is relevant for modern applications where an investigation of mechanisms of electronic and thermal transport at nanoscale remains an interesting problem. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.

A numerical study of multi-parameter full waveform inversion with iterative regularization using multi-frequency vibroseis data

We study the inverse boundary value problem for time-harmonic elastic waves, for the recovery of P- and S-wave speeds from vibroseis data or the Neumann-to-Dirichlet map. Our study is based on our recent result pertaining to the uniqueness and a conditional Lipschitz stability estimate for parametrizations on unstructured tetrahedral meshes of this inverse boundary value problem. With the conditional Lipschitz stability estimate, we design a procedure for full waveform inversion (FWI) with iterative regularization. The iterative regularization is implemented by projecting gradients, after scaling, onto subspaces associated with the mentioned parametrizations yielding Lipschitz stability. The procedure is illustrated in computational experiments using the continuous Galerkin finite element method of recovering the rough shapes and wave speeds of geological bodies from simple starting models, near and far from the boundary, that is, the free surface.

Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures

Consider the scattering of a time-harmonic elastic plane wave by a periodic rigid surface. The elastic wave propagation is governed by the two-dimensional Navier equation. Based on a Dirichlet-to-Neumann (DtN) map, a transparent boundary condition (TBC) is introduced to reduce the scattering problem into a boundary value problem in a bounded domain. By using the finite element method, the discrete problem is considered, where the TBC is replaced by the truncated DtN map. A new duality argument is developed to derive the a posteriori error estimate, which contains both the finite element approximation error and the DtN truncation error. An a posteriori error estimate based adaptive finite element algorithm is developed to solve the elastic surface scattering problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Revisiting terrain-following Boussinesq equations on a highly variable periodic bed

On re-deriving the Boussinesq-type equations in a terrain-following coordinate system based on the conformal map for an arbitrarily (submerged) periodic bed, it is pointed out that some aspects of the existing approach need to be clarified, specifically the use of velocity potential gradient in the mapped plane in connection with the transformation of velocity vector, and the appropriate frequency–wavenumber relation that is from the solution to the linearized Boussinesq equations. It is shown that as the bed undulation height increases, this relationship increasingly departs from the flat-bottom dispersion relation that has previously been assumed for the terrain-following Boussinesq systems. Over a highly variable periodic bed, the waveforms of linear time harmonic waves already have features reminiscent of nonlinear waveforms, which should be distinguished from the subsequent nonlinear evolution.

Multiscale Finite Element Method for scattering problem in heterogeneous domain

In this paper, we consider the wave scattering problem in heterogeneous domain. The mathematical model is described by the Helmholtz equation for time-harmonic wave propagation with absorbing boundary condition. We construct a coarse grid approximation for the efficient numerical solution on coarse grids using the Multiscale Finite Element Method (MsFEM), where we construct a multiscale space using solution of some local problems in each local domain. Numerical results are presented to illustrate the performance of the method.

Direct and inverse electromagnetic scattering problems for bi-anisotropic media**This paper is dedicated to the memory of Professor Armin Lechleiter.

This paper is concerned with the direct and inverse scattering of time-harmonic electromagnetic waves from bi-anisotropic media. For the direct problem, we establish an integro-differential equation formulation, its Fredholm property, and the uniqueness of a weak solution. Using this integro-differential formulation we study a fast spectral Galerkin method for the numerical solution to the direct problem. Numerical examples are presented and convergence of the spectral method is proved via Gårding estimates for (strongly singular) integro-differential equations. We solve the inverse problem of recovering bi-anisotropic scatterers from far-field data using orthogonality sampling methods. These methods aim to construct imaging functionals which are robust to noise, computationally cheap, and require data for only one or a few incident fields. We provide some theoretical analysis as well as numerical simulations for the proposed imaging functionals.

A quasi-optimal non-overlapping domain decomposition method for two-dimensional time-harmonic elastic wave problems

This article presents the construction of a new non-overlapping domain decomposition method (DDM) for two-dimensional elastic scattering problems. The method relies on a high-order Transmission Boundary Condition (TBC) between sub-domains, which accurately approximates the exact Dirichlet-to-Neumann map. First, we explain the derivation of this new TBC in the context of a non-overlapping DDM. Next, a mode-by-mode convergence study for a model problem is presented, which shows the new method to be quasi-optimal, i.e. with an optimal convergence rate for evanescent modes and an improved convergence rate for the other modes compared to the standard low-order Lysmer-Kuhlemeyer TBC. Finally, the effectiveness of the new DDM is demonstrated in a finite element context by analyzing the behavior of the method on high-frequency elastodynamic simulations.

Propagating waves in elastic material with voids subjected to electro-magnetic interactions

Constitutive relations and field equations are developed for an elastic solid with voids subjected to electro-magnetic field. The linearized form of the relations and equations are presented separately when medium is subjected to a large magnetic field and when it is subjected to a large electric field. The possibility of propagation of time harmonic plane waves in an infinite elastic solid with voids has been explored. It is found that when the medium is subjected to large magnetic field, there exist two coupled longitudinal waves propagating with distinct speeds and a transverse wave mode. However, when the medium is subjected to a large electric field, there may propagate five basic waves comprising of four coupled longitudinal waves propagating with distinct speeds and a lone transverse wave. The effects of magnetic and electric fields are observed on the propagation characteristics of the existing waves. Under the limiting cases of frequency and for different electric conductive materials, the speeds of various waves are investigated. The phase speeds of different waves and their corresponding attenuations have been computed against the frequency parameter and depicted graphically for a specific material.

A near-field inverse scattering problem for an elastic ellipsoid

The scattering problem of time-harmonic elastic waves from a triaxial rigid ellipsoidal obstacle is considered. The direct scattering problem is formulated. A method using near-field data for solving the corresponding inverse scattering problem in order to specify the size and the orientation of the ellipsoid is described. A matrix with elements given in terms of measurements of the leading low-frequency coefficient of the scattered field is constructed. The eigenvalues of this measurement matrix provide information for the semi-axes of the ellipsoid whereas the eigenvectors together with a rotation matrix whose elements are given in terms of the Euler angles are used to specify the orientation of the ellipsoid. Corresponding results for geometrically degenerate cases of the ellipsoid such as spheroids, spheres, needles and disks are obtained for appropriate values of the geometrical parameters.

A priori estimates of attraction basins for nonlinear least squares, with application to Helmholtz seismic inverse problem

In this paper, we provide an a priori optimizability analysis of nonlinear least squares problems that are solved by local optimization algorithms. We define attraction (convergence) basins where the misfit functional is guaranteed to have only one local—and hence global—stationary point, provided the data error is below some tolerable error level. We use geometry in the data space (strictly quasiconvex sets) in order to compute the size of the attraction basin (in the parameter space) and the associated tolerable error level (in the data space). These estimates are defined a priori, i.e. they do not involve any least squares minimization problems, and only depend on the forward map. This methodology is applied to the comparison of the optimizability properties of two methods for the seismic inverse problem for a time-harmonic wave equation: the full waveform inversion (FWI) and its migration-based travel time (MBTT) reformulation. Computing the size of the attraction basins for the two approaches allows us to quantify the benefits of the latter, which can alleviate the requirement of low-frequency data for reconstruction of the background velocity model.

Acoustic excitation and detection of guided waves in a submerged aluminum plate using an angular spectrum approach

An angular spectrum approach can be used to obtain high-resolution measurements of the acoustic plane wave reflection and transmission coefficients corresponding to a given planar sample of material. Such an approach involves exciting the sample with a compact broadband source located in the near field and then scanning the resulting reflected or transmitted field on a plane parallel to the sample surface. Fourier transforms in both space and time then yield a description of the acoustic field in terms of its angular spectrum, or expansion into time-harmonic plane waves propagating throughout a range of frequencies and angles. To demonstrate this method, measurements made on a thick aluminum plate submerged in water are shown. In this case the reflection and transmission coefficients are intricately structured, with individual features corresponding to different symmetric and antisymmetric guided wave modes propagating within the plate. The coincidence effect, by which the plate is nearly transparent acoustically at just those frequency-angle pairs that excite guided waves, is clearly seen in the measurements. Measurements made on other materials, as well as possible extensions to this method, will also be discussed. [Research supported by ONR.]