Time-harmonic Waves Research Articles

Convergence of the PML method for elastic wave scattering problems

In this paper we study the convergence of the perfectly matched layer (PML) method for solving the time harmonic elastic wave scattering problems. We introduce a simple condition on the PML complex coordinate stretching function to guarantee the ellipticity of the PML operator. We also introduce a new boundary condition at the outer boundary of the PML layer which allows us to extend the reflection argument of Bramble and Pasciak to prove the stability of the PML problem in the truncated domain. The exponential convergence of the PML method in terms of the thickness of the PML layer and the strength of PML medium property is proved. Numerical results are included.

GetDDM: An open framework for testing optimized Schwarz methods for time-harmonic wave problems

We present an open finite element framework, called GetDDM, for testing optimized Schwarz domain decomposition techniques for time-harmonic wave problems. After a review of Schwarz domain decomposition methods and associated transmission conditions, we discuss the implementation, based on the open source software GetDP and Gmsh. The solver, along with ready-to-use examples for Helmholtz and Maxwell’s equations, is freely available online for further testing. Program summaryProgram title: GetDDMCatalogue identifier: AEZZ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZZ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU GPL v2No. of lines in distributed program, including test data, etc.: 1426800No. of bytes in distributed program, including test data, etc.: 12362781Distribution format: tar.gzProgramming language: Gmsh (http://gmsh.info) and GetDP (http://getdp.info).Computer: PC, Mac, Tablets, Computer clusters.Operating system: Linux, Windows, MacOSX.Has the code been vectorized or parallelized?: YesRAM: From 512 Megabytes upwardsClassification: 4.3, 4.12, 6.5, 10.Nature of problem: Computing the solution of large scale time-harmonic acoustic and electromagnetic wave problems.Solution method: Finite element method with optimized Schwarz domain decomposition method.Running time: From a few seconds for simple problems to several days for large-scale simulations.

Domain Decomposition Methods for Time-Harmonic Electromagnetic Waves With High-Order Whitney Forms

Classically, the domain decomposition methods (DDMs) for time-harmonic electromagnetic wave propagation problems make use of the standard, low-order, Nedelec basis functions. This paper analyzes the convergence rate of DDM when higher order finite elements are used for both volume and interface discretizations, in particular when different orders are used in the volume and on the interfaces.

Adaption for 2-D Edge Elements in the Nonconforming Voxel Finite-Element Method

The nonconforming voxel finite-element (FE) method avoids many of the difficulties of traditional FE mesh generation using a nested grid of rectangular elements. It models arbitrary boundary shapes by adaptively refining the mesh at the boundaries. Here, the adaption is extended to reduce errors arising from elements that are away from boundaries, but are too big to represent the field adequately. The application is to 2-D edge elements for solving the time-harmonic wave equation. Results for a miter-bend waveguide and a dielectric post resonator demonstrate the effectiveness of the new algorithm.

Constructive Formulations of Resonant Maxwell's Equations

In this article we propose new constructive weak formulations for resonant time-harmonic wave equations with singular solutions. Our approach follows the limiting absorption principle and combines standard weak formulations of PDEs with properties of elementary special functions adapted to the singularity of the solutions, called manufactured solutions. We show the well-posedness of several formulations obtained by these means for the limit problem in dimension one, and propose a generalization in dimension two.

Near-field imaging of periodic interfaces in multilayered media

This paper is concerned with the inverse problem of scattering of time-harmonic electromagnetic waves by a penetrable multilayered periodic structure. The structure separates the whole space into three regions: the medium above and below the structure is assumed to be homogeneous but with different wave numbers, and the medium inside the structure is assumed to be inhomogeneous characterized by the refractive index. We prove, under certain conditions, that the factorization method can be used to reconstruct the upper interface from the scattered near-field data measured only above the structure and generated by a countably infinite number of downward propagating incident waves merely from the top region, leading to a fast imaging algorithm. A similar result can be obtained for reconstructing the lower interface from the scattered near-field data measured only below the structure, corresponding to a countably infinite number of upward propagating incident waves also merely from the bottom region. Thus, our approach is applicable, even if one merely has access to the object under investigation from one side. Finally, numerical examples are presented to illustrate the effectiveness of the inversion algorithm.

An efficient high-order Nyström scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

This text proposes a fast, rapidly convergent Nystr\"{o}m method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper.

Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves

Consider a time-harmonic acoustic wave incident onto a doubly periodic (biperiodic) interface from a homogeneous compressible inviscid fluid. The region below the interface is supposed to be an isotropic linearly elastic solid. This paper is concerned with the inverse fluid-solid interaction (FSI) problem of recovering the unbounded periodic interface separating the fluid and solid. We provide a theoretical justification of the factorization method for precisely characterizing the region occupied by the elastic solid by utilizing the scattered acoustic waves measured in the fluid. A computational criterion and a uniqueness result are presented with infinitely many incident acoustic waves having common quasiperiodicity parameters. Numerical examples in 2D are demonstrated to show the validity and accuracy of the inversion algorithm.

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.

Modeling and simulation of wave scattering of multiple inhomogeneities in composite media

This article presents an analytical–numerical method for the multiple scattering problem of elastic composite media with interacting inhomogeneities under time-harmonic antiplane elastic incident waves. The main focus is on the detailed evaluation of the effectiveness and accuracy of the method in the determination of the local dynamic behaviour of such composite media with significant numbers of inhomogeneities. The method is based on the eigenfunction expansion (Fourier expansion) of single inhomogeneity problem and the use of a pseudo-incident wave technique, which allows the accurate determination of local stress field caused by the interaction. The accuracy and effectiveness of the method for dealing with multiple interaction problems are discussed in detail. Illustrative examples under different loading and geometric conditions are considered to study the local dynamic field, the effective properties of the composite media based on self-consistent material modelling, and the behaviour of stop-band of wave propagating for periodic inhomogeneity arrangements.

Asymptotic analysis for time harmonic wave problems with small wave number

We study the asymptotic behavior of the solution to some time harmonic wave problems when the wave number is taken as a small asymptotic parameter. Our basic strategy is to introduce suitable Lagrangian multipliers into the governing equations, and transforming them into saddle point problems. These saddle point problems are uniformly invertible with respect to the wave number k∈[0,k0], with k0 being an arbitrary but fixed positive number. The asymptotic expansion is then derived by the standard regular perturbation technique.

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.

Dynamic Stress Intensity Factors of Three Collinear Cracks in an Orthotropic Plate Subjected to Time-Harmonic Disturbance

AbstractAbstract-The dynamic stresses around three collinear cracks in an infinite orthotropic plate are solved. In this configuration, two equal cracks are situated symmetrically on either side of a central crack, and time-harmonic elastic waves impinge perpendicularly to the cracks. The problem is solved by superimposing two types of solutions. One solution is that for a crack in an infinite orthotropic plate, and the other is for two collinear cracks. The unknown coefficients in the superimposed solution are determined by applying the boundary conditions at the surfaces of the three cracks using the Schmidt method. The dynamic stress intensity factors are calculated numerically for an orthotropic plate that corresponds to the elastic properties of a boron-epoxy composite.

A weak Galerkin method for diffraction gratings

This paper concerns a weak Galerkin method (WGM) for the diffraction of a time-harmonic incident wave impinging upon a one-dimensional periodic grating structure. The existence and uniqueness of the weak Galerkin solution to the grating problem are established using a variational approach. The convergence rate of the proposed WGM is systematically analyzed. Numerical simulations are presented to verify the efficiency of the WGM for solving grating problems.

Application of scattering theory to P-wave amplitude fluctuations in the crust

The amplitudes of high-frequency seismic waves generated by local and/or regional earthquakes vary from site to site, even at similar hypocentral distances. It had been suggested that, in addition to local site effects (e.g., variable attenuation and amplification in surficial layers), complex wave propagation in inhomogeneous crustal media is responsible for this observation. To quantitatively investigate this effect, we performed observational, theoretical, and numerical studies on the characteristics of seismic amplitude fluctuations in inhomogeneous crust. Our observations of P-wave amplitude for small to moderately sized crustal earthquakes revealed that fluctuations in P-wave amplitude increase with increasing frequency and hypocentral distance, with large fluctuations showing up to ten-times difference between the largest and the smallest P-wave amplitudes. Based on our theoretical investigation, we developed an equation to evaluate the amplitude fluctuations of time-harmonic waves that radiated isotropically from a point source and propagated spherically in acoustic von Karman-type random media. Our equation predicted relationships between amplitude fluctuations and observational parameters (e.g., wave frequency and hypocentral distance). Our numerical investigation, which was based on the finite difference method, enabled us to investigate the characteristics of wave propagation in both acoustic and elastic random inhomogeneous media using a variety of source time functions. The numerical simulations indicate that amplitude fluctuation characteristics differ a little between medium types (i.e., acoustic or elastic) or source time function durations. These results confirm the applicability of our analytical equation to practical seismic data analysis.

Optimized Schwarz algorithms for solving time-harmonic Maxwell’s equations discretized by a hybridizable discontinuous Galerkin method

This work is concerned with the development of numerical methods for the simulation of time-harmonic electromagnetic wave propagation problems. A hybridizable discontinuous Galerkin (HDG) method is adopted for the discretization of the two-dimensional time-harmonic Maxwell’s equations on a triangular mesh. A distinguishing feature of the present work is that this discretization method is employed at the subdomain level in the framework of a Schwarz-type domain decomposition algorithm (DDM). We show that HDG method naturally couples with a Schwarz method relying on optimized transmission conditions. The presented numerical results show the effectiveness of the optimized DDM-HDG method.

Scattering of a scalar time-harmonic wave by a penetrable obstacle with a thin layer

This work looks at the asymptotic behaviour of the solution to the Helmholtz equation in a penetrable domain of$\mathbb{R}$3with a thin layer of thickness δ which tends to 0. We use the method of multi-scale expansion to derive and justify an asymptotic expansion of the solution with respect to the thickness δ up to any order. We then provide approximate transmission conditions of order two defined on an interface located inside the thin layer, with accuracy up toO(δ2), which allow one to take into account the influence of the thin layer.

On the wave propagation in the time differential dual-phase-lag thermoelastic model

We study the propagation of plane time harmonic waves in the infinite space filled by a time differential dual-phase-lag thermoelastic material. There are six possible basic waves travelling with distinct speeds, out of which, two are shear waves, and the remaining four are dilatational waves. The shear waves are found to be uncoupled, undamped in time and travels independently with the speed that is unaffected by the thermal effects. All the other possible four dilatational waves are found to be coupled, damped in time and dispersive due to the presence of thermal properties of the material. In fact, there is a damped in time longitudinal quasi-elastic wave whose amplitude decreases exponentially to zero when the time is going to infinity. There is also a quasi-thermal mode, like the classical purely thermal disturbance, which is a standing wave decaying exponentially to zero when the time goes to infinity. Furthermore, there are two possible longitudinal quasi-thermal waves that are damped in time with different decreasing rates or there is one plane harmonic in time longitudinal thermal wave, depending on the values of the time delays. The surface wave problem is studied for a half space filled by a dual-phase-lag thermoelastic material. The surface of the half space is free of traction and it is free to exchange heat with the ambient medium. The dispersion relation is written in an explicit way and the secular equation is established. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.

Acoustic Interaction Forces and Torques Acting on Suspended Spheres in an Ideal Fluid.

In this paper, the acoustic interaction forces and torques exerted by an arbitrary time-harmonic wave on a set of N objects suspended in an inviscid fluid are theoretically analyzed. We utilize the partial-wave expansion method with translational addition theorem and re-expansion of multipole series to solve the related multiple scattering problem. We show that the acoustic interaction force and torque can be obtained using the farfield radiation force and torque formulas. To exemplify the method, we calculate the interaction forces exerted by an external traveling and standing plane wave on an arrangement of two and three olive-oil droplets in water. The droplets' radii are comparable to the wavelength (i.e., Mie scattering regime). The results show that the acoustic interaction forces present an oscillatory spatial distribution which follows the pattern formed by interference between the external and rescattered waves. In addition, acoustic interaction torques arise on the absorbing droplets whenever a nonsymmetric wavefront is formed by the external and rescattered waves' interference.

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.