Engineers modelling noise and vibration in vehicles and buildings hit a wall at around 500 hertz, where standard simulation methods become too slow to be useful. This project tackles that frequency ceiling by fixing a hidden flaw in an alternative technique called Dynamical Energy Analysis (DEA). DEA works by tracking energy flow rather than individual waves, making it far more efficient for large structures like ships and tractors. But a recent mathematical study revealed that DEA loses accuracy at corners and edges, where the underlying equations break down. The researchers plan to build a data-driven version of DEA that automatically detects these problem zones and splits the data into smaller, well-behaved groups. Within each group, the energy flow can be approximated with high precision, then reassembled into a complete picture. If successful, the method will produce faster, more accurate simulations for the same computational cost. This matters because quieter cars, trains, and aircraft depend on engineers’ ability to model high-frequency noise and vibration reliably—something current tools cannot do at scale. The work is a direct application of fundamental mathematical theory to an industrial bottleneck.
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Modelling high-frequency wave fields such as sound, vibration and electromagnetic waves is a challenging task. Wave simulations for large-scale, complex structures such as aeroplanes, trains or buildings are typically based on a class of methods collectively known as finite element analysis. However, these methods are only efficient at low frequencies, where the typical dimensions of the structure being modelled do not significantly exceed the wavelength. Noise and vibration modelling in the automotive industry, for example, can be performed reliably with finite element analysis only up to around 500Hz. An alternative technique, termed Dynamical Energy Analysis (DEA), has been developed in Nottingham and is based on simulating the energy flow through the structure, rather than the oscillating waves that give rise to the limitations of finite element approaches. DEA has been developed to the point where it can be applied to the large complex structures arising in industry applications, such as a large container ship and a tractor. The method is now used both in the engineering community and by industry. DEA exhibits a rich underlying mathematical structure, formulated in terms of an operator, known as the transfer operator, originally arising in the theory of chaotic dynamical systems. However, a recent rigorous study of this underlying mathematical structure discovered a critical issue within the transfer operator when applied to model the energy flow through structures with corners (in two-dimensions) or edges (in three-dimensions). This discovery has implications for the accuracy of DEA, limiting the rate at which DEA becomes more accurate as the model is refined. To advance the applicability of the method further it is necessary to find an efficient way to overcome this limitation. The aim of this proposal is to build upon recent advancements in data-driven methodologies to develop the first data-driven formulation of DEA that is free from the accuracy limitation within current DEA implementations. Here, the transfer operator will be constructed using data providing the position and flow direction of particles as they travel between the boundaries of small regions or elements (typically simple polygons in two-dimensions), which collectively form the complex structure to be modelled. The major advantage of a data-driven DEA approach is that we can exploit data clustering algorithms to automatically subdivide in tandem the element boundaries and the flow directions, whenever the particles arrive at corners or edges. In doing so, the data will be split into groups within which the particle flow behaves well, in the sense that a small change in the initial particle position or direction will only give a small change in the position and direction after the particle has travelled across the element to another part of the boundary. The transfer operator may then be constructed from the data within each well-behaved region separately, where we can approximate the energy flow with high accuracy. These approximations can then be combined afterwards to build up a picture of the energy flow throughout the entire complex structure. The success of our approach will provide the first data-driven implementation of DEA with significant accuracy improvements on current DEA implementations, and consequently, faster running simulations to achieve a fixed accuracy target. The project constitutes a prime example where the knowledge arising from applied mathematical research can be channelled into industrial applications.
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