Modelling and simulation is becoming ever more important in designing mechanical structures –such as cars, airplanes or ships. Both the static and dynamic properties, but also the interaction of a structure with the surrounding environment – be it aerodynamic properties or crash behaviour – is now routinely investigated through simulations. Virtual testing has thus become an essential part of the design cycle. In particular, the Finite Element Method (FEM) is used universally across industries and a range of sophisticated commercial software tools exist. There is also a need for modelling *noise and vibration* in the mid-to-high frequency regime, mostly for passenger safety and comfort. However, a straightforward application of FEM at high frequencies (starting for cars typically between 500Hz and 1kHz) is problematic as results become unreliable and the model sizes become prohibitively large. A new dynamical theory—*Dynamical Energy Analysis (DEA)* – can deal with the full complexity of the structure on a reasonable computational time scale and has the potential to make a full ray tracing approach viable for modelling structure-borne vibrations including reflection, transmission and shell effects. Integral formulations for the propagation of ray densities are thus becoming an interesting alternative to statistical approaches such as the *Statistical Energy Analysis (SEA)* or similar Power Balance Methods due to their ability to work directly on FE-meshes – with a resulting ease of implementation. However, there is plenty of ground to cover before this method has been fully developed and its potential for applications has been exhausted.

**DEA – Numerical Implementation**

In 2009 new approach called *Dynamical Energy Analysis (DEA)* has been proposed in Nottingham determining the distribution of mechanical and acousticwave energy in complex built-up structures. The technique interpolates between standard *Statistical Energy Analysis (SEA)* and full ray tracing containing both these methods as limiting cases. By writing the flow of ray trajectories in terms of linear phase space operators, it is suggested to reformulate ray-tracing algorithms in terms of boundary operators containing only short ray segments. SEA can now be identified as a low-resolution, ray-tracing algorithm and typical SEA assumptions can be quantified in terms of the properties of the ray dynamics. The new technique enhances the applicability of standard SEA considerably by systematically incorporating dynamical correlations wherever necessary. Some of the inefficiencies inherent in typical ray-tracing methods can be avoided using only a limited amount of the geometrical ray information. The new dynamical theory thus provides a universal approach towards determining wave energy distributions in complex structures in the high-frequency limit.

**Related Publications:**

- High-frequency structure- and air-borne sound transmission for a tractor model using Dynamical Energy Analysis, Timo Hartmann, Satoshi Morita and Gregor Tanner, to appear in Wave Motion (2018)
- Boundary element Dynamical Energy Analysis: A versatile high-frequency method applicable for two or three dimensional problems, D. J. Chappell, and G. Tanner, Proc. 18th Int. Congress on Sound and Vibrations (ICSV18, Rio de Janeiro, 2011), 8 pages
- Dynamical energy analysis for built-up acoustic systems at high frequencies, D. J. Chappell, S. Giani and G. Tanner, Journal of the Acoustical Society of America 130, 1420 (2011)
- Dynamical Energy Analysis – determining wave energy distributions in vibro-acoustical structures in the high-frequency regime, G. Tanner, Journal of Sound and Vibration 320, issues 4-5,1023-1038 (2009)

**Discrete Flow Mapping Technique (DFM) – Vibro-acoustics on FE meshes**

Simulations of the vibro-acoustic performance of automobiles are routinely carried out in various design stages. To understand the transmission of structure-borne sound in cars, it is necessary to have effective and efficient modelling tools to support the structural design process ideally before a prototype vehicle is built. The major difficulty in modelling structure-borne sound lies in the complex geometry of the car structure. The Finite Element Method (FEM) can describe geometric details of the car structure with sufficient accuracy in the low frequency region, typically below 500 Hz. High frequency analysis using FEM requires extremely fine meshes of the body structure to capture the shorter wavelengths and, at the current time, such analysis poses significant computational challenges. Dynamical Energy Analysis (DEA) combined with the Discrete Flow Mapping technique (DFM) has recently been introduced as a mesh-based high frequency method modelling structure borne sound for complex built-up structures. This has proven to enhance vibro-acoustic simulations considerably by making it possible to work directly on *existing finite element meshes*, circumventing time-consuming and costly re-modelling strategies. In addition, DFM provides detailed spatial information about the vibrational energy distribution within a complex structure in the mid-to-high frequency range. Progress in the development of the DEA method towards handling complex FEM-meshes including Rigid Body Elements has been achieved. In addition, structure borne transmission paths due to spot welds have been considered.

**Related Publications:**

- BAJARS, J., CHAPPELL, D.J., HARTMANN, T. and TANNER, G., 2017. Improved Approximation of Phase-Space Densities on Triangulated Domains Using Discrete Flow Mapping with p-Refinement. Journal of Scientific Computing. ISSN 0885-7474
- Comparison of Methods for Modelling Mid-to-High Frequency Vibro-Acoustic Energy Distributions in a Vehicle Floor Structure,

T. Hartmann, G. Tanner, G. Xie and D. J. Chappell, SAE International, Technical Report 2016-01-1853 (doi: 10.4271/2016-01-1853), presented at 9th Int. Styrian Noise, Vibration & Harshness Congress (ISNVH16). - Transport of phase space densities through tetrahedral meshes using discrete flow mapping, J. Bajars, D. J. Chappell, N. Søendergaard and G. Tanner, Journal of Computational Physics 328, 95–108 (2016)
- Dynamical energy analysis on mesh grids – a new tool for describing the vibro-acoustic response of complex mechanical structures, D. J. Chappell, D. Löechel, N. Søndergaard and G. Tanner, Wave Motion 51, 589-597 (2014)

**Structure borne sound propagation in composites **

Composites have superior structural characteristics compared to steel or aluminium, but can exhibit poorer dynamic and acoustic performance levels. The study of the acoustic transmission in such structures helps to understand how to reduce the noise footprint on aircrafts, for example. Existing methods of analysis such as Finite Element Method (FEM) and Statistical Energy Analysis (SEA) suffer from limitations when used for modelling geometrically complex structures in the high frequency regime. In that regard, a new method – Dynamic Energy Analysis (DEA) in the form of Discrete Flow Mapping is used for high-frequency vibration analysis of aircraft panels. Aircraft panels consist of elements that can be modelled as waveguides such as plates and cylinders coupled together through different types of joints. The coupling is represented in terms of reflection/transmission matrices that can be obtained by solving the wave equation numerically using the Wave Finite Element method (WFE). Then, by incorporating WFE coupling models into the DEA method we compute the acoustic energy distribution within the structure.

**Tunnelling corrections to wave transmissions on shell structures**

In the mid-to-high frequency regime, ray tracing in the form of DEA propagation is well suited for capturing the distribution of vibroacoustic energy in complex built-up structures. However, the effects of curvature can only be treated approximately. We present a method here that captures wave effects in a ray-tracing treatment on curved plates. The equations of a ray dynamics for energy transport on arbitrarily curved and inhomogeneous smooth thin shells canbe obtained via the Eikonal approximation from the underlying wave equations. We analyse mid-frequency effects below the ring frequency of curved plates for a cylindrical region smoothly connected to two flat plates using Donnell shell theory. We perform ray tracing based on Hamilton’s equations derived in the short wavelength regime for bending, shear and pressure incident waves. Rays incident on the curved shell struc-ture may be reflected or transmitted. Simple ray tracing gives either total reflection or total transmission; the solution of the full wave equations shows in contrast a smooth transition and exhibits resonant states in the waist of the cylindrical region. In one-dimension, both the smooth transition and the resonant states can be treated using the Wentzel-Kramers-Brillouin (WKB) approximation extended to complex rays. We use graph models to account for resonant tunnelling in such curved plates. For classically transmitted bending rays, we find complex rays which connect them to the resonant states that are formed in the curved region. Similarly, for classically reflected rays, we identify those complex rays that connect to the phase space of transmitting trajectories. We successfully find a theoretical expression for calculating the scattering matrix for bending rays which accounts for resonant tunnelling mediated by resonant states. Our model agrees well with full wave solutions.

**Related Publications:**

- Tunnelling corrections to wave transmissions on shell structures, Neekar M. Mohammed, Stephen C. Creagh, Gregor Tanner and David J. Chappell, Proceedings of ISMA 2018, Leuven, Belgium (2018).

**Uncertainty quantification of phase-space flow methods**

Vibrational and acoustic energy distributions of wave fields in the high- frequency regime are often modeled using flow transport equations. In real live applications, the flow of rays or non-interacting particles is often driven by an uncertain force or velocity field and the dynamics are determined only up to a degree of uncertainty. A boundary integral equation description of wave energy flow along uncertain trajectories in finite two-dimensional domains has been developped, which is based on the truncated normal distribution, and interpolates between a deterministic and a completely random description of the trajectory propagation. The properties of the Gaussian probability density function appearing in the model are applied to derive expressions for the variance of a propagated initial Gaussian density in the weak noise case. Numerical experiments are performed to illustrate these findings and to study the properties of the stationary density, which is obtained in the limit of infinitely many reflections at the boundary.

**Related Publications:**

- Uncertainty quantification for phase-space boundary integral models of ray propagation, David J. Chappell and Gregor Tanner, to appear in Wave Motion (2018).
- BAJARS, J. and CHAPPELL, D.J., 2018. A boundary integral method for modelling vibroacoustic energy distributions in uncertain built up structures. Journal of Computational Physics, 373, pp. 130-147. ISSN 0021-9991
- A boundary integral formalism for stochastic ray tracing in billiards, D. J. Chappell and G. Tanner, CHAOS 24, 043137 (2014)