Diffusive reflection of wave-fields from rough surfaces is ubiquitous in applications, providing an important propagation mechanism in wireless communication systems, for example. We describe this phenomenon here using the formalism of Wigner functions, which allows us to represent incoherent incident and reflected wave fields in a form that has a natural limit as phase-space densities of rays. A propagator of Wigner functions, such as described in this paper, can therefore effectively be used to incorporate intrinsically interference-dominated rough surface reflection into practical simulation algorithms for ray tracing. We demonstrate this approach by incorporating the Wigner propagator into an application of the Dynamical Energy Analysis (DEA) method, which allows efficient calculation of ray densities in enclosed, multi-reflective environments.

We experimentally realize an angular filter using a microwave network of coaxial cables, guided by the theoretical framework of quantum graphs with beyond-nearest-neighbor coupling. Our setup implements a resonant metamaterial interface that selectively transmits waves only at discrete output directions. The experimental results confirm the presence of angular filtering predicted in quantum graph models. A signal introduced at one side of a finite square lattice network is transmitted through the filter only along certain directional components at specific frequencies. The effect is caused by a resonance-induced switching of boundary conditions at the filter interface. Our findings validate theoretical predictions and demonstrate the practical applicability of graph-based metamaterial concepts in real-world wave systems.

Modelling the response of complex built-up structures under the influence of distributed and correlated high-frequency force fields is an important aspect in many structural dynamics applications. A prime example of such a forcing is the pressure field applied by a Turbulent Boundary Layer (TBL) on an aircraft in flight. Previous studies focus on the total vibrational power input into the structure and do not consider spatial variations or directional components of the energy input introduced due to correlated forcing. We will close this gap by demonstrating how correlated force fields can be implemented as the source term in a Dynamical Energy Analysis (DEA) treatment determining the resulting vibrational energy distribution in a complex structure. Using Wigner-transformation techniques, we convert force-correlation functions (such as those routinely used to describe TBL excitations) into directional ray-source terms which then provide the source for a DEA treatment. Results are presented for the vibrational energy distribution across a flat plate excited by a fully formed, stationary TBL under a variety of flow conditions. We note that even though the excitation is spatially uniform across the plate, there is a preference for energy to flow in the downstream direction. This leads to a marked enhanced of the vibrational excitation at the trailing edge of the plate.

We leverage quantum graph theory to quickly and accurately characterize acoustic metamaterials comprising networks of interconnected pipes. Anisotropic bond lengths are incorporated in the model that correspond to space-coiled acoustic structures to exhibit dispersion spectra reminiscent of hyperbolic metamaterials. We construct two metasurfaces with embedded graph structure and, motivated by the graph theory, infer and fine-tune their dispersive properties to engineer nonresonant negative refraction of acoustic surface waves at their interface. Agreement between the graph model, full-wave simulations, and experiments bolsters quantum graph theory as a new paradigm for metamaterial design.

Predicting the vibroacoustic behaviour of structures in contact with water and at high frequencies can be a complex task due to the fluid-structure interaction. Computation time increases with frequency and the complexity of the structure. Being able to predict the sound radiated by a naval vehicle in contact with water could help in improving passengers' comfort as well as the impact on the marine fauna. We use a prediction method called Dynamic Energy Analysis (DEA), so far emloyed mainly to assess the vibrational energy levels in the structure itself. DEA is a phase-space based ray tracing method determining energy densities in terms of the ray densities in a structure. The method can be applied on FEM-meshes and thus needs no SEA type substructuring. We expand the DEA methodology here to take into account radiation into the fluid in contact with the structure. From structure borne DEA calculations, we can determine directional sound radiation patterns both inside and outside the structure. We will demonstrate the new technique using some simple model systems.

Dynamical Energy Analysis (DEA) is an approach used to compute the vibro-acoustic response of complex structures to externally applied high-frequency excitation. DEA tracks the global energy flow in a structure in terms of a local ray tracing approach implemented on meshes. Due to its in-built directional set-up, the approach is ideally suited for modelling the response of structures to directional excitation. Examples thereof are dipole sources or - more generally - correlated and distributed source excitation such as due to a Turbulent Boundary Layer (TBL) excitation across an aircraft wing during flight. We will demonstrate here how such directional sources are implemented in a DEA setting. TBL induced sources or general correlated sources can be described in terms of appropriate force or wave correlation functions (CF). Using Wigner-transformation, these CFs can be associated with a directional energy flow source. Vibrational energy field results are presented for a pair of phase-locked point forces. Results are also presented for 9 point-sources, which demonstrate the ability to channel the energy flow away from the sources by carefully choosing their relative phases. An implementation of this approach within DEA is also presented, with results demonstrating strong agreement with direct calculations.

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.

Reconfigurable intelligent surfaces (RISs) got an increasing amount of attention recently for their capability to provide remarkable performance enhancements in the actual and next 6G generation wireless systems. These programmable metasurfaces can control the electromagnetic propagation and achieve the desired wavefront by tuning the local reflection phase of its elements. Accurately estimation of the EM propagation in the RIS-assisted radio channel represents a challenge and becomes crucial for Telecom operators for the management and the optimized allocation of the radio resources. In this context, the authors propose the use of a raytracing method operating in the phase-space setting, the Dynamical Energy Analysis (DEA), as a coverage radio planning tool. DEA can take into account the EM interaction between RISs and the surrounding environment. The leading characteristics of DEA are discussed highlighting the motivations that make it a suitable tool for these purposes. Moreover, some preliminary results of the RIS integration within the DEA algorithm will be presented in some numerical examples.

For layered metamaterial devices, the reflection and transmission coefficients at an interface typically depend on the properties of the coupling between different layers. In this paper, we set out to engineer the reflection/transmission behaviour at boundaries to obtain desirable properties such as achieving total reflection and transmission. Based on the quantum graph formulation for modelling metamaterials developed in Sci. Rep. 12(1), 18006 (2022), we tailor the interface reflection and transmission coefficients by patterning the boundary with resonant elements at each interface vertex. By tuning the internal lengths of the resonant elements, we demonstrate both minimization and maximization of the reflection coefficient via a scattering formulation. In addition, we present an interface set-up incorporating beyond-nearest-neighbour connections, which yields narrow-band transmission for certain angles only, creating an angular filtering interface.

In this work we present a three step procedure for generating a closed form expression of the Green’s function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard (2001 Phys. Rev. E 65 016205) and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate. The derivation of the Green’s function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green’s function is derived. The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green’s functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.

The use of guided waves to identify damage has become a popular method due to its robustness and fast execution, as well as the advantage of being able to inspect large areas and detect minor structural defects. When a travelling wave on a plate interacts with a defect, it generates a scattered field that will depend on the defects geometry. By analysing the scattered field, one can thus characterize the type and size of the plate damage. A Bayesian framework based on a guided waves interaction model for damage identification of infinite plate for the first time is presented here. A semi-analytical approach based on the lowest order plate theories is adopted to obtain the scattering features for damage geometries with circular symmetry, resulting in an efficient inversion procedure. Subsequently, ultrasound experiments are performed on a large aluminium plate with a circular indentation to generate wave reflection and transmission coefficients. With the aid of signal processing techniques, the effectiveness and efficiency of the proposed approach are verified. A full finite element model is used to test the damage identification scheme. Finally, the scattering coefficients are reconstructed, reliably matching the experimental results. The framework supports digital twin technology of structural health monitoring.

Since the turn of the century, metamaterials have gained a large amount of attention due to their potential for possessing highly nontrivial and exotic properties-such as cloaking or perfect lensing. There has been a great push to create reliable mathematical models that accurately describe the required material composition. Here, we consider a quantum graph approach to metamaterial design. An infinite square periodic quantum graph, constructed from vertices and edges, acts as a paradigm for a 2D metamaterial. Wave transport occurs along the edges with vertices acting as scatterers modelling sub-wavelength resonant elements. These resonant elements are constructed with the help of finite quantum graphs attached to each vertex of the lattice with customisable properties controlled by a unitary scattering matrix. The metamaterial properties are understood and engineered by manipulating the band diagram of the periodic structure. The engineered properties are then demonstrated in terms of the reflection and transmission behaviour of Gaussian beam solutions at an interface between two different metamaterials. We extend this treatment to N layered metamaterials using the Transfer Matrix Method. We demonstrate both positive and negative refraction and beam steering. Our proposed quantum graph modelling technique is very flexible and can be easily adjusted making it an ideal design tool for creating metamaterials with exotic band diagram properties or testing promising multi-layer set ups and wave steering effects.

Reconfigurable intelligent surfaces are planar structures that dynamically change their reflection or refraction behaviour to engineer complex propagation environments. A physics-based modelling framework is formulated that accounts for the scattered electromagnetic field by an array of reconfigurable unit cells. The model is grounded on finitesize electric/magnetic surface current densities, and develops a wave-dynamical phase space, or angular-positional, representation of scattered waves. The phase-space representation facilitates the integration of intelligent surfaces within Eulerian ray-tracing methods, such as the Dynamical Energy Analysis. The framework is used for evaluating and optimising the design of beam-splitters and anomalous reflectors. A Wigner transform approach is used to devise the field-based phase-space representation. Obtained results are the first step towards the development of a flexible and efficient coverage planning tool for multi-RIS mobile networks.

Reconfigurable intelligent surfaces have been recently investigated for their potentials to offer significant performance improvements in the next generation wireless telecommunication systems (5G and beyond / 6G). Intelligent surfaces are programmed to control the electromagnetic propagation and obtain the desired wavefront by tuning the local reflection phase of unit elements. Predicting the electromagnetic propagation in the RIS-assisted wireless channel accurately is a significant challenge for researchers and becomes crucial for Telecom operators to properly allocate the radio resources. We propose the use of an Eulerian ray-tracing method, the Dynamical Energy Analysis (DEA), as a coverage planning tool capable of account for the EM interaction between reconfigurable intelligent surfaces and the surrounding environment. The main characteristics that make DEA suitable for this purpose are discussed and some preliminary results of the reflective surface integration within the DEA code will be presented.

Controlling and simulating the sound radiating from complex structures is of importance in many engineering applications. We calculate the radiated acoustic power from plates with diffuse bending vibrations. We characterise the diffuse field by a two-point correlation function (CF) of normal velocities. Given the relation between field–field CFs and ray-dynamical phase space densities, the approach taken here offers a basis for coupling structure borne ray-tracing techniques with acoustic radiation. At the same time, it caters for stochastic, noisy driving of such systems. The results for the radiation efficiency of a plate are presented in an asymptotic form analogous to the Weyl formula for the density of states. Leading contributions from the plate interior and its boundary are derived, with corner corrections also being given for particular boundary conditions and right-angled corners. A notable feature of this analysis is that the bulk contribution vanishes below a critical frequency, and the asymptotic estimate of radiated power then leads with a boundary contribution. This is shown to agree well with a more traditional calculation based on modal analysis in the special case of a rectangular plate.

We present a numerical method for computing reflection and transmission coefficients at joints connecting composite laminated plates. The method is based on modelling joints with finite elements with boundary conditions given by the solutions of the wave finite element method for the plates in the infinite half-spaces connected to the joint. There are no restrictions on the number of plates, inter-plate angles, and material parameters of individual layers forming the composite. An L-shaped laminated plate junction is discussed in more detail. Comparisons of numerically predicted scattering coefficients with semi-analytical solutions for the selected structures are presented. The results obtained are essential for statistical energy analysis and dynamical energy analysis based calculations of the wave energy distribution in full built-up structure.

Future aircraft concepts utilizing innovative lightweight structures and novel propulsion concepts are a necessity for long term sustainable air travel. These concepts pose new challenges for the vibro-acoustic assessment of cabin structures and the associated noise impact on passengers. Finite Element (FE) models derived from aircraft pre-design data are not optimized for use in acoustic analyses, i.e. the mesh is too coarse to provide meaningful results while setting up Statistical Energy Analysis models for this specific purpose is adding another time-consuming step. A possible alternative, Discrete Energy Analysis (DEA), is evaluated. This method allows to calculate the acoustic behavior of thin-walled structures in higher frequency ranges simply using existing FE meshes. In this paper an experimental lightweight aluminum structure and its respective FE model is investigated for a frequency range up to 5000 Hz. A comparison in terms of vibrational energy between DEA, FE and measurement results are presented. Finally, a lower-bound frequency range is identified in which DEA and FEM correlate and thus allow a substitution for further simulations at higher frequencies.

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.

We present a semi-analytical method for computing the reflection and transmission coefficients at joints connecting an arbitrary number of semi-infinite orthotropic plates based on the line-junction approximation. We use a wave approach and describe reflection, transmission and mode conversion between eigenmodes of the vibro-acoustic equations for orthotropic plates. A detailed derivation is presented here for the first time for an arbitrary number of plates meeting at the junction and without restrictions on the orientation of the principal material axes both with respect to the junction and with respect to the orientation in different plates. The approach is discussed for two specific example configurations, namely an L- and a T-shaped orthotropic plate junction. Furthermore, the scattering coefficients for a rib-stiffened orthotropic plate are derived, and the occurrence of resonance phenomena is discussed.

The importance of filters in signal processing abound in real-life applications such as audio electronics and power distribution networks. We introduce an efficient concept of constructing reconfigurable cable networks for interference control. The method utilises quantum-graph formalisms on composite nodes to implement reconfigurable, compact, wideband filters. By using a cascade of loop networks as building blocks, we show that such meta-networks can provide a flexible way of suppressing unwanted signals thereby increasing the efficiency of the underlying networks.Numerical results show that microwave interference can be filtered to allow only specific narrow (or wide) band frequencies to be fully transmitted while suppressing the other frequency bands. For example, a wave of frequency 0≤ f≤ 6 GHz can be filtered using a cascade of three-loop networks with cable lengths 0.0001 ≤ l ≤ 0.05 m. By tuning one of the constituent cables, we are able to achieve maximum power transmission on a specific set of frequencies while completely suppressing signals of unwanted frequencies. The present paper shows how to implement narrow-, medium- and wide-band bandpass filters by adopting a simple and easy-to-design cable topology for the reconfigurable filter. The proposed solution can be easily integrated with solutions, including varactor diode and phase-shifter based architectures, to achieve practical implementations.