Asymptotic modelling of micro-structured acoustic devices including thermoviscous effects

There is renewed interest in using structured acoustic devices to filter, guide and absorb sound, especially on small scales and incorporating new ideas originating in the field of electromagnetic metamaterials. Analytical modelling of such devices, however, has often relied on heuristics and the use of `fudge' parameters to fit experiments - especially when including thermoviscous effects, which play a singular role in many acoustic metamaterials and are key to sound absorption. I will present several new analytical models, derived using matched asymptotics with thermoviscous effects systematically accounted for. I will first revisit the classical problem of calculating the acoustic impedance of a cylindrical orifice. Building on this, I will present a detailed asymptotic model of a three-dimensional Helmholtz resonator embedded in a wall, as well as acoustic metasurfaces formed of finite and infinite arrays of such resonators. Lastly, I will also discuss the important role of thin thermoviscous boundary layers in experiments of extraordinary acoustic transmission through narrow slits. (Joint work with Ory Schnitzer and Jacob Holley.)

Asymptotic approximations for Bloch waves and topological mode steering in a planar array of Neumann scatterers

We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet-Bloch waves through an infinite array of scatterers. The approach lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on topical examples from topological wave physics.

Advanced Artificial Structures For The Control of Acoustic and Mechanic Energy

The control of acoustic energy is a challenging problem with a broad range of applications. In this talk we will review some specific mechanism to achieve this control by means of artificial structures specially designed for this purpose. It will be shown how engineered gratings can be used to modulate the flow of acoustic and elastic energy towards specific directions, and a very efficient inverse design method will be presented. Some applications will be summarized, and the experimental realization of an acoustic carpet cloak based on these structures will also be presented.

Sommerfeld integral method for discrete diffraction problems

Two diffraction problems on a 2D square lattice are studied: finding the Green's function and scattering by a half-line. The governing equation in both cases is the Helmholtz equation with a standard discrete Laplacian. For the Green's function of an entire plane, we build the field in the form of a plane wave decomposition. This decomposition is rewritten as a contour integral of an analytical differential 1-form on a complex manifold, which is the lattice dispersion diagram (the solution of the dispersion equation). The dispersion diagram of the lattice is, topologically, a torus, and this makes possible to develop an invariant representation of the field. For the diffraction problems, an analogue of the Sommerfeld integral is built. This is an integral of a Sommerfeld transformant (a two-valued function on the torus) along a system of appropriate integrals.

As a result, we obtain new representations of the discrete wave fields. Potentially, these methods can be applied to a wider class of problems.

Ensemble average waves in random materials of any geometry

Which is more useful: knowing the effective properties of a material, or its effective wavenumbers? In a low frequency regime these are essentially the same, but when the wavelength becomes comparable to the microstructure's size, they are not. Using a broad range of frequencies is needed to characterize materials and to design for broad wave speed control and attenuation.

Suppose we perform an experiment that measures the scattered field from a material with a random microstructure. We could then find what effective properties would lead to the same scattered field we measured. So far so good. But what if we took that same microstructure and molded it into a different geometry and then repeated the experiment? The bad news is that even when using the same frequency we would potentially find a different set of effective properties. In contrast, in this talk I show how effective wavenumbers are inherently related to the material microstructure and not its geometry. I will also show how to calculate the average scattered field, including the field scattered from a pipe geometry filled with particles and a spherical droplet filled with particles.

Thermo-Visco-Elastic effects in Wave Propagation

Recent work in the metamaterial literature has shown the importance of taking into account the presence of visco-thermal losses through dissipation for the accurate description of acoustic fields in a variety of metamaterial designs of practical interest. In general, the background fluid in which the material is aimed to operate plays an important role in the design and associated mathematical modelling. This is for example illustrated in underwater structures which are typically subject to high hydrostatic pressure loads, and hence fluid-structure interaction (FSI) effects must be taken into account, as opposed to standard in-air scenarios where FSI is often negligible. Many interesting solid metamaterials (such as membrane type media), are of a (thermo-) viscoelastic nature. It would therefore be convenient to develop a framework that allows the study of the acoustics of thermo- viscoelastic continua, including both solids and fluids. The derivation of such a model will be introduced and put into practice by considering the canonical problem of the interaction of two thermo-viscoelastic media separated by an interface. In particular, differences with conventional acoustic models will be highlighted. This is a joint work with A. L. Gower (University of Sheffield), P. A. Cotterill, R. C. Assier, W. J. Parnell (University of Manchester) and D. Nigro (Thales UK).

Convergence Properties of Dynamical Energy Analysis - A Ray-based Method Using Transfer Operators

Describing the distribution of vibrational energy in real world applications is challenging, especially in the mid-to-high frequency regime. A very promising way is using densities of rays and creating a transfer operator T by a method called Dynamical Energy Analysis. This has been used in a range of real-world scenarios ranging from gear boxes over car bodies to whole ship hulls, see for example [1]. The role of this operator T is to propagate the intensity of the vibrational excitation across a given structure. Using local properties of the structure, this propagation can account for different material properties and geometrical aspects. For thin shells for example, it describes the vibrations in terms of pressure, shear, and bending wave modes. In order to compare predictions with experimental data this tool has to be used numerically. This makes it necessary to represent T in terms of a finite basis. To have an efficient implementation with a good convergence and small errors it is necessary to choose an optimal set of basis functions. The talk will cover convergence properties of this method as well comparing them with a simple analytical model. [1] Hartmann, Morita, Tanner, Chappell Wave Motion, 87, 2019, 132-150.

Asymptotic solutions of the plasmonic eigenvalue problem

A flat interface between a metal and a dielectric supports surface electromagnetic waves, typically in the visible range. Such surface waves have the special property that their wavelength can be squeezed down to nanometric scales, small in comparison to the wavelength of light in free space. For this reason, subwavelength metallic nanoparticles support quasi-static "plasmonic" modes, governed by a purely geometric "plasmonic eigenvalue problem," whose resonant excitation by incident radiation results in enhanced absorption and scattering. In particular, many applications of plasmonic resonance, such as targeted heating and bio-sensing based on nonlinear optics, employ multiple-scale particle geometries in order to enhance and localise electric fields on nanometric scales. Motivated by this, in this talk I will discuss the asymptotic properties of multiple-scale plasmonic particles including close-to-touching cylinders and spheres, and slender particles (work on slender particles is joint with Matias Ruiz). Time permitting, I will briefly comment on other asymptotic results in plasmonics, including the accumulation of modes at the surface-plasmon frequency and nonlocal effects.

Acoustic imaging of low-frequency sources with extended resolution

Acoustic imaging is to identify the locations and strengths of the sound sources, based on which the analysis of sound generation mechanisms is possible. The resolution is often dependent on the frequency of the sound source. At low frequency, a practical issue is that various closely located sources may not be distinguishable due to the large wavelength, leading to challenges for correct interpretation of the acoustic physics. One key stage in our proposed extended-resolution acoustic imaging method is to introduce a high-frequency incident wave to interact with the target sources. The nonlinear interaction between different source components can induce sound waves at an alternative high frequency. The induced sound carries the information of the target low-frequency, which can be rigorously estimated base on the acoustic analogy theory. Also, the wavelength of the induced sound is relatively small (depends on the frequency of the incident wave) and is beneficial for the high-resolution imaging of the low-frequency sources. We will demonstrate that sources with a separation distance smaller than 1/10 of the wavelength can be identified. This talk is based on a joint work with W. Q. Chen and X. Huang at Peking University, China.

Edge Diffraction of Acoustic Waves by Periodic Composite Metamaterials: The Hollow Wedge

Generally in metamaterial research, some form of infinite periodicity is assumed which allows us to restrict the study to a small portion known as the unit cell. This has led to many studies which increase the complexity of the unit cell and reconstruct the global scattering using the periodicity of the metamaterial. An alternative approach looks into the case where infinite periodicity is no longer assumed. This means that the metamaterial will have well-defined boundaries that can symbolise many different interfaces such as edges and corners. In this presentation, the scattering of an acoustic pressure wave by a hollowed out wedge is studied where, for simplicity, the unit cells will be sound-soft cylinders with infinite height and a small radius. This configuration can also be viewed as two separate semi-infinite gratings with two sets of scattering coefficients to determine. We will construct an iterative scheme from the resulting infinite system of equations and find a solution using the discrete Wiener-Hopf technique. We shall also discuss some tools that are useful for computations of the Wiener-Hopf kernel such as tail-end asymptotics and rational approximations. Comparison to COMSOL simulations are made and resonant cases are identified. This is joint work with Anastasia Kisil and Raphael Assier.

Guided wave propagation in nonlinear plates

Elastic waves in plates have been intensively studied over last decades due to their practical importance and wide applicability of thin-walled components in light-weight structures. Lamb waves display complex and multi-modal dispersion characteristics that make them attractive for various purposes, e.g. NDT and SHM, but more challenging in the context of signal analysis. Recently, nonlinear features of guided waves in plates have been investigated for their applications in high-resolution imaging, more accurate damage detection and various types of acoustic devices. In the following we propose a methodology for analyzing wave propagation in nonlinear plates i.e., a layer of nonlinear material bounded between two parallel traction-free boundaries. The proposed approach employs a perturbation-based strategy for sequentially solving the nonlinear elastodynamic set of equations. We show that the nonlinearity induces frequency shifts that depend on the primary wave amplitude, resulting in amplitude-dependent dispersion surfaces. It is also found that the nonlinear nature of the medium impacts other wave propagation phenomena, such as internal resonances.

Swimming of waving plates

When two or more fliers or swimmers move together, their interactions can significantly affect the characteristics of the surrounding flow. Indeed, it is well known that many natural swimmers exploit these effects to enhance their propulsive efficiency. This raises the question of when these swimmers are operating in co-operation or competition; i.e. do the interaction effects help or hinder the swimmers. We use conformal maps and multiply connected function theory to build a model for these interactions. Our model is based on thin aerofoil theory and requires equivalent assumptions such as attached flow, small-amplitude motions and linearised wakes. Accordingly, our approach is very general and permits consideration of a range of wing motions (pitching, heaving, undulatory) and configurations (tandem, in-line, periodic, ground effect). Unlike previous approaches, our model is not based on the assumption that the swimmers are far apart and thus interact only weakly. We focus on the (doubly connected) case where there are two interacting swimmers and find that our results show excellent agreement with experimental data. Specifically, our model recovers the equilibrium configurations observed in recent experiments and suggests the existence of new stable configurations.

Sound Propagation in Shear Flow. A Quest for Adiabatic Invariants

Adiabatic invariants are the holy grail in a WKB analysis of waves in a slowly varying medium. If one exists, it serves as an exact integral for the slowly varying amplitude of the wave. This is no exception for acoustic modes in a slowly varying duct with slowly varying mean flow. Adiabatic invariants are invariants under slow variation, not any variation. Their existence ensues sometimes, but not only, from the more stringent conservation of energy. Acoustic energy in mean flow is not always conserved: it is conserved in potential flow, but not in vortical (i.e. shear) flow where the acoustic field exchanges energy with the mean flow. Adiabatic invariants are therefore common for modes in slowly varying potential flows, but so far unknown in sheared flows.

We found that: (i) in 2D shear flow the modes satisfy in general an incomplete adiabatic invariant; (ii) this reduces to a complete one for linear shear flow. This result makes the WKB approximation for a mode in a slowly varying duct almost as simple as the solution for a mode in a straight duct. Journal of Fluid Mechanics, 906, 2021-01-10, A23; https://doi.org/10.1017/jfm.2020.687

Wiener-Hopfing the wake of an elastic cylinder

Computer-controlled metal forming is currently held back by a lack of suitable modelling. As one example, a computer-controlled flexible metal sheet spinning process recently developed by the Allwood group at the University of Cambridge is now being used by industry, but real-time control has been foiled by a lack of predictive modelling of the physics. Existing industrial metal forming produces large amounts of CO2 emissions, so it is vital that we iron this out.

Motivated by this application, in this talk I will describe initial analytic work where we model the metal media as being linear elastic, and derive a model for a rigid cylinder rolling along an elastic half-space. The frictional between roller and half-space gives a mixture of boundary conditions, leading to a matrix Wiener-Hopf problem, which may be solved by implementing the iterative method for triangular matrix Wiener-Hopf equations with exponential factors. Finally, I shall discuss the free-boundary problem associated with the contact points. This is joint work with Dr Ed Brambley.

Graded arrays for spatial frequency separation and amplification of water waves

Wave-energy converters extracting energy from ocean waves are known to suffer from poor efficiency. We propose structures capable of substantially amplifying water waves over a broad range of frequencies at selected locations, with the idea of enhanced energy extraction. The structures consist of full or C-shaped bottom-mounted cylinders arranged in one-dimensional or two-dimensional arrays, with the cylinder properties or the array spacing graded along the array. Using linear potential-flow theory, it is shown that the energy carried by a plane incident wave is amplified within specified locations, for wavelengths comparable to the array length, and for a range of incident directions. Transfer-matrix analysis is used to analyse the large amplifications and we also show results from recent wave-flume experiments confirming the amplification phenomenon in practice.

Scattering of high-frequency whispering gallery waves by boundary inflection: asymptotics and integral representations

The talk is on an open canonical problem of high-frequency diffraction: the problem of a whis[1]pering gallery asymptotic mode's Helmholtz scattering by a boundary inflection. The related key inner boundary-value problem for a Schr¨odinger equation on a half-line with a potential linear in both space and time was first formulated and analysed by M.M. Popov starting from 1970-s, and appears fundamental for describing transitions from modal to scattered asymptotic patterns. The talk briefly reviews the background, and then reports a recent result in [1] proving that the solution past the inflection point has a "searchlight" asymptotics corresponding to a beam concentrated near the limit ray. We also review some most recent progress on a reduction of the problem to well-posed boundary integral equations. As a result, a representation for the solution is obtained in form of a limit of locally uniformly and absolutely convergent Neumann series, and an integral representation is derived for the searchlight amplitude. In conclusion, some further prospects and challenges are briefly discussed. Some parts of the work are joint with Ilia Kamotski, and with Shiza Naqvi

Tuneable topological edge modes in systems of subwavelength resonators

Our goal is to advance the development of wave-guiding subwavelength crystals (i.e. high-contrast metamaterials) by developing designs whose properties are stable with respect to geometric imperfections. Using layer-potential formulations and asymptotic techniques, we have studied the topological properties of chains of resonator dimers and seen that stable edge modes exist at the interfaces of topological indices. We will also examine more recent work which builds on this by proving results which describe how edge mode frequencies cross the subwavelength band gap when dislocations are introduced to the initial periodic array. We study infinite chains of resonators, using the method of fictitious sources, and conduct a stability analysis through numerical experiments on the corresponding truncated finite structure. This approach gives an intuitive insight into the mechanisms that underpin the existence of robust localized edge modes in topological crystals and reveals an analytic way to fine tune their properties.

Non-polynomial discretizations of wave problems: accuracy and stability

Many old and recent discretization strategies for wave equations exhibit a kind of creativity, in that the basis functions are chosen to be wavelike rather than polynomial-like. The most explicit examples are Trefftz methods and the method of fundamental solutions, but various other methods exist based on plane waves. A wavelike discretization for a wave problem seems intuitively reasonable, but one quickly loses mathematical analysis. Is there a common ground? We attempt, from the point of view of numerical analysis, to formulate some ground rules. In the process, our analysis offers some guiding design principles. It also allows us to give partial answers to the following types of questions. Are there limits to the creativity, or is there room for more? Should one aim for a basis, or can this be relaxed? Is ill-conditioning a concern? If so, when is it and why is it (or when is it not and why is it not)? Our answers are very incomplete when it comes to wave problems. However, for the simpler problem of approximating a function using anything other than polynomials, the theory is rather precise.

The unified transform method: beyond circular domains

In this talk, we present a new approach for desingularisation of the Cauchy kernel for convex polygons, as well as discuss extensions beyond circular domains. Our approach builds on the transform pairs obtained via the Unified Transform Method. The formulation can be used to solve mixed boundary value problems for elliptic problems. We briefly discuss how it can be implemented to solve problems in fluid dynamics and diffraction theory. This is joint work with Jesse Hulse (Syracuse), Loredana Lanzani (Syracuse) and Stefan Llewellyn Smith (UCSD).

Recording unavailable.

The Helmholtz boundary element method does not suffer from the pollution effect

In d dimensions, approximating an arbitrary function oscillating with frequency less than or equal to k requires ~ k^d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k and in d dimensions) suffers from the pollution effect if, as k increases, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than k^d for domain-based formulations, such as finite element methods, and k^{d-1} for boundary-based formulations, such as boundary element methods).

It is well known that the h-version of the finite element method (FEM) suffers from the pollution effect. In contrast, at least empirically, the h-version of the boundary element method (BEM) does not suffer from the pollution effect, but this has not been proved up till now.

In this talk, I will discuss recent results (obtained with Jeffrey Galkowski) showing that the h-BEM does not suffer from the pollution effect in certain common situations.

Bempp - What's next?

Over the last few years our open-source boundary element software package Bempp has been used in a variety of applications across engineering and the physical sciences. Examples include high-frequency scattering, electromagnetic simulations of ice crystals, high-intensity focused ultrasound, and others. In this talk we highlight some key challenges in these applications, and provide an overview of what is next in store on the way to developing large-scale parallel high-frequency boundary element solvers.