High-frequency wave energy transport is investigated in bounded domains with piecewise constant wave speed using a radiative transfer equation based model. We develop a powerful and efficient discontinuous Galerkin framework including both Fourier and discrete ordinate discretisations in the direction variable, which is widely applicable for wave problems arising within acoustics, elasticity and electromagnetics. Key features of the discontinuous Galerkin scheme implementation include fast exact quadrature for piecewise polynomial basis functions on general polygonal elements, together with an efficient parallel implementation. The scheme is benchmarked against both exact solutions and previous simulation results, demonstrating optimal convergence of the method with refinement of the grid as well as demonstrating convergence when increasing the spatial polynomial degree (alongside refinement of the directional discretisation) on a fixed coarse polygonal grid. As a final test, the scheme is applied to an example with more intricate geometric features requiring finer meshing and having millions of degrees of freedom. As well as producing converged results within competitive simulation times, the method was also used to investigate the inclusion of evanescent waves within the reflection and transmission laws for flexural plate motion between regions of different plate thickness. The expected trend was observed, with the inclusion of evanescent waves becoming less significant as the excitation frequency is increased.

A new approach for solving the optical inverse problem of quantitative photoacoustic tomography is introduced, which interpolates between the well-known diffusion approximation and a radiative transfer equation (RTE) based model. The proposed formulation combines a spatial finite volume scheme with a truncated Fourier expansion in the direction variable for the RTE. The finite volume scheme provides a natural and simple approach for representing piecewise constant image data modelled using transport equations. The truncated Fourier expansion in the direction variable facilitates the interpolation between the diffusion approximation at low order, and the full radiative transfer model as the truncation limit N ->infinity. It is therefore possible to tune the precision of the model to the demands of the imaging application, taking N = 1 for cases when the diffusion approximation would suffice and increasing the number of terms otherwise. We will then utilise the non-linear optimisation functionality of Matlab to address the corresponding large-scale non-linear inverse problem using gradient based quasi-Newton minimisation via the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm. Numerical experiments for two test-cases of increasing complexity and resolution will be presented, and the effect of logarithmically rescaling the problem data on the accuracy of the reconstructed solutions will be investigated. We will focus on cases where the diffusion approximation is not sufficient to demonstrate that our approach can provide significant accuracy gains with only a modest increase in the number of Fourier terms included.

We examine the scattering of Ostrovsky wave packets, generated from an incident solitary wave, in a two-layered waveguide with a delamination in the centre and soft (imperfect) bonding on either side of the centre. The layers of the waveguide are assumed to consist of different materials, and the strains are described by a system of coupled Boussinesq equations. A semi-analytical approach consisting of matched asymptotic multiple-scale expansions is applied, leading to Ostrovsky equations in soft-bonded regions and Korteweg-de Vries (KdV) equations in the delaminated region. This semi-analytical method has good agreement with direct numerical simulations, validating the approach as well as providing a significant reduction in the computation time. In the delaminated regions, Ostrovsky wave packets evolve into a train of solitary waves, which subsequently evolve into Ostrovsky wave packets in the second bonded region. Analysis of the phase shift in the wave packet, introduced from the delaminated region, allows us to predict both the position and the length of the delamination; the first time this has been achieved using nonlinear waves. These results motivate experiments to validate the theoretical results, with the aim of creating a tool to monitor the integrity of layered structures.

Dynamical Energy Analysis was introduced in 2009 as a novel method for predicting high-frequency acoustic and vibrational energy distributions in complex engineering structures. In this paper we introduce the first time-dependent Dynamical Energy Analysis method. Time-domain models are important in numerous applications including sound simulation in room acoustics, predicting shock-responses in structural mechanics and modelling electromagnetic scattering from conductors. The first step is to reformulate Dynamical Energy Analysis in the time-domain by means of a convolution integral operator. We are then able to employ the Convolution Quadrature method to provide a link between the previous frequency-domain implementations of Dynamical Energy Analysis and fully time-dependent solutions by means of the Z-transform. By combining a modified multistep Convolution Quadrature approach for the time discretisation, together with Galerkin and Petrov-Galerkin methods for the space and momentum discretisations, respectively, we are able to accurately track the propagation of high-frequency transient signals through phase-space. The implementation here is detailed for finite two-dimensional spatial domains and we demonstrate the versatility of our approach by performing a range of numerical experiments for regular, non-convex and irregular geometries as well as different types of wave source.

Ray-tracing is a well established approach for modelling wave propagation at high frequencies, in which the ray trajectories are defined by a Hamiltonian system of ODEs. An approximation of the wave amplitude is then derived from estimating the density of rays in the neighbourhood of a given evaluation point. An alternative approach is to formulate the ray-tracing model directly in terms of the ray density in phase-space using the Liouville equation. The solutions may then be expressed in integral form using the Frobenius-Perron (F-P) operator, which is a transfer operator transporting the ray density along the trajectories. The classical approach for discretising such operators dates back to 1960 and the work of Stanislaw Ulam. The convergence of the Ulam method has been established in some cases, typically in low dimensional settings with continuous densities and hyperbolic dynamics. In this chapter, we outline some recent work investigating the convergence of the Ulam method for ray tracing in triangular billiards, where the dynamics are parabolic and the flow map contains jump discontinuities.

In this paper we examine the effect of delamination on wave scattering, with the aim of creating a control measure for layered waveguides of various bonding types. Previous works have considered specific widths of solitary waves for the simulations, without analysing the effect of changing the soliton parameters. We consider two multi-layered structures: one containing delamination ‘sandwiched’ by perfect bonding and one containing delamination but ‘sandwiched’ by soft bonding. These structures are modelled by coupled Boussinesq-type equations. Matched asymptotic multiple-scale expansions lead to coupled Ostrovsky equations in soft bonded regions and Korteweg-de Vries equations in the perfectly bonded and delaminated region. We use the Inverse Scattering Transform to predict the behaviour in the delaminated regions. In both cases, numerical analysis shows that we can predict the delamination length by changes in the wave structure, and that these changes depend upon the Full Width at Half Magnitude (FWHM) of the incident soliton. In the case of perfect bonding, we derive a theoretical prediction for the change and confirm this numerically. For the soft bonding case, we numerically identify a similar relationship using the change in amplitude. Therefore we only need to compute one curve to determine the behaviour for any incident solitary wave, creating a framework for designing measurement campaigns for rigorously testing the integrity of layered structures.

In this chapter, we compare the performance of two recently proposed hybrid methods for numerically solving the wave equation in two spatial dimensions. The convolution quadrature (CQ) method is employed for the time discretisation, which can be used to transform the original time-domain problem into a system of frequency domain Helmholtz problems with complex wavenumbers.

Ray flow methods are an efficient tool to estimate vibro-acoustic or electromagnetic energy transport in complex domains at high-frequencies.Here, a Petrov-Galerkin discretization of a phase-space boundary integral equation for transporting wave energy densities on two-dimensional surfaces is proposed.The directional dependence of the energy density is approximated at each point on the boundary in terms of a finite local set of directions propagating into the domain.The direction of propagation can be preserved for transport across multi-component domains when the directions within the local set are inherited from a global direction set.The range of applicability and computational cost of the method will be explored through a series of numerical experiments, including wave problems from both acoustics and elasticity in both single and multi-component domains.The domain geometries considered range from both regular and irregular polygons to curved surfaces, including a cast aluminium shock tower from a Range Rover car.

A spatially periodic voltage was used to create a dielectrophoresis induced periodic micro wrinkle deformation on the surface of a liquid film. Optical Coherence Tomography provided the equilibrium wrinkle profile at submicron accuracy. The dynamic wrinkle amplitude was derived from optical diffraction analysis during sub-millisecond wrinkle formation and decay, after abruptly increasing or reducing the voltage, respectively. The decay time constant closely followed the film thickness dependence expected for surface tension driven viscous levelling. Modelling of the system using numerical solution of the Stokes flow equations with electrostatic forcing predicted that wrinkle formation was faster than decay, in accord with observations.

Dynamical energy analysis (DEA) is a computational method to address high-frequency vibro-acoustics in terms of ray densities. It has been used to describe wave equations governing structure-borne sound in two-dimensional shell elements as well as three-dimensional electrodynamics. To describe either of those problems, the wave equation is reformulated as a propagation of boundary densities. These densities are expressed by finite dimensional approximations. All use-cases have in common that they describe the resulting linear problem using a very large matrix which is block-sparse, often real-valued, but non-symmetric. In order to efficiently use DEA, it is therefore important to also address the performance of solving the corresponding linear system. We will cover three aspects in order to reduce the computational time: The use of preconditioners, properly chosen initial conditions, and choice of iterative solvers. Especially the aspect of potentially reusing preconditioners for different input parameters is investigated.

We propose two hybrid convolution quadrature based discretizations of the wave equation on interior domains with broadband Neumann boundary data or source terms. The convolution quadrature method transforms the time-domain wave problem into a series of Helmholtz problems with complex-valued wavenumbers, in which the boundary data and solutions are connected to those of the original problem through the Z-transform. The hybrid method terminology refers specifically to the use of different approximations of these Helmholtz problems, depending on the frequency. For lower frequencies, we employ the boundary element method, while for more oscillatory problems, we develop two alternative high frequency approximations based on plane wave decompositions of the acoustic field on the boundary. In the first approach, we apply dynamical energy analysis to numerically approximate the plane wave amplitudes. The phases will then be reconstructed using a novel approach based on matching the boundary element solution to the plane wave ansatz in the frequency region where we switch between the low and high frequency methods. The second high frequency method is based on applying the Neumann-to-Dirichlet map for plane waves to the given boundary data. Finally, we investigate the effectiveness of both hybrid approaches across a range of numerical experiments.

The computation of wave-energy distributions in the mid-to-high frequency regime can be reduced to ray-tracing calculations. Solving the ray-tracing problem in terms of an operator equation for the energy density leads to an inhomogeneous equation which involves a Perron-Frobenius operator defined on a suitable Sobolev space. Even for fairly simple geometries, let alone realistic scenarios such as typical boundary value problems in room acoustics or for mechanical vibrations, numerical approximations are necessary. Here we study the convergence of approximation schemes by rigorous methods. For circular billiards we prove that convergence of finite-rank approximations using a Fourier basis follows a power law where the power depends on the smoothness of the source distribution driving the system. The relevance of our studies for more general geometries is illustrated by numerical examples.

In this paper we address the calculation of vibrational energy distributions in built-up mechanical structures at high-frequencies. Dynamical Energy Analysis (DEA) has been established as a mesh-based ray-tracing method working at mid-to-high excitation frequencies for structure-borne sound on complex vehicle structures. It extends both ray-tracing and statistical approaches like Statistical Energy Analysis (SEA). Here we investigate the influence of stiffeners on the damping in these computations. As these pose a mid-frequency problem in most applications, it is necessary to be able to well approximate their influence in DEA.

A recently proposed phase-space boundary integral model for the stochastic propagation of ray densities is presented and, for the first time, explicit connections between this model and parametric uncertainties arising in the underlying physical model are derived. In particular, an asymptotic analysis for a weak noise perturbation of the propagation speed is used to derive expressions for the probability distribution of the phase-space boundary coordinates after transport along uncertain, and in general curved, ray trajectories. Furthermore, models are presented for incorporating geometric uncertainties in terms of both the location of an edge within a polygonal domain, as well as small scale geometric fluctuations giving rise to rough boundary reflections. Uncertain source terms are also considered in the form of stochastically distributed point sources and uncertain boundary data. A series of numerical experiments is then performed to illustrate these uncertainty models in two-dimensional convex polygonal domains. (c) 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license. (http://creativecommons.org/licenses/by/4.0/)

The effect of an externally applied electric field on the motion of an interface between two viscous dielectric fluids is investigated. We first develop a powerful, efficient and widely applicable boundary integral method to compute the interface dynamics in a general multiphysics model comprising coupled Laplace and Stokes flow problems in a periodic half-space. In particular, we exploit the relevant Stokes and Laplace Green's functions to reduce the problem to one defined on the interfacial part of the domain alone. Secondly, motivated by recent experimental work that seeks to underpin the development of switchable liquid optical devices, we concentrate on a fluid-air interface and derive asymptotic approximations suitable to describe the behaviour of a thin film of fluid above an array of electrodes. In this case, the problem is reduced to a single nonlinear partial differential equation describing the film height, coupled to the electrostatic problem via suitable numerical solution or via an asymptotic formula for electrostatic forcing. Comparison against numerical simulations of the full problem shows that the reduced models successfully capture key features of the film dynamics in appropriate regimes; all three approaches are shown to reproduce experimental results.

Dynamical Energy Analysis (DEA) has been established as a mesh-based ray tracing method working at mid-to-high excitation frequencies for structure-borne sound on complex vehicle structures. Unless standard ray-tracing used, for example, in room acoustics and following long rays through a series of specular reflections, DEA is based on an integral representation by (a) reformulating the energy transport problem in terms of ray densities and linear matrix equations and by (b) applying the ray tracing directly on preexisting meshes used also in, for example, FEM calculations. The advantage is that DEA can be used as a drop-in replacement for FEM tool working on the same meshes but determining results at very high frequencies. In order to achieve this, a so-called transfer matrix has to be created and later solved for the energy equilibrium distribution. In this presentation, we address recent developments in making DEA an efficient black-box tool for mid- to high-frequency vibrations including convergence issues and suitable preprocessor applications, sound radiation from DEA calculations as well as verification and validation studies.

Dynamical Energy Analysis (DEA) is a mesh-based high frequency method for modelling structure borne sound in complex built-up structures. Vibro-acoustic simulations are performed directly on finite element meshes, circumventing the need for re-modelling strategies. DEA provides detailed spatial information about the vibrational energy distribution within a complex structure in the mid-to-high frequency range. We will present here progress in the development of the DEA method towards handling complex FE-meshes including Rigid Body Elements and sound radiation. We also provide, for the first time, a detailed comparison of the simulations with measurements on a complex engineering structure consisting of the chassis and cabin of a tractor. Both structure borne vibrations and sound pressure levels (SPL) inside the cabin were considered. For the latter, a combined DEA/SEA analysis has been developed. The simulation results compare favourably with measurement results, both for vibration levels measured across the structure and for SPLs inside the cabin.

In this paper mid-frequency effects below the ring frequency of curved plates for a cylindrical region smoothly connected to two flat plates using Donnell shell theory has been analysed. We perform ray tracing based on Hamilton's equations derived in the short wavelength regime for bending incident waves. Simple ray tracing gives either total reflection or total transmission; the numerical solution of the full wave equations shows in contrast a smooth transition and exhibits resonant states. In this paper a method that captures wave effects in a ray-tracing treatment on curved plates is presented. We use graph models to account for resonant tunnelling in such curved plates. We successfully find a theoretical expression for the scattering matrix for such bending rays which accounts for tunnelling mediated by resonant states. Our model agrees well with the full wave solution.

A phase-space boundary integral method is developed for modelling stochastic high-frequency acoustic and vibrational energy transport in both single and multi-domain problems. The numerical implementation is carried out using the collocation method in both the position and momentum phase-space variables. One of the major developments of this work is the systematic convergence study, which demonstrates that the proposed numerical schemes exhibit convergence rates that could be expected from theoretical estimates under the right conditions. For the discretisation with respect to the momentum variable, we employ spectrally convergent basis approximations using both Legendre polynomials and Gaussian radial basis functions. The former have the advantage of being simpler to apply in general without the need for preconditioning techniques. The Gaussian basis is introduced with the aim of achieving more efficient computations in the weak noise case with near-deterministic dynamics. Numerical results for a series of coupled domain problems are presented, and demonstrate the potential for future applications to larger scale problems from industry.