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. The 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 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**

We propose a new approach towards determining the distribution of mechanical and acoustic wave 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 range of 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—dynamical energy analysis (DEA)—thus provides a universal approach towards determining wave energy distributions in complex structures in the high-frequency limit.

**Related Publications:**

**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. We will present here progress in the development of the DEA method towards handling complex FEM-meshes including Rigid Body Elements. In addition, structure borne transmission paths due to spot welds are considered.

**Related Publications:**

**Vibro-acoustics on network of beams **

We use networks of beams/plates to model the propagation of noise and vibration in large structures. Using fourth order beam equations on these graph-like structures, we propose an extension of quantum graphs to the elastic case. The fourth order beam equations introduce evanescent waves into the system. Formulating the problem in terms of scattering matrices we note that the transfer operator is non-unitary due to the presence of these evanescent modes. Despite this non-unitarity we are able to derive a functional equation, which guarantees real eigenvalues. The flux-conservation conditions that underly the functional equation may also be used to derive alternative state-counting “Weyl formulas” which distinguish between generic modes and “edge-modes”, which are localised on junctions between plates while decaying evanescently into the plates themselves. The transfer operator can also be used to investigate statistical properties correlation functions, which we use as atool for understanding fluctuations about the mean solution. Ray tracing methods such as Dynamical Energy Analysis (DEA) emerge as a limit of this approach while statistical characteristics are common to more complex, harder-to-compute systems.