Time-harmonic Waves Research Articles

On efficient computation of highly oscillatory retarded potential integral equations

ABSTRACTThis paper presents an efficient numerical method for retarded potential integral equations with highly oscillatory spatially time-harmonic incident waves, which is based on inverse Fourier transforms and efficient algorithms for the highly oscillatory Volterra integral equation of the first kind. From the integral equation, it leads to an efficient approximation by applying the Clenshaw–Curtis-type method which costs the same operations independent of large values of frequencies. Applying inverse Fourier transforms yields numerical results on solving the retarded potential integral equations. Preliminary numerical results show the efficiency and accuracy of the approximations.

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.

On the far-field computation of acoustic radiation forces.

It is known that the steady acoustic radiation force on a scatterer due to incident time-harmonic waves can be calculated by evaluating certain integrals of velocity potentials over a sphere surrounding the scatterer. The goal is to evaluate these integrals using far-field approximations and appropriate limits. Previous derivations are corrected, clarified, and generalized. Similar corrections are made to textbook derivations of optical theorems.

On scattering of waves on square lattice half-plane with mixed boundary condition

The method developed by Wiener and Hopf is applied to the problem of scattering of time-harmonic waves by the surface of a semi-infinite square lattice with mixed boundary condition. Three distinct configurations of the boundary are analyzed in the paper. The far-field approximation of the exact solution is provided along with some graphical illustrations and numerical calculations. Potential applications of the analysis include the scattering of elastic, phononic, photonic and electronic waves by surfaces which are exposed, in patches, simultaneously to different kinds of environmental conditions.

Sound insulation properties in low-density, mechanically strong and ductile nanoporous polyurea aerogels

Aerogels are quasi-stable, nanoporous, low-density, three-dimensional assemblies of nanoparticles. In this paper, an extremely high sound transmission loss for a family of ductile polyurea aerogels (e.g., over 30dB within 1 to 4kHz at bulk density 0.25g/cm3 and 5mm thickness) is reported. The fundamental mechanisms behind the aerogel acoustic attenuations are investigated. Sharing striking similarities with acoustic metamaterials, initially, aerogels are studied via a one-dimensional multi degree-of-freedom mass-spring system. Different effects such as spring constant disparity are investigated in regards to the structural vibration wave transmission loss. Results are given for different configurations consistent with the aerogel nano/microstructures. A significant wave attenuation is observed by considering a random spring distribution. In the next step towards modeling such a complex hierarchical and random structural material, the continuum Biot's dynamic theory of poroelasticity is implemented to analyze the experimental sound transmission loss results. In this framework, a two-dimensional plane strain analysis is considered for the interaction of a steady state time-harmonic plane acoustic wave with an infinite aerogel layer immersed in and saturated with air. The effects of bulk density and thickness on the aerogel sound transmission loss are elucidated. By comparing the theoretical results with the experimental observations, this study develops a qualitative/quantitative basis for the dynamics of the aerogel nanoparticle network as well as the air flow and solid vibroacoustic interactions. This basis provides a better understanding on the overall acoustic properties of the aerogels that might also be helpful in the design of the future hierarchical materials.

Invisibility and Perfect Reflectivity in Waveguides with Finite Length Branches

We consider a time-harmonic wave problem, appearing for example in water-waves and in acoustics, in a setting such that the analysis reduces to the study of a 2D waveguide problem with a Neumann boundary condition. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains one branch of finite length. We analyse the behaviour of the complex scattering coefficients R, T as the length of the branch increases and we exhibit situations where non reflectivity (R = 0, |T| = 1), perfect reflectivity (|R| = 1, T = 0) or perfect invisibility (R = 0, T = 1) hold. Numerical experiments allow us to illustrate the different results.

Electromagnetic wave scattering by random surfaces: Shape holomorphy

For time-harmonic electromagnetic waves scattered by either perfectly conducting or dielectric bounded obstacles, we show that the fields depend holomorphically on the shape of the scatterer. In the presence of random geometrical perturbations, our results imply strong measurability of the fields, in weighted spaces in the exterior of the scatterer. These findings are key to prove dimension-independent convergence rates of sparse approximation techniques of polynomial chaos type for forward and inverse computational uncertainty quantification. Also, our shape-holomorphy results imply parsimonious approximate representations of the corresponding parametric solution families, which are produced, for example, by greedy strategies such as model order reduction or reduced basis approximations. Finally, the presently proved shape holomorphy results imply convergence of shape Taylor expansions of far-field patterns for fixed amplitude domain perturbations in a vicinity of the nominal domain, thereby extending the widely used asymptotic linearizations employed in first-order, second moment domain uncertainty quantification.

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.

Time-Harmonic Ultrasound elastography of the Descending Abdominal Aorta: Initial Results

Stiffening of central large vessels is considered a key pathophysiologic factor within the cardiovascular system. Current diagnostic parameters such as pulse wave velocity (PWV) indirectly measure aortic stiffness, a hallmark of coronary diseases. The aim of the present study was to perform elastography of the proximal abdominal aorta based on externally induced time-harmonic shear waves. Experiments were performed in 30 healthy volunteers (25 young, 5 old, >50 y) and 5 patients with longstanding hypertension (PWV >10 m/s). B-Mode-guided sonographic time-harmonic elastography was used for measurement of externally induced shear waves at 30-Hz vibration frequency. Thirty-hertz shear wave amplitudes (SWAs) within the abdominal aorta were measured and displayed in real time and processed offline for differences in SWA between systole and diastole (ΔSWA). Data were analyzed using the Kruskal–Wallis test and receiver operating characteristic curve analysis. The change in SWA over the cardiac cycle was reduced significantly in all patients as assessed with ΔSWA (volunteers: mean = 10 ± 5 μm, patients: mean = 4 ± 1 μm; p < 0.001). The best separation of healthy volunteers from patients was obtained with a ΔSWA threshold of 4.7 μm, resulting in a sensitivity of 0.9 and a specificity of 1.0, with an overall area under the curve of 0.96. Time harmonic elastography of the abdominal aorta is feasible and shows promise for the exploitation of time-varying shear wave amplitudes as a diagnostic marker for aortic wall stiffening. Patients with elevated PWVs suggesting increased aortic wall stiffness were best identified by ΔSWA—a parameter that could be related to the ability of the vessel walls to distend on passages of the pulse wave.

Effects of elastic bed on hydrodynamic forces for a submerged sphere in an ocean of finite depth

In this paper, we consider a hydroelastic model to examine the radiation of waves by a submerged sphere for both heave and sway motions in a single-layer fluid flowing over an infinitely extended elastic bottom surface in an ocean of finite depth. The elastic bottom is modeled as a thin elastic plate and is based on the Euler–Bernoulli beam equation. The effect of the presence of surface tension at the free-surface is neglected. In such situation, there exist two modes of time-harmonic waves: the one with a lower wavenumber (surface mode) propagates along the free-surface and the other with higher wavenumber (flexural mode) propagates along the elastic bottom surface. Based on the small amplitude wave theory and by using the multipole expansion method, we find the particular solution for the problem of wave radiation by a submerged sphere of finite depth. Furthermore, this method eliminates the need to use large and cumbersome numerical packages for the solution of such problem and leads to an infinite system of linear algebraic equations which are easily solved numerically by any standard technique. The added-mass and damping coefficients for both heave and sway motions are derived and plotted for different submersion depths of the sphere and flexural rigidity of the elastic bottom surface. It is observed that, whenever the sphere nearer to the elastic bed, the added-mass move toward to a constant value of 1, which is approximately twice of the value of added-mass of a moving sphere in a single-layer fluid flowing over a rigid and flat bottom surface.

An adaptive DPG method for high frequency time-harmonic wave propagation problems

The Discontinuous Petrov–Galerkin (DPG) method for high frequency wave propagation problems is discussed. The DPG method, with its attractive uniform (mesh and wavenumber independent) pre-asymptotic stability property, allows for a fully automatic adaptive hp-algorithm that can be initiated from very coarse meshes. Moreover, DPG always delivers a Hermitian positive definite system, suggesting the use of the Conjugate Gradient algorithm for its solution. We present a new iterative solution scheme which capitalizes on these attractive properties of DPG. This novel solver is integrated within the adaptive procedure by constructing a two-grid-like preconditioner for the Conjugate Gradient method that exploits information from previous meshes. The construction of our preconditioner is discussed, and its efficacy is illustrated with an example of a 2D acoustics problem. Our results show that the proposed iterative algorithm converges at a rate independent of the mesh and the wavenumber.

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.

The numerical solution of the acoustic wave scattering from penetrable obstacles by a least-squares method

This paper documents a numerical method for a two dimensional time-harmonic wave scattering problem by penetrable obstacles. The Fourier–Bessel function combining a multipole expansion is used to give an approximation of the scattering field. This method is based on the least-squares technique. Especially, we find a simple function to control the errors, and then give the theoretical results of the presented method. The continuity across the element boundaries is enforced by minimizing a simple quadratic functional. This method does not need to truncate the domain and could obtain high accuracy by increasing the number of basis functions with even coarse mesh. At last, we give some examples to illustrate the effectiveness of the approach including the solution domain being multiple or even multi-connected.

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.

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.

Decoupling elastic waves and its applications

In this paper, we consider time-harmonic elastic wave scattering governed by the Lamé system. It is known that the elastic wave field can be decomposed into the shear and compressional parts, namely, the pressure and shear waves that are generally coexisting, but propagating at different speeds. We consider the third or fourth kind impenetrable scatterer and derive two geometric conditions, respectively, related to the mean and Gaussian curvatures of the boundary surface of the scatterer that can ensure the decoupling of the shear and pressure waves. The decoupling results are new to the literature and are of significant interest for their own sake. As an interesting application, we apply the decoupling results to the uniqueness and stability analysis for inverse elastic scattering problems in determining polyhedral scatterers by a minimal number of far-field measurements.

High order surface radiation conditions for time-harmonic waves in exterior domains

We formulate a new family of high order on-surface radiation conditions to approximate the outgoing solution to the Helmholtz equation in exterior domains. Motivated by the pseudo-differential expansion of the Dirichlet-to-Neumann operator developed by Antoine et al. (1999), we design a systematic procedure to apply pseudo-differential symbols of arbitrarily high order. Numerical results are presented to illustrate the performance of the proposed method for solving both the Dirichlet and the Neumann boundary value problems. Possible improvements and extensions are also discussed.

Wave propagation in magneto-electro-elastic multilayered plates with nonlocal effect

In this paper, analytical solutions for propagation of time-harmonic waves in three-dimensional, transversely isotropic, magnetoelectroelastic and multilayered plates with nonlocal effect are derived. We first convert the time-harmonic wave problem into a linear eigenvalue system, from which we obtain the general solutions of the extended displacements and stresses. The solutions are then employed to derive the propagator matrix which connects the field variables at the upper and lower interfaces of each layer. Making use of the continuity conditions of the physical quantities across the interface, the global propagator relation is assembled by propagating the solutions in each layer from the bottom to the top of the layered plate. From the global propagator matrix, the dispersion equation is obtained by imposing the traction-free boundary conditions on both the top and bottom surfaces of the layered plate. Dispersion curves and mode shapes in layered plates made of piezoelectric BaTiO3 and magnetostrictive CoFe2O4 materials are presented to show the influence of the nonlocal parameter, stacking sequence, as well as the orientation of incident wave on the time-harmonic field response.