An efficient high-order algorithm for acoustic scattering from penetrable thin structures in three dimensions

This paper presents a high-order accelerated algorithm for the solution of the integral-equation formulation of volumetric scattering problems. The scheme is particularly well suited to the analysis of “thin” structures as they arise in certain applications (e.g., material coatings); in addition, it is also designed to be used in conjunction with existing low-order FFT-based codes to upgrade their order of accuracy through a suitable treatment of material interfaces. The high-order convergence of the new procedure is attained through a combination of changes of parametric variables (to resolve the singularities of the Green function) and “partitions of unity” (to allow for a simple implementation of spectrally accurate quadratures away from singular points). Accelerated evaluations of the interaction between degrees of freedom, on the other hand, are accomplished by incorporating (two-face) equivalent source approximations on Cartesian grids. A detailed account of the main algorithmic components of the scheme are presented, together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results

Asymptotic Behavior for a Coalescence Problem

Consider spherical particles of volume x having paint on a fraction y of their surface area. The particles are assumed to be homogeneously distributed at each time t, so that one can introduce the density number n (x, y, t). When collision between two particles occurs, the particles will coalesce if and only if they happen to touch each other, at impact, at points which do not belong to the painted portions of their surfaces. Introducing a dynamics for this model, we study the evolution of n (x, y, t) and, in particular, the asymptotic behavior of the mass x n (x, y, t) dx as t → ∞

On the O(1) Solution of Multiple-Scattering Problems

In this paper, we present a multiple-scattering solver for nonconvex geometries such as those obtained as the union of a finite number of convex surfaces. For a prescribed error tolerance, this algorithm exhibits a fixed computational cost for arbitrarily high frequencies. At the core of the method is an extension of the method of stationary phase, together with the use of an ansatz for the unknown density in a combined-field boundary integral formulation

Quasistatic Motion of a Capillary Drop I. The Two-Dimensional Case

AbstractA theory is presented for analyzing the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia. The theory is developed here on the two-dimensional version of the relevant free-boundary model for Stokes equations. As we show, the two-dimensional problem presents most of the difficulties expected from a projected three-dimensional study while allowing for simpler manipulation of the spherical harmonics. Within this context we show that if the free-boundary initiates close to a circler=1+ελ0(θ),|ε|small, then there exists a global-in-time solution with free boundaryr=1+λ(θ, t, ε)=1+∑n⩾1λn(θ, t)εn, which approaches a circle exponentially fast as t→∞. Moreover, we prove that if λ0(θ) is analytic (resp. C∞) in θ, then the velocity u(x, t, ε), the pressure p(x, t, ε), and the free boundary λ are all jointly analytic (resp. C∞) in (x, ε)

An integral representation of the Green’s function for a linear array of acoustic point sources

We present a new algorithm for the evaluation of the periodized Green’s function fora linear array of acoustic point sources such as those arising in the analysis of linearray loudspeakers. A variety of classical algorithms (based on spatial and spectralrepresentations, Ewald transformation, etc.) have been implemented in the past toevaluate these acoustic fields. However as we show, these methods become unstableand/or impractically expensive as the frequency of use of the sources increases. Herewe introduce a new numerical scheme that overcomes some of these limitations allowingfor simulations at unprecedentally large frequencies. The method is based ona new integral representation derived from the classic spatial form, and on suitablefurther manipulations of the relevant integrands to render the integrals amenable toefficient and accurate approximations through standard quadrature formulas. Weinclude a variety of numerical results that demonstrate that our algorithm comparesfavorably with several classical method both for points close to the line where thepoles are located and at high-frequencies while remaining competitive with them inevery other instance

A combined parabolic-integral equation approach to the acoustic simulation of vibro-acoustic imaging

This paper aims to model ultrasound vibro-acoustography to improve our understanding of the underlying physics of the technique thus facilitating the collection of better images. Ultrasound vibro-acoustography is a novel imaging technique combining the resolution of high-frequency imaging with the clean (speckle-free) images obtained with lower frequency techniques. The challenge in modeling such an experiment is in the variety of scales important to the final image. In contrast to other approaches for modeling such problems, we break the experiment into three parts: high-frequency propagation, non-linear interaction and the propagation of the low-frequency acoustic emission. We then apply different modeling strategies to each part. For the high-frequency propagation we choose a parabolic approximation as the field has a strong preferred direction and small propagation angles. The non-linear interaction is calculated directly with Fourier methods for computing derivatives. Because of the low-frequency omnidirectional nature of the acoustic emission field and the piecewise constant medium we model the low-frequency field with a surface integral approach. We use our model to compare with experimental data and to visualize the relevant fields at points in the experiment where laboratory data is difficult to collect, in particular the source of the low-frequency field. To simulate experimental conditions we perform the simulations with the two frequencies 3 and 3.05 MHz with an inclusion of varying velocity submerged in water

High-Order Boundary Perturbation Methods

Perturbation theory is among the most useful and successful analytical tools in applied mathematics. Countless examples of enlightening perturbation analyses have been performed for a wide variety of models in areas ranging from fluid, solid, and quantum mechanics to chemical kinetics and physiology. The field of electromagnetic and acoustic wave propagation is certainly no exception. Many studies of these processes have been based on perturbative calculations where the role of the variation parameter has been played by the wavelength of radiation, material constants, or geometric characteristics. It is this latter instance of geometric perturbations in problems of wave propagation that we shall review in the present chapter. Use of geometric perturbation theory is advantageous in the treatment of configurations which, however complex, can be viewed as deviations from simpler ones—those for which solutions are known or can be obtained easily. Many uses of such methods exist, including, among others, applications to optics, oceanic and terrain scattering, SAR imaging and remote sensing, and diffraction from ablated, eroded, or deformed objects; see, e.g., [47, 52, 56, 59, 62]. The analysis of the scattering processes involved in such applications poses challenging computational problems that require resolution of the interplay between highly oscillatory waves and interfaces. In the case of oceanic scattering, for instance, nonlinear water wave interactions and capillarity effects give rise to highly oscillatory modulated wave trains that are responsible for the most substantial portions of the scattering returns [35]

Field enhancement and saturation of milimeter waves inside a metallic nanogap

This paper investigates the millimeter electromagnetic waves passing through a metal nanogap. Based upon the study of a perfect electrical conductor model, we show that the electric field enhancement inside the gap saturates as the gap size approaches zero, and the ultimate enhancement strength is inversely proportional to the thickness of the metal film. In addition, no significant enhancement can be gained by decreasing the gap size further if the aspect ratio between the dimensions of the underlying geometric structure exceeds approximately 100