Organisers
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Anastasia Kisil, The University of Manchester
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Georg Maierhofer, Sorbonne University
About:
Monthly online seminar series which started April 2020.
The focus of this seminar series is on waves interaction with complex media, membranes and metamaterials. The applications include acoustics, aero- and hydro-acoustics, elastic, electromagnetic waves. Key challenge areas to be addressed include the analysis of propagation in complex domains (e.g. with multiscale structure and/or complicated geometries), the development of effective methods for high frequency problems (in particular, the study of canonical problems arising in the Geometrical Theory of Diffraction), and the rigorous analysis and validation of computational methods.
Seminars will be held on the first Tuesday of the month 16.00-18.00 UK time. There will be two 25 minute talks (plus 5 min for questions) followed by optional informal discussion with one of the speakers in a breakout room (30 minutes).
The talks will be recorded (when permission is given) and made available (via links) on this webpage. Recordings from the seminar series are available to watch here.
[1] Matthew Colbrook, Anastasia Kisil “A Mathieu function boundary spectral method for diffraction by multiple variable poro-elastic plates, with applications to metamaterials and acoustics”, in preparation. Code developed by Matthew is available here.
Programme:
| Speaker One |
| Speaker Two |
| Breakout Rooms |
| End |
Past Events:
Extended Filon quadrature methods for high-frequency wave scattering
Motivated by collocation methods for wave scattering problems in 2D, we study the efficient approximation of highly oscillatory integrals in the presence of singularities and stationary points arising from the Green's function of the Helmholtz equation. From asymptotic theory we know that, for large frequencies, these integrals are dominated by contributions at corners, singular and stationary points. We exploit the understanding of this asymptotic behaviour to design bespoke quadrature methods based on the classical Filon method for oscillatory integrals. These methods allow for numerical approximation at uniform cost both for small and for very large frequencies. We demonstrate how the required Chebyshev moments can be stably computed based on a duality to a spectral method applied to Bessel's equation. Our design for this algorithm has significant potential for further generalisations that would allow Filon methods to be constructed for a wide range of integrals involving special functions. In this talk we will place special emphasis on the application of these methods to numerical wave scattering. They provide a flexible and frequency-independent way of assembling the collocation matrix for hybrid methods in high-frequency wave scattering problems on convex polygonal shapes, and we will demonstrate this favourable performance on several numerical examples.
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.)
Lax-Phillips Scattering Theory for Simple Wave Scattering
Lax-Philips scattering theory is a method to solve for scattering as an expansion over the singularities of the analytic extension of the scattering problem to complex frequencies. I will show how a complete theory can be developed in the case of simple scattering problems. Even for the simplest case, it requires a non-trivial generalised eigenfunction transformation to project into the space of analytic functions on the real line. The scattering operator in this space is simply the complex exponential. I will illustrate how this theory can be used to find a numerical solution, and I will demonstrate the method by applying it to the vibration of ice shelves.
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.
Scattering, Acoustic Black Holes and Mathieu Functions: A boundary spectral method for diffraction by multiple variable poro-elastic plates
Many fluid dynamical and acoustic problems can be modelled by PDEs on unbounded domains with complicated boundary conditions. Accurate and fast solutions of these systems is key to predicting the effect of physical variables/parameters and external forces, and thus crucial for providing insight into a wide range of problems. In this talk I will focus on the problem of Helmholtz scattering off multiple plates in arbitrary locations, with boundary conditions that model variable elasticity and porosity. Such boundary conditions present a considerable challenge to current methods.
The use of local Mathieu function expansions and their asymptotics leads to an efficient and robust high-order boundary spectral method, suitable for a wide range of frequencies, and tackling the ODE boundary conditions. The solution representation directly provides a sine series approximation of the far-field directivity and can be evaluated near or on the scatterers, meaning that the near field can be computed stably and efficiently. As an example, it is shown that a power-law decrease to zero in stiffness parameters gives rise to unexpected scattering and aeroacoustic effects similar to an acoustic black hole metamaterial. This talk is based on joint work with Anastasia Kisil.
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.
Recent advances on the preconditioning of 3D fast Boundary Element Solvers for 3D acoustics and elastodynamics
Recent works in the Boundary Element Method (BEM) community have been devoted to the derivation of fast techniques to perform the matrix vector product needed in the iterative solver. Fast BEMs are now very mature. However, it has been shown that the number of iterations can significantly hinder the overall efficiency of fast BEMs. The derivation of robust preconditioners is now inevitable to increase the size of the problems that can be considered. I will present some recent works on analytic and algebraic preconditioners for fast BEMs.
Exchange pulse corresponding to phase synchronism in a flexible plate loaded by gas
A problem of excitation of a sonic pulse in a system comprised by an elastic plate and a gas is studied. The source is localized in time and space, so the problem is non-stationary. It is known that such a system can have a phase synchronism point in the spectral domain, i.e. the values of frequency and wavenumber belonging both to the dispersion diagram of the plate and the gas. We demonstrate that such a point leads to appearing of an ``exchange pulse'', that is a quasi-monochromatic long pulse in the gas. In the talk, we write down an integral representation of the sonic field and discuss a method of asymptotic evaluation of the 2D Fourier integral. We demonstrate the term responsible for the exchange pulse. The work is motivated by an experiment made by one of the authors, who recorded sound produced by kicking of a thin (about 3 cm) layer of ice on a pond. The ice plays the role of the plate, and the gas is the air. The experimental signals generally support the theory presented in the talk. The work is supported by RFBR grant 19-29-06048 MK.
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.
Complex-scaling method for the plasmonic resonances of a 2D subwavelength particle with corners
It is well-known that a metallic particle can support surface plasmonic modes. For a subwavelength particle, these modes correspond to negative values of the permittivity, which are solutions of a self-adjoint eigenvalue problem. In this work, we are interested in the finite element computation of plasmonic modes in the case of a 2D particle whose boundary is smooth except for one corner. While a smooth particle has a discrete sequence of plasmonic eigenvalues, the corner leads to the presence of an essential spectrum, due to the existence of hyper-oscillating waves at the corner, the so called black-hole waves. Following our previous works, we introduce a complex scaling at the corner, and solve the complex-scaled eigenvalue problem. Numerical results show that the complex scaling deforms the essential spectrum, so as to unveil both embedded eigenvalues and complex plasmonic resonances. The later are analogous to scattering resonances with the local behavior at the corner playing the role of the behavior at infinity. We illustrate in particular the study of Li and Shipman (J. Integral Equations and Appl., 31(4), 2019), which proved the existence of embedded eigenvalues for the Neumann-Poincaré operator for a geometry with reflectional symmetry.
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).
Streamwise varying porosity and trailing-edge noise
Whilst it is known that porosity can significantly reduce the aerodynamic noise scattered by sharp edges, previous studies focus only on structures which are uniformly porous, or have an impermeable section and uniformly permeable section. The air flow resistance through birds wings (related to the porosity) however varies along the chord, and chordwise variations from barn owl wings and common buzzard wings are measurably different. Unsuprisingly, the owl's distribution of porosity is predicted to produce less trailing edge noise. By considering similar monotonic variations of porosity, we illustrate that it is not only what the edge values of porosity are, but how you get between them that impacts the total scattered noise.
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.
High-frequency homogenisation in 1D periodic media with imperfect interfaces of the spring-mass type
In this work, the concept of high-frequency homogenisation is extended to the case of one-dimensional periodic media that have an imperfect interface at the edges of the periodic cell. Indeed, when considering the propagation of elastic waves, displacement and stress discontinuities of the spring-mass types are allowed across the borders of the periodic cell. The high-frequency homogenisation of such media is carried out about the edges of the Brillouin zone that do not correspong to angular frequencies that are small, contrary to the classical low-frequency homogenization framework. At these edges, the dispersion diagram displays band-gaps, i.e. regions in the angular frequency space where waves do not propagate. A two-scale asymptotic expansion method is applied in order to approximate how the dispersion relation will behave around these zones. Asymptotic expansions are thus provided for the higher branches of the dispersion diagram and the resulting wavefield. The limiting case of two branches of the dispersion diagram that intersect with a non-zero slope (Dirac point) is also studied. The intermediate case of narrow bandgaps is also considered in order to obtain a uniform approximation that remains valid in the Dirac point limit. The examples of a monolayerd and a bilayered material are treated by this homogenised approach and a Bloch-Floquet approach in order to illustrate the validity of the method presented.
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.
Homogenization of an array of Helmholtz resonators: application to perfect absorption
We inspect the influence of the spacing on the resonance of a periodic arrangement of Helmholtz resonators. An effective problem is used which captures accurately the properties of the resonant array within a large range of frequency. It is shown that the strength of the resonance is enhanced when the array becomes sparser. This degree of freedom on the radiative damping is of particular interest since it does not affect the resonance frequency nor the damping due to losses within each resonator. We show that it can be used for the design of a perfect absorbing walls.
No recording available.
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.
Application of Uniform Geometric Theory of Diffraction (UGTD) to Investigations of Aircraft Power Plant Noise Reduction by Means of Shielding Effect
In this talk the results of aircraft power plant noise reduction by means of shielding effect will be presented. Based on the Uniform Geometric Theory of Diffraction (UGTD) expressions for the diffraction of sound on an infinite half-plane for point dipole and quadrupole sources are obtained. The obtained expressions were used to calculate the shielding efficiency for the round single-stream jet noise and azimuthal modes radiated from a cylindrical channel. The jet noise correlation theory was used as a model which describing the jet noise radiation, and the azimuthal modes radiation from a cylindrical channel was described based on the exact Weinstein solution. The calculations were performed for a flat acoustically absolutely rigid rectangular and polygonal shields which modeling the Blended-Wing-Body (BWB) airframe. The comparison of the calculated and experimental data showed a good agreement which made it possible to develop a method for calculating external noise reduction for aircraft that implement the shielding effect. This talk is based on a joint work with Viktor. F. Kopiev, Nikolay. N. Ostrikov.
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.
The complex-scaled half-space matching method and boundary integral equations in the complex domain
The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to PML or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretisation localised around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, wellposedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results.
This is joint work with: Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss, and Christophe Hazard (CNRS-INRIA-ENSTA, Paris), Karl-Mikael Perfekt (Reading), and Yohanes Tjandrawidjaja (Technische Universitaet Dortmund).
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.
Application of the Wiener-Hopf technique to the problem of jet-wing interaction noise
Significant progress has been made in engine noise reduction due to increasing bypass ratio (BPR) accompanying by the decrease of the jet exhaust velocity. However, modern engines with high and ultra-high BPR have progressively larger diameters and thus require tight integration into the airframe to provide a reasonable ground clearance. Close location of the turbulent jet and the wing leads to a significant low-frequency extra noise related to the jet-wing interaction which should be taken into account when assessing community noise. We have formulated and solved 2D and 3D model problems clarifying the physics of this phenomenon. In each of the problems we have used the Wiener-Hopf technique to obtain analytical solution. In the first problem, we consider the interaction of a plane acoustic wave with a two-dimensional model of a nozzle edge located near a wing trailing edge. The nozzle edge and the trailing edge are simulated by two staggered parallel half-planes with different flow velocities on both sides of the "nozzle". Shear layer behind the nozzle edge is represented by a vortex sheet. The matrix Wiener-Hopf equation is solved in conjunction with the full Kutta condition applied at the edges. Factorization of the kernel matrix is performed by the combination of Padé approximation and pole removal technique. Analytical results show strong dependence of the scattered acoustic field on the mutual position of the wing trailing edge and the nozzle edge. In the second model problem, a more realistic configuration with a round jet in the vicinity of a wing is considered. Low-frequency pressure fluctuations in the jet near field are represented in terms of a linear superposition of Gaussian-type wave packets of different azimuthal order. The problem of the scattering of these perturbations on a rigid half-plane, mimicking the wing, is solved by means of the Wiener-Hopf technique, and then 2D steepest descent method is used to calculate far-field asymptotic. It is shown that this model correctly captures main installation noise characteristics observed in experiments.
This talk is based on a joint work with Oleg Bychkov and Victor Kopiev.
Numerical Modelling of Scattering by Fractals
Fractals provide a natural mathematical model for the multi-scale roughness of many natural and man-made scattering objects, such as ice crystals in the atmosphere and fractal antennas/transducers. Fractals have features at all length scales, so discretisation poses some interesting challenges. In this talk I will present a range of novel discretisation techniques for scattering by fractals, alongside some neat numerical experiments.
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
Fast iterative BEM for high-frequency scattering problems in elastodynamics
The numerical solution of high-frequency scattering problems of time-harmonic elastic waves by a three-dimensional obstacle is a challenging task. The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is inevitable to increase the size of the problems that can be considered. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. We propose to combine an approximate adjoint Dirichlet-to-Neumann (respectively Neumann-to-Dirichlet) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet (respectively Neumann) exterior scattering problems. We compare standard and preconditioned Combined Field Integral Equations (CFIEs).
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.
Analysis and Design of Inhomogeneous Lenses for Millimeter Wave Communications
Latest technological advancements in additional manufacturing allow the easy fabrication of inhomogeneous dielectric materials where the local dielectric permittivity can be tailored at each voxel, thus enabling the realization of inhomogeneous lens antennas for millimeter-wave communications. In standard homogeneous lenses, rays follow broken lines with kinks at the lens input and output surfaces given by the air-dielectric interface refraction (but simply go straight inside the lens), thus reshaping the wavefront. Instead, in inhomogeneous lenses, rays follow arbitrary curved paths inside the lens because they are smoothly and continuously bent by the refractive index gradient. Therefore, lens interfaces can simply be flat while many degrees of freedom are still available for the design of the lens, which controls not only the wavefront but also the beam amplitude distribution. In the seminar, a review of a Geometrical Optics description of the electromagnetic wave propagation through an inhomogeneous medium will be presented (including a novel transport equation for the ray divergence) which is used as a tool for the analysis of inhomogeneous lenses. In addition, the lens design problem will also be addressed which consists in retrieving the refractive index distribution which drives the rays along prescribed paths.
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.
Expected reflection of ultrasound by randomly rough defects for inspection qualification
The characteristics of planar defects (no loss of material volume) that may arise during industrial plant operation are difficult to predict. Inspection modelling is increasingly used to design and qualify ultrasonic inspections and while modelling of smooth defects is relatively mature and validated, issues have remained in the treatment of rough defect species. In this talk, an approach to predict the expected surface reflection from a rough defect using a theoretical statistical model will be presented. Given only the frequency, angle of incidence and two statistical parameter values used to characterise the defects, the expected reflection amplitude is obtained for any scattering angle and size of defect, for both compression and shear waves. The method is applicable for inspections of isotropic media that feature surface reflections such as Pulse-Echo or Pitch-Catch. The new model predicts increases of up to 20dB in signal amplitude in comparison with models presently used in industry. All mode conversions are included and validations using numerical and experimental methods were performed.
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
The finite product method in approximation theory, and some applications
Many well-known functions in mathematics can be written as infinite products of simple factors. These include all the basic functions of trigonometry, of which Euler's infinite product for the sine is the best known. Unfortunately, truncations of these expressions to finite products are not normally of use, because of Runge's phenomenon, which is the presence of enormous unwanted oscillations near the boundaries of the domain of interest.
In this talk, it will be shown that in a class of applied problems in wave propagation, these high-amplitude oscillations cancel out exactly, to leave an extremely useful family of finite-product approximations, whose high accuracy and range of validity are extraordinary. The talk includes a full account of Runge's phenomenon (for researchers new to the topic), a simple proof of the exact cancellation, using only Stirling's approximation to the Gamma Function (with the `one-twelfth correction'), and some examples of wave propagation in which the resulting finite-product approximations have been put to good use by the speaker and Professor S. V. Sorokin.
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.
Rayleigh-Bloch waves above the cut-off
When a plane incident wave interacts with a long line array of identical equally-spaced scatterers it would be reasonable to assume the solution away from the array ends is approximately the same as the corresponding infinite array. But this is not necessarily the case, as homogeneous solutions of the infinite-array problem known as Rayleigh-Bloch waves can be excited by the incident wave and propagate along the array, possibly dominating the local wave field and generating resonances. For acoustic sound-hard scatterers, classic solution methods have been used to study Rayleigh-Bloch waves for a wide class of scatterer shapes up to a cut-off frequency and connect them to primary resonances on finite arrays. I will present a transfer operator method, use it to study Rayleigh-Bloch waves above the cut-off and connect them to higher-order resonances on finite arrays.
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.
Analytical continuation of two-dimensional wave fields
In this talk, we will consider a wide range of two-dimensional diffraction problems and ask ourselves the following question: what happens to the solution of these diffraction problems when the space variables are complexified?
We will devise a systematic way of analytically continuing these physical solutions as functions of two complex variables (and explain what we mean by this!). The singularities of the resulting (multi-valued) analytical continuation will be studied and we will show that it is possible to describe all the infinitely many possible values of this function in terms of a finite set of basis functions. Each basis function will be expressed explicitly as a Green's integral over a double-eight (or Pochhammer) contour. Finally, we will show how this finite basis property can be used to obtain precious information on the actual physical solution of the diffraction problem at hand.
This is a joint work with Dr Andrey Shanin from Moscow State University.
Reference:
R.C. Assier and A.V. Shanin, Analytical continuation of two-dimensional wave fields, Proc. Roy. Soc. A, 477:2020081, 2021 (open access)
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.
High frequency scattering by multiple obstacles
Standard Boundary Element Methods (BEM) for scattering problems, with piecewise polynomial approximation spaces, have a computational cost that grows with frequency. Recent Hybrid Numerical Asymptotic (HNA) BEMs with enriched approximation spaces consisting of the products of piecewise polynomials with carefully chosen oscillatory functions have shown to be effective in overcoming this limitation for a range of problems, mostly (with some recent exceptions) focused on single convex scatterers or very specific non-convex or multiple scattering configurations. Here we present a novel HNABEM approach to the problem of 2D scattering by a pair of screens in an arbitrary configuration, which we anticipate may serve as a building block towards algorithms for general multiple scattering problems with computational cost independent of frequency.
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.
The Hydrodynamic Instability in Quadratic Sheared Flow over Acoustic Linings
Jet engines, with increasing noise restrictions, make use of acoustic linings to reduce sound attenuation. Modelling these reduces the engine to a duct and the lining to an impedance boundary condition.
Making use of the Fourier transformed Euler equations, one can locate 'wave modes' that act as poles in contribution to Fourier inversion by residues. These wave modes can be located by numerical methods and in many cases include an unstable mode with a growing contribution. Under a sheared flow a branch cut, known as the critical layer, is also present but is often ignored.
Solving the problem analytically using Frobenius series solutions for a quadratic shear allows us to track the poles as we vary the system parameters and find that the unstable mode may become stable. This occurs by the pole moving through the branch cut and onto another Riemann sheet, where numerical methods would no longer be able to locate it. The pole's contribution when this occurs is absorbed by the critical layer branch cut, suggesting the critical layer branch cut cannot be ignored.
Nonlinear waves: caught between asymptotic analysis and high performance computing
We explore a collection of related scenarios in which nonlinear travelling waves arise in multi-layer fluid systems, and are subsequently investigated with both analytical and computational tools. We begin by interrogating two-fluid Couette flows using a novel evolution equation whose dynamics is validated by direct numerical simulation (DNS). The evolution equation incorporates inertial effects at arbitrary Reynolds numbers through a non-local term arising from the coupling between the two fluid regions, and is valid when one of the layers is thin. The equation predicts asymmetric solutions and exhibits bistability, features that are essential observations in the experiments of Barthelet et al.(J. Fluid Mech., vol. 303, 1995, pp. 23–53), without any ad hoc modifications. Comparisons between model solutions and DNS show excellent agreement at Reynolds numbers of up to O(1000) found in the experiments [1]. The modelling framework is then generalised to account for pressure-driven channel (Poiseuille) flows, also incorporating the effects of gravity and allowing the study of the competition between multiple physical effects [2]. Finally, while previous topics focus on capturing and understanding the resulting wave behaviour in these systems, our ultimate goal is one of robust controllability. In the context of liquid films falling down inclined surfaces, we explore a hierarchical approach in which strategically deployed reduced-order models for the liquid-gas interface (ranging from Kuramoto-Sivanshinsky equations, to Benney and weighted residual approaches) work in concert with the DNS environment in order to navigate complex parameter spaces efficiently and generate new predictive capabilities [3].
[1] Kalogirou A., Cimpeanu R., Keaveny E.E., Papageorgiou D.T., Capturing nonlinear dynamics of two-fluid Couette flows with asymptotic models, JFM 806, R1, 2016.
[2] Kalogirou A., Cimpeanu R., Blyth M., Asymptotic modelling and direct numerical simulations of multi-layer pressure-drive flows, EJMB 80, 2020.
[3] Cimpeanu R., Gomes S.N., Papageorgiou D.T., Active control of liquid film flows: beyond reduced-order models, Nonlinear Dynamics 104, 2021.
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.