UKAN+ Mathematical Acoustics Workshop

Title: Silence is bliss in a Platonic relationship: An overview of acoustic control, cloaking, low frequency transparency and elastostatic neutrality
Abstract: Over the last two decades a huge amount of work has focussed on cloaking – the ability to make an object invisible to a probing wave – via metamaterials of either passive or active type. Acoustic control via active sources has been a topic of interest for many decades now however, with the concept of active noise control invented by Paul Lueg and Henri Coanda in the 1930s. In this talk we will give a brief overview of acoustic control and cloaking, starting with Lueg’s work and the controversy associated with his patents. We will discuss modern developments in active acoustic control and specifically on active acoustic cloaking, including some recent theoretical work by the author and colleagues on a specific distribution of active sources at the vertices of Platonic solids. We show that this distribution provides some simplifications of the calculations for the required active source amplitudes in order to ensure a compact quiet zone.

Active control requires energy input to the system of course and in contrast, passive metamaterials allow the concept of cloaking via either redirection due to induced anisotropy and inhomogeneity, or wave interference at specific frequencies. We give an overview of this now well-known concept, with some examples. We then describe how the cloak can have quite simple homogeneous properties at very low frequencies. Finally, we provide what we hope is an interesting link of the low frequency problem to elastostatics (after all there should be one!). This allows us to make some connections to composite materials design to reduce the possibility of failure under load via the concept of a neutral inclusion. Given that the talk is late in the day I will endeavour to make this an overview talk, with a peppering of equations where necessary.

Title: Convergence of iterative solvers for the Helmholtz equation at high frequency.
Abstract: Many interesting applications require the solution of the linear wave equation in heterogeneous media and/or complicated geometry – for example forward and inverse scattering problems in acoustics or electromagnetics. When the data oscillates on a restricted range of frequencies, application of Fourier transform can remove the time dependence, leading to the Helmholtz equation – an indefinite linear second order elliptic PDE. However (at high frequency), this equation is non-coercive (in standard settings) and has highly oscillatory solutions.

To compute solutions, fine discretizations (finite element methods) are required, resulting in (sparse) systems of linear equations with millions of unknowns and highly indefinite system matrices. Because of the system size and structure, application of modern direct methods (i.e., clever variants of Gaussian elimination) are problematic, so there is great interest in finding iterative methods which compute the solution via a sequence of `local’ approximations, (so-called `domain
decomposition’ methods). Such methods are well-suited to implementation on modern parallel hardware and the search for fast convergent iterative methods for the Helmholtz equation (guaranteed by theory) is thus a very active current research topic in numerical analysis/scientific computing.

In the low-frequency case, the PDE behaves like the Poisson equation, the linear systems are symmetric positive definite, and many `optimal’ methods (the most famous being `multigrid’) are available for fast iterative solution. However these methods generally fail at high-frequency.

In the talk I’ll present some theory of the Helmholtz equation and its discretization, and a simple iterative procedure which forms the basis of several successful practical large-scale solvers. Then I’ll present some recent theory which explains the convergence properties of this method.

The techniques of analysis involve combining the theory of the Helmholtz equation at high frequency with numerical analysis of finite element and domain decomposition methods.

The work is joint with Shihua Gong (Chinese University of Hong Kong Shenzhen) and Euan Spence (Bath). Some results are also joint with Martin Gander (Geneva) and David Lafontaine (Toulouse).

Title: Log Kernel on a Square.
Abstract: A high performance technique for solving inhomogeneous PDEs is to combine convolution with a Green’s function for finding a particular solution with a boundary element method for computing the remaining homogenous solution. This is trivially parallelisable: if one subdivides the domain into elements (triangles / squares) then computing the convolution over each element is completely independent. There is, of course, a major drawback: Green’s functions in 2D have logarithmic singularities and so one has to apparently wrestle with singular quadrature. This talk over turns this drawback for rectangular elements: we show that convolution of log kernels with tensor products of Legendre polynomials satisfy simple recurrence relationships that enable one to effectively compute them explicitly, both on, near, and far away from the element.

We also review related results on intervals, corners, and curves. Finally, we demonstrate the techniques applied to Bessel kernels arising in acoustics, where the approach avoids the pollution effect.

Title: Multimode Multiple Scattering in Soft Media
Abstract: We will consider the propagation of ultrasound in nano/microparticle suspensions consisting of randomly distributed solid particles in a viscous liquid which is a system relevant to industrial process monitoring and to the characterisation of soft materials e.g. biological media, and potentially offers opportunities for interesting metamaterials. An important and interesting characteristic of these systems is the existence of two wavenumbers of very different magnitudes in the continuous phase – the compressional and shear modes. For typical liquids at MHz frequencies, and with colloidal particles, the dimensionless compressional wavenumber (based on particle size) may be small whereas the dimensionless shear wavenumber can range from small to large and is always much larger than the dimensionless compressional wavenumber. In addition, we have another length scale that affects the multiple scattering, i.e. the interparticle separation; its effects are usually accounted for through the concentration or number density but it may be helpful to consider these effects in terms of additional dimensionless wavenumbers relating to the interparticle distance.
In this talk, I present recent progress on the mathematical analysis of the system, highlighting the interpretation of the results in terms of the length scales and dimensionless wavenumbers and identifying the dominant contributions to the effective wavenumber. The analysis is based on ensemble averaged multiple scattering formulations using on multipole expansions and the T-matrix for individual scatterers.

Title: An open-waveguide resonator with extraordinary properties.
Abstract: We propose a simple geometric modification of the classical open-waveguide resonator which we show exhibits isolated dipolar scattering resonances of extremely high Q-factor and field enhancement (exponential in the waveguide’s aspect ratio). These ‘super-resonances’ are in addition to the conventional monopolar ‘quarter-wavelength’ resonances of an open waveguide (Q-factor and enhancement algebraic in the aspect ratio), which are virtually unaffected by the modification. The design, which is inspired by the physics of whispering-gallery modes and bound states in the continuum (embedded trapped modes), has many potential benefits—it enables low-multipolar-order super-scattering without requiring a large volume, or high-index materials, and is easy to tune. We will use matched asymptotic expansions, together with conformal mappings and integral balances, to analytically characterise the leaky eigenmodes of the proposed configuration, and their excitation by a plane wave, and compare the analytical predictions to numerical simulations. (Joint work with Richard Porter, University of Bristol.)

Title: Micro-resonant homogenisation: error analysis and two-scale interpolation operator.
Abstract: A motivation comes from wave propagation in periodic media with `soft’ inclusions surrounded by `stiff’ matrices, where the inclusions serve as subwavelength micro-resonators. This corresponds to PDE models with a special critical scaling between the small periodicity and the high contrast, and the resulting two scale approximations for the solutions remain intrinsically two-scale reflecting the underlying resonance effect. Mathematically this leads to studying asymptotic properties of Floquet-Bloch waves and spectrum. To construct approximations with controllably small errors, one first needs to recast potentially oscillatory input data as two-scale functions. This is achieved by specialising an abstract general theory recently developed by us [1] to a class of such problems, and results in a new two-scale interpolation operator which is a two-scale version of the classical Whittaker-Shannon interpolation formula. Joint work with Ilia Kamotski and Shane Cooper.

[1] S Cooper, I Kamotski, VP Smyshlyaev, Uniform asymptotics for a family of degenerating variational problems and error estimates in homogenisation theory. https://arxiv.org/abs/2307.13151

Title: Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem.
Abstract: When solving the Helmholtz transmission problem with boundary integral equations, for certain coefficients and geometries the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the underlying PDE problem. This talk will explain why and when these “spurious quasi-resonances” occur, and will propose modified boundary integral equations that are not affected by them.

Title: Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies.
Abstract: For $h$-FEM discretisations of the Helmholtz equation with wavenumber $k$, we obtain $k$-explicit analogues of the classic local FEM error bounds of [1], [2], [3, §9], showing that these bounds hold with constants independent of $k$, provided one works in Sobolev norms weighted with $k$ in the natural way.
We prove two main results: (i) a bound on the local $H^1$ error by the best approximation error plus the $L^2$ error, both on a slightly larger set, and (ii) the bound in (i) but now with the $L^2$ error replaced by the error in a negative Sobolev norm. The result (i) is valid for shape-regular triangulations, and is the $k$-explicit analogue of the main result of [1]. The result (ii) is valid when the mesh is locally quasi-uniform on the scale of the wavelength (i.e., on the scale of $k^{-1}$) and is the $k$-explicit analogue of the results of [2], [3, §9].
Since our Sobolev spaces are weighted with $k$ in the natural way, the result (ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo low frequencies (i.e., frequencies$≲k$). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error.

[1] A. Demlow, J. Guzmán, and A. H. Schatz. Local energy estimates for the finite element method on sharply varying grids. Mathematics of Computation, 80(273):1–9, 2011.
[2] J. A. Nitsche and A. H. Schatz. Interior estimates for Ritz-Galerkin methods. Mathematics of Computation, 28(128):937–958, 1974.
[3] L. B. Wahlbin. Local behavior in finite element methods. In Handbook of numerical analysis, Vol. II, pages 353–522. North-Holland, Amsterdam, 1991.

Title: Fast approximation of far-field patterns for polygonal scatterers.
Abstract: For problems of time-harmonic scattering by polygonal obstacles, ’embedding formulae’ express the far-field pattern induced by any incident plane wave in terms of the far-field patterns for a relatively small (frequency-independent) set of ‘canonical’ incident angles. Although these remarkable formulae are exact in theory, they are highly sensitive to numerical errors in the canonical far-field patterns, producing arbitrarily large numerical errors at certain observation angles. This makes embedding formulae unsuitable for practical applications.

In recent work with Steve Langdon, we have derived a reformulation of the embedding formula for rational polygons, derived initially by Biggs ’06, based on earlier work of Craster et al. ’03. Our new approach uses a complex contour integral representation for the far-field pattern. If the contour is chosen correctly, the numerical error scales linearly with the error of the canonical far-field patterns – a property not enjoyed by previous embedding formulae.

In this talk, I will introduce embedding formulae, demonstrating why these break down in practice. I will then present our new approach alongside error estimates. Then, I will discuss efficient and accurate implementation techniques, ending with some numerical experiments at high frequencies.

Title: A two Complex Variable Approach to the Right-Angled No-Contrast Penetrable Wedge Diffraction Problem.
Abstract: We study the diffraction problem resulting from the interaction of a time-harmonic plane-wave with a right-angled no-contrast penetrable wedge, by following a two-complex-variable approach. Central to this approach is that the physical problem leads to a Wiener-Hopf type equation in two complex variables. This equation involves some unknown ‘spectral’ functions. We present these spectral functions’ singularities and discuss their physical importance. Namely, we can exploit our knowledge on the singularity structure to obtain closed-form far-field asymptotics of the diffracted physical fields, wherein the (cylindrical and lateral) diffraction coefficients are expressed in terms of these spectral functions.

Title: Diffraction by a set of collinear cracks on a square lattice: an iterative Wiener-Hopf method approach.
Abstract: The diffraction of a time-harmonic plane wave on collinear finite defects in a square lattice is studied. This problem is reduced to a matrix Wiener-Hopf equation with infinitely growing factors. This work aims to adapt the recently developed iterative Wiener-Hopf method to this situation. The method was motivated by wave scattering in continuous media but can also be employed in a discrete lattice setting. The complexity of the algorithm is independent of the length of the crack, so it can provide high effectiveness for numerical implementation and fast convergence.

Title: High frequency homogenization for periodic dispersive media.
Abstract: High-frequency homogenization is used to study dispersive media, containing inclusions placed periodically, for which the properties of the material depend on the frequency (Lorentz or Drude model with damping, for example). Effective properties are obtained near a given point of the dispersion diagram in frequency-wavenumber space in both one and two dimensions. The asymptotic approximations of the dispersion diagrams, and the wavefields, so obtained are then validated via comparison with finite element method simulations.