Time-harmonic Waves Research Articles

Locating a complex inhomogeneous medium with an approximate factorization method

Consider the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous medium with complex refractive index. We show that an approximate factorization method can be applied to reconstruct the support of the complex inhomogeneous medium from the far-field data. Numerical examples are also provided to illustrate the practicability of the inversion algorithm.

Direct error in constitutive equation formulation for inverse problems in time harmonic elastography

Shear wave imaging techniques allow the evaluation of rigidity and viscosity of tissues locally within a material. From an inverse problem perspective, the approach is quite attractive insofar as it provides a densely sampled displacement field in the interior of the object from which to invert for material properties. We consider several challenges related to elastic wave inverse problems arising in acoustic radiation force imaging. First, we validate an axisymmetric viscoelasticity model suitable for some applications of acoustic radiation force imaging. Second, we consider reconstructing lateral displacement components from measured axial displacement components. Finally, we present a new variational formulation, the direct error in constitutive equation formulation, for inverse problems in time harmonic viscoelastic wave propagation with full-field data. The formulation relies on minimizing the error in the constitutive equation with a momentum equation constraint. Numerical results on model problems show that the formulation is capable of handling discontinuous and noisy strain fields and also converging with mesh refinement for continuous and discontinuous material property distributions. Applications to MRE and ARFI measured wave data are considered.

Measuring the transmission coefficient of a steel plate using an angular spectrum approach

Ultrasonic measurements were performed on a thin steel plate submerged in water in order to obtain the plane-wave transmission coefficient as a function of angle and frequency. An angular spectrum approach was utilized wherein the acoustic field parallel to the plate was scanned with sufficient spatial accuracy to obtain a frequency-wavenumber description of the transmitted field, representing an expansion of the field into time-harmonic plane waves propagating within a range of angles. Isotropy of the plate material and symmetry of the measurement configuration allowed for a vast reduction of the number of scan points required in order to obtain the transmitted angular spectrum. Once the transmitted acoustic data is acquired, the specific method of windowing the data in space and time can have a drastic effect on the resulting angular spectrum. With sufficient care, a broadband estimation of the transmission coefficient with a high angular resolution is obtained. Finally, inverse methods were utilized in order to estimate the thickness and material properties of the plate using the measured transmission coefficient data. [Work supported by ONR.]

A multiscale continuous Galerkin method for stochastic simulation and robust design of photonic crystals

We present a multiscale continuous Galerkin (MSCG) method for the fast and accurate stochastic simulation and optimization of time-harmonic wave propagation through photonic crystals. The MSCG method exploits repeated patterns in the geometry to drastically decrease computational cost and incorporates the following ingredients: (1) a reference domain formulation that allows us to treat geometric variability resulting from manufacturing uncertainties; (2) a reduced basis approximation to solve the parametrized local subproblems; (3) a gradient computation of the objective function; and (4) a model and variance reduction technique that enables the accelerated computation of statistical outputs by exploiting the statistical correlation between the MSCG solution and the reduced basis approximation. The proposed method is thus well suited for both deterministic and stochastic simulations, as well as robust design of photonic crystals. We provide convergence and cost analysis of the MSCG method, as well as a simulation results for a waveguide T-splitter and a Z-bend to illustrate its advantages for stochastic simulation and robust design.

Optimal Control Theory Applied to Unintended Source Control and Field Shaping for Time-Harmonic Electromagnetic Waves

This paper addresses the experimental control of time-harmonic electromagnetic waves. The underlying optimal control problem aims to find a set of values to apply to control sources, located in the controlled region or on its boundary, allowing any electromagnetic field caused by an undesired noise source to be suppressed or replaced by any other field map. After a presentation of the control method, illustrated with a few numerical applications, two experimental setups are considered. The first setup aims to control the voltage within a mixed microstrip/coaxial transmission line. In the second setup, the electric field inside a cross-shaped waveguide network is controlled. For both setups, the working frequency is chosen in the ultra-high frequency range, and a comparison is made to the results given by the equivalent numerical models.

Inverse transmission scattering problem via a Dirichlet-to-Neumann map

In this paper, we consider an inverse scattering problem of a time-harmonic wave from a penetrable obstacle. A Dirichlet-to-Neumann map is defined to convert this problem into an exterior boundary value problem with an impedance-like boundary condition. The problem is then solved without the need of solving an interior problem which is inevitable in the usual way. This process reduces the size of the problem and thus enables an efficient treatment of this inverse scattering problem.

Plane waves in nonlocal thermoelastic solid with voids

This work is concerned with the propagation of time harmonic plane waves in an infinite nonlocal thermoelastic solid having void pores. Three sets of coupled dilatational waves and an independent transverse wave may travel with distinct speeds in the medium. All these waves are found to be dispersive in nature, but the coupled dilatational waves are attenuating, while transverse wave is nonattenuating. Coupled dilatational waves are found to be influenced by the presence of voids, thermal field and elastic nonlocal parameter. While the transverse wave is found to be influenced by the nonlocal parameter, but independent of void and thermal parameters. For a particular model, the effects of frequency, void parameters, thermal parameter and nonlocality have been studied numerically on the phase speeds, attenuation coefficients and specific losses of all the propagating waves. All the computed results obtained have been depicted graphically and explained.

Waveform of gravity and capillary-gravity waves over a bathymetry

The exact Floquet theory reveals that the waveforms (eigenfunctions) of linear eigenwaves over a variable bed have complex features that are reminiscent of nonlinear waveforms. This is in contrast to the long-standing view that linear time harmonic waves are of permanent sinusoidal form.

Boundary-integral representations for ship motions in regular waves

Alternative boundary-integral representations of linear potential flow around a ship that travels at a constant speed in regular (time-harmonic) waves in deep water are considered. In particular, the Neumann–Michell (NM) boundary-integral representation of steady flow around a ship traveling in calm water is extended to ship motions in regular waves. This extension is straightforward, but points to a complicated and unresolved fundamental issue that is essentially related to the formulation of a linear flow model. Indeed, the NM boundary-integral flow representation for ship motions in regular waves given in the study suggests that the classical Neumann–Kelvin (NK) flow model may be inadequate at a ship waterline. Alternative NM boundary-integral representations of time-harmonic ship waves that are consistent with the previously given NM representation of ship waves in calm water, and yield Fourier–Kochin representations of ship waves that are well suited for numerical computations, are given.

An edge-based smoothed finite element method for wave scattering by an obstacle in elastic media

Scattering of a time-harmonic plane elastic wave by a rigid obstacle embedded in an isotropic homogeneous elastic medium has wide applications in science and engineering. This paper presents a novel method to study elastic wave scattering problems satisfying the Navier equation and Helmholtz equations with coupled boundary obtained by Helmholtz decomposition. We derive first smoothed Galerkin weak forms of Navier equation and Helmholtz equations to create effective smoothed finite element method (S-FEM) models. On the top of a three-noded triangular mesh, an edge-based S-FEM (ES-FEM-T3) combining with the perfectly matched layer (PML) technique is then established for solving the elastic wave scattering in bounded domains with PML to eliminate wave reflections. Some numerical experiments demonstrate that ES-FEM-T3 is more stable and accurate than the standard FEM for the elastic wave scattering.

Quadratic quantities in acoustics: Scattering cross-section and radiation force

Time-harmonic acoustic waves interact with a scatterer: this interaction is calculated using linear, first-order theory. However, there are quadratic, second-order quantities that are of interest. These include the scattering cross-section and the steady radiation force; these quantities can be expressed as integrals of products of first-order quantities over a sphere. These integrals are evaluated exactly. The results are infinite series of products of the coefficients in the spherical multipole expansions of the incident and scattered fields; they do not depend on the radius of the spherical integration surface. For a specific scattering problem, the coefficients can be connected by an appropriate T-matrix. In most previous work, the spherical surface is moved to infinity so that far-field quantities can be introduced: it is shown that this process is not straightforward and it may introduce spurious difficulties.

The outgoing time-harmonic electromagnetic wave in a half-space with non-absorbing impedance boundary condition

We show existence and uniqueness of the outgoing solution for the Maxwell problem with an impedance boundary condition of Leontovitch type in a half-space. Due to the presence of surface waves guided by an infinite surface, the established radiation condition differs from the classical one when approaching the boundary of the half-space. This specific radiation pattern is derived from an accurate asymptotic analysis of the Green’s dyad associated to this problem.

Near-Field Imaging of an Unbounded Elastic Rough Surface with a Direct Imaging Method

This paper is concerned with the inverse scattering problem of time-harmonic elastic waves by an unbounded rigid rough surface. A direct imaging method is developed to reconstruct the unbounded rou...

Inverse Elastic Scattering for a Random Source

Consider the inverse random source scattering problem for the two-dimensional time-harmonic elastic wave equation with an inhomogeneous, anisotropic mass density. The source is modeled as a microlocally isotropic generalized Gaussian random function whose covariance operator is a classical pseudo-differential operator. The goal is to recover the principle symbol of the covariance operator from the displacement measured in a domain away from the source. For such a distributional source, we show that the direct problem has a unique solution by introducing an equivalent Lippmann--Schwinger integral equation. For the inverse problem, we demonstrate that, with probability one, the principle symbol of the covariance operator can be uniquely determined by the amplitude of the displacement averaged over the frequency band, generated by a single realization of the random source. The analysis employs the Born approximation, asymptotic expansions of the Green tensor, and microlocal analysis of the Fourier integral operators.

Characterization of the acoustic fields scattered by a cluster of small holes

We deal with the time-harmonic acoustic waves scattered by a large number of small holes, with radius a , a ≪ 1 , arbitrarily distributed in a bounded part of the homogeneous background R 3 . We assume no periodicity in distributing these holes. Usin

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.

A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

Solving time-harmonic wave propagation problems by iterative methods is a difficult task, and over the last two decades an important research effort has gone into developing preconditioners for the simplest representative of such wave propagation problems, the Helmholtz equation. A specific class of these new preconditioners is considered here. They were developed by researchers with various backgrounds using formulations and notations that are very different, and all are among the most promising preconditioners for the Helmholtz equation. \indent The goal of the present article is to show that this class of preconditioners is based on a common mathematical principle, and that they can all be formulated in the context of domain decomposition methods known as optimized Schwarz methods. This common formulation allows us to explain in detail how and why all these methods work. The domain decomposition formulation also allows us to avoid technicalities in the implementation description we give of these recent methods. \indent The equivalence of these methods with optimized Schwarz methods translates at the discrete level into equivalence with approximate block LU decomposition preconditioners, and in each case we give the algebraic version, including a detailed description of the approximations used. While we choose to use the Helmholtz equation for which these methods were developed, our notation is completely general and the algorithms we give are written for an arbitrary second-order elliptic operator. The algebraic versions are even more general, assuming only a connectivity pattern in the discretization matrix. \indent All the new methods studied here are based on sequential decomposition of the problem in space into a sequence of subproblems, and they have in their optimal form the property to lead to nilpotent iterations, like an exact block LU factorization. Using our domain decomposition formulation, we finally present an algorithm for two-dimensional decompositions, i.e., decompositions that contain cross points, which is still nilpotent in its optimal form. Its approximation is currently an active area of research, and it would have been difficult to discover such an algorithm without the domain decomposition framework.

Survey and Review

The Survey and Review article in this issue is “A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods,” by Martin J. Gander and Hui Zhang. The Helmholtz equation appears in the study of time-harmonic waves \(u(x) \exp(-iømega t)\) arising in acoustics, electromagnetism, and other fields. In its simplest form the equation reads \((\Delta+k^2) u = f\), where (\Delta\) denotes the Laplacian operator and the constant \(k\) is the wavenumber, and therefore is reminiscent of the Poisson equation (\Delta u =f\). However, this similarity is only superficial, and the numerical solution of boundary value problems, often posed in unbounded domains, involving the Helmholtz equation is challenging. Classical iterative solvers are not suitable, and the recent literature has devoted much effort to the development of novel solvers. New iterative methods have been suggested by authors with a wide range of backgrounds and motivations and invoke different discretization strategies; some of them have been formulated at the linear algebra level and others at the level of the differential equation. Gander and Zhang survey the new developments using a unifying point of view. The main insight is given in section 2: the forward and backward substitution sweeps of the block LU factorization of the discrete system correspond at the continuous level to sweeps from left to right and right to left of an optimal Schwarz domain decomposition method. The article will be useful not only to those working directly with the Helmholtz and related partial differential equations, but also to anybody interested in iterative solvers for large linear systems.

A Framework for Simulation of Multiple Elastic Scattering in Two Dimensions

Consider the elastic scattering of a time-harmonic wave by multiple well separated rigid particles in two dimensions. To avoid using the complex Green's tensor of the elastic wave equation, we utilize the Helmholtz decomposition to convert the boundary value problem of the elastic wave equation into a coupled boundary value problem of Helmholtz equations. Based on single, double, and combined layer potentials with the simpler Green's function of the Helmholtz equation, we present three different boundary integral equations for the coupled boundary value problem. The well-posedness of the new integral equations are established. Computationally, a scattering matrix based method is proposed to evaluate the elastic wave for arbitrarily shaped particles. The method uses the local expansion for the incident wave and the multipole expansion for the scattered wave. The linear system of algebraic equations is solved by GMRES with fast multipole method (FMM) acceleration. Numerical results show that the method is fast and highly accurate for solving the elastic scattering problem with multiple particles.

Anisotropic adaptivity of the p-FEM for time-harmonic acoustic wave propagation

This paper deals with the a-priori assignment of polynomial order in the p-version of the FEM for the efficient simulation of time-harmonic acoustics problems. Anisotropic p-refinement, allowing direction-dependent polynomial approximations is examined, including for unstructured meshes with curved elements. A methodology to automatically choose the order repartition in the model is proposed and verified. These features are important for two main reasons. First, they allow to better control the accuracy on distorted elements with large aspect ratios. This in turn makes the numerical model less sensitive to the quality of the finite element mesh. Secondly, this allows to deal efficiently with problems where the wave properties are anisotropic. The restriction on the numerical resolution can be relaxed to obtain significant reductions in the computational cost compared to classical, isotropic order adaptivity. The paper presents several examples of the efficiency and robustness of the method for the propagation of acoustic waves on distorted meshes and/or in the presence of strong background mean flows.