Time-harmonic Waves Research Articles

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.

Plane wave enriched Partition of Unity Isogeometric Analysis (PUIGA) for 2D-Helmholtz problems

In general, numerical concerns in simulation of time harmonic acoustic waves are mainly due to the dispersion errors and the demands in modeling the problems with higher number of degrees of freedom per wavelength to obtain acceptable results. As a remedy, plane wave enriched Partition of Unity Finite Element Method (PUFEM) was earlier proposed by Melenk and Babuška (1996). Even though, PUFEM gives accurate results with lesser degrees of freedom and shows exponential convergence, it suffers from the approximate representation of geometry and leads to the boundary representation induced errors. In this regard, plane wave enriched PUFEM under Isogeometric framework named Partition of Unity Isogeometric Analysis (PUIGA) is proposed for accurate geometry representation and smooth solution approximation which can lead to a better solution. Several benchmark problems are considered in the present study, to illustrate the computational effectiveness of PUIGA.

Electromagnetic energy within an isolated C60 molecule

By using the equation of motion for the dynamic response of the σ and π electrons in the carbon nanostructure, the classical Poynting theorem of conservation of electromagnetic energy is extended so as to be applicable to derive an exact expression for the time-averaged electromagnetic energy within an isolated C60 molecule, which is irradiated by a plane and time-harmonic electromagnetic wave. Numerical results show that the stored energy exhibits resonant enhancement near resonances of extinction cross section of the system.

Localization in semi-infinite herringbone waveguides.

The paper includes novel results for the scattering and localization of a time-harmonic flexural wave by a semi-infinite herringbone waveguide of rigid pins embedded within an elastic Kirchhoff plate. The analytical model takes into account the orientation and spacing of the constituent parts of the herringbone system, and incorporates dipole approximations for the case of closely spaced pins. Illustrative examples are provided, together with the predictive theoretical analysis of the localized waveforms.

Scattering of flexural waves in Euler–Bernoulli beams by short-range potentials

Time-harmonic flexural waves on a beam and on two elastically coupled beams with short-range localized imperfections in the mass distribution and in the position dependant coupling are considered. Thus scattering of an incident wave solution to the Euler–Bernoulli equation by a Dirac delta function and its derivative up to order three is studied, and the possible physical interpretations are outlined. Reflected, transmitted and evanescent waves exist, and their scattering data are determined. For and , the scattering problem is solved by standard integration. For and , the standard integration procedure does not work and solutions are obtained by regularization. In the latter case the scatterer is in general nontransparent and only partially penetrable at discrete resonances. The first few of these as well as their scattering data are determined numerically.

Kinetic-j: A computational kernel for solving the linearized Vlasov equation applied to calculations of the kinetic, configuration space plasma current for time harmonic wave electric fields

We present the Kinetic-j code, a computational kernel for evaluating the linearized Vlasov equation with application to calculating the kinetic plasma response (current) to an applied time harmonic wave electric field. This code addresses the need for a configuration space evaluation of the plasma current to enable kinetic full-wave solvers for waves in hot plasmas to move beyond the limitations of the traditional Fourier spectral methods. We benchmark the kernel via comparison with the standard k→-space forms of the hot plasma conductivity tensor. Program summaryProgram Title:Kinetic-jProgram Files doi:http://dx.doi.org/10.17632/w6znwd42z8.1Licensing provisions: MITProgramming language: C++ / IDL / CUDA / PythonNature of problem: Configuration space evaluation of the linear, kinetic plasma current.Solution method: Direct numerical evaluation of the linearized solution of the Vlasov equation, i.e., provided a time harmonic wave electric field as input, evaluate the unperturbed Lorentz orbit of each phase-space (position and velocity) point trajectory and integrate over its time history, then take the first velocity moment to calculate current.Additional comments: This initial version is 1x-3v, and accepts input fields in either Cartesian or cylindrical coordinates. The code is hosted at https://github.com/ORNL-Fusion/kineticj.

Direct and inverse elastic scattering from anisotropic media

Assume that a time-harmonic elastic wave is incident onto a penetrable anisotropic elastic body embedded in a homogeneous isotropic background medium. The scattering problem is reduced to a truncated domain. Uniqueness and existence of weak solutions are proved by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the boundary of the scatterer is derived. As an application, a descent algorithm is designed for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are presented to show the validity and accuracy of the algorithm.

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.

Preconditioning Methods for Thin Scattering Structures Based on Asymptotic Results

We present a method to precondition the discretized Lippmann--Schwinger integral equations to model scattering of time-harmonic acoustic waves through a thin inhomogeneous scattering medium. The preconditioner is based on asymptotic results as the thickness of the third component direction goes to zero and requires solving a two dimensional formulation of the problem at the preconditioning step.

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.

The Limiting Absorption Principle and a Radiation Condition for the Scattering by a Periodic Layer

Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These nonzero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply nonuniqueness of any solution to the scattering problem. In this paper, we consider a medium that is defined in the upper two-dimensional half-space by a penetrable and periodic contrast. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. Our method of proof seems to be new: By the Floquet--Bloch transform we first reduce the scattering problem to a finite-dimensional one that is set in the linear space spanned by all surface waves. In this space, we then compute explicitly which modes propagate along the periodic structure to the left or to the right. This finally yields a representation for our limiting absorption solution which leads to a proper extension of the well known upward propagating radiation condition. Finally, we prove the uniqueness of a solution under this radiation condition.

Non reflection and perfect reflection via Fano resonance in waveguides

We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter e and of the frequency λ is in general not continuous at a point $(e, λ) = (0,λ_0$) where trapped modes exist. In particular, we prove that for a given $e = 0$ small, the scattering matrix exhibits a rapid change for frequencies varying in a neighbourhood of $λ_0$. We use this property to construct examples of waveguides such that the energy of an incident wave propagating through the structure is perfectly transmitted (non reflection) or perfectly reflected in monomode regime. We provide numerical results to illustrate our theorems.

Perfectly Matched Layer Boundary Integral Equation Method for Wave Scattering in a Layered Medium

For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive, since they are formulated on lower-dimension boundaries or interfaces, and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on the Green's function of the background medium must evaluate the expensive Sommefeld integrals. Alternative BIE methods based on the free-space Green's function give rise to integral equations on unbounded interfaces which are not easy to truncate, since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. The PMLs are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space. Our PML-based BIE method uses the Green's function of the PML-transformed free space to define the boundary integral operators. The method is efficient, since the Green's function of the PML-transformed free space is easy to evaluate and the PMLs are very effective in truncating the unbounded interfaces. Numerical examples are presented to validate our method and demonstrate its accuracy.

Second-Kind Boundary Integral Equations for Scattering at Composite Partly Impenetrable Objects

We consider acoustic scattering of time-harmonic waves at objects composed of several homogeneous parts. Some of those may be impenetrable, giving rise to Dirichlet boundary conditions on their surfaces. We start from the second-kind boundary integral approach of [X. Claeys, and R. Hiptmair, and E. Spindler. A second-kind Galerkin boundary element method for scattering at composite objects. BIT Numerical Mathematics, 55(1):33-57, 2015] and extend it to this setting. Based on so-called global multi-potentials, we derive variational second-kind boundary integral equations posed in $L^2(\Sigma)$, where $\Sigma$ denotes the union of material interfaces. To suppress spurious resonances, we introduce a combined-field version (CFIE) of our new method. Thorough numerical tests highlight the low and mesh-independent condition numbers of Galerkin matrices obtained with discontinuous piecewise polynomial boundary element spaces. They also confirm competitive accuracy of the numerical solution in comparison with the widely used first-kind single-trace approach.

Improved sweeping preconditioners for domain decomposition algorithms applied to time-harmonic Helmholtz and Maxwell problems

Sweeping-type algorithms have recently gained a lot of interest for the solution of highfrequency time-harmonic wave problems, in particular when used in combination with perfectly matched layers. However, an inherent problem with sweeping approaches is the sequential nature of the process, which makes them inadequate for efficient implementation on parallel computers. We propose several improvements to the double-sweep preconditioner originally presented in [18], which uses sweeping as a matrix-free preconditioner for a Schwarz domain decomposition method. Similarly, the improved preconditioners are based on approximations of the inverse of the Schwarz iteration operator: the general methodology is to apply well-known algebraic techniques to the operator seen as a matrix, which in turn is processed to obtain equivalent matrix-free routines that we use as preconditioners. A notable feature of the new variants is the introduction of partial sweeps that can be performed concurrently in order to make a better usage of the resources. As these modifications still leave some unexploited computational power, we also propose to combine them with right-hand side pipelining to further improve parallelism and achieve significant speed-ups. Examples are presented on high-frequency Helmholtz and Maxwell problems, in two and three dimensions, to demonstrate the properties of our improvements on parallel computer architectures.

Looking Back on Inverse Scattering Theory

We present an essay on the mathematical development of inverse scattering theory for time-harmonic waves over the past fifty years together with some personal memories of our participation in these...

Modeling and simulation of an acoustic well stimulation method

This paper presents a mathematical model and a numerical procedure to simulate an acoustic well stimulation (AWS) method for enhancing the permeability of the rock formation surrounding oil and gas wells. The AWS method considered herein aims to exploit the well-known permeability-enhancing effect of mechanical vibrations in acoustically porous materials, by transmitting time-harmonic sound waves from a sound source device—placed inside the well—to the well perforations made into the formation. The efficiency of the AWS is assessed by quantifying the amount of acoustic energy transmitted from the source device to the rock formation in terms of the emission frequency and the well configuration. A simple methodology to find optimal emission frequencies for a given well configuration is presented. The proposed model is based on the Helmholtz equation, a sound-hard boundary condition at the casing, and an impedance boundary condition that effectively accounts for the porous solid–fluid interaction at the interface between the rock formation and the well perforations. Exact non-reflecting boundary conditions derived from Dirichlet-to-Neumann maps are utilized to truncate the circular cylindrical waveguides considered in the model. The resulting boundary value problem is then numerically solved by means of the finite element method. A variety of numerical examples are presented in order to demonstrate the effectiveness of the proposed procedure for finding optimal emission frequencies.

Recovering an elastic obstacle containing embedded objects by the acoustic far-field measurements

Consider the inverse scattering problem of time-harmonic acoustic waves by a 3D bounded elastic obstacle which may contain embedded impenetrable obstacles inside. We propose a novel and simple technique to show that the elastic obstacle can be uniquely recovered by the acoustic far-field pattern at a fixed frequency, disregarding its contents. Our method is based on constructing a well-posed modified interior transmission problem on a small domain and makes use of an a priori estimate for both the acoustic and elastic wave fields in the usual H1-norm. In the case when there is no obstacle embedded inside the elastic body, our method gives a much simpler proof for the uniqueness result obtained previously in the literature (Natroshvili et al 2000 Rend. Mat. Serie VII 20 57–92; Monk and Selgas 2009 Inverse Problems Imaging 3 173–98).

A meshfree method with plane waves for elastic wave propagation problems

In this paper, we address the meshfree numerical solution of time-harmonic linear elastic wave propagation problems in homogeneous media. In particular, we analyze the asymptotic behavior of the method of fundamental solutions (MFS) with source points located far away from the domain of interest. The asymptotic MFS is shown to be equivalent to a Trefftz method, here referred to as the plane waves method (PWM), based on superposition of shear and compressional elastic plane waves with different directions of propagation. Several numerical examples are included in order to illustrate the equivalence between the asymptotic MFS and the PWM. The convergence and stability of the PWM are also analyzed in smooth settings.

An Adaptive Finite Element PML Method for the Acoustic-Elastic Interaction in Three Dimensions

AbstractConsider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable, and isotropic elastic solid, which is immersed in a homogeneous compressible air or fluid. The paper concerns the numerical solution for such an acoustic-elastic interaction problem in three dimensions. An exact transparent boundary condition (TBC) is developed to reduce the 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 using a PML equivalent TBC. An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.