Modelling high-frequency wave fields ranging from noise and vibration to electromagnetic waves is a challenging task. Wave simulations for large-scale, complex structures such as aeroplanes, cars or buildings are mainly based on a class of methods, known as finite element techniques, which are efficient only at low frequencies with typical length-scales of the structure being comparable to or smaller than the wavelength. Noise and vibration modelling in the automotive industry, for example, can be performed reliably with finite element techniques only up to 500Hz. An alternative technique, termed Dynamical Energy Analysis (DEA), has recently been developed in Nottingham and is based on computing energy flow equations. It has been refined to be applicable to real scale structures such as a large container ship or a tractor model from Yanmar Co, Ltd, a tractor manufacturer from Japan. 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 transfer operator, originally arising in the theory of chaotic dynamical systems. In order to advance the applicability of the method further, a thorough mathematical analysis is needed. The aim of this proposal is to exploit advanced tools from functional analysis to put DEA on sound foundations and, at the same time, improve the efficiency of the method further in a systematic way. This is facilitated by recent progress in transfer operator methods and numerical analysis. The former allows for an increased flexibility in constructing new function spaces on which the operator has good spectral properties, the latter is achieved using block compression and reordering techniques for the DEA matrix based on matrix graph algorithms to improve solver efficiency and enhance parallelism. The project members have the expertise to bring these diverse fields together with Queen Mary University of London leading in transfer operator techniques, the University of Nottingham bringing in detailed knowledge on current implementations of DEA and Nottingham Trent University having the numerical analysis skills in the context of energy flow equations. The project thus constitutes a prime example where pure mathematics informs applied mathematics and the arising knowledge is channelled straight into industrial applications.

The use of scale resolving simulations (SRS) for single phase flow applications has already shown dramatic accuracy benefits. The term SRS encompasses methods resolving a greater spectrum of turbulence e.g. large eddy simulation (LES), quasi-direct numerical simulation and hybrid methods e.g. detached eddy simulation (DES). The purpose of this work is to extend these methods for multi-phase applications. The use of SRS for single-phase turbulent flows is an area of fluids mechanics that has been widely studied for the past twenty years but SRS of multi-phase flows remains a very understudied area. The project will develop a massively parallel, high-order, fully implicit (temporal and spatial), multi-phase scale resolving methodology and perform simulations of (1) a representative aero-engine bearing chamber, (2) a representative transmission system gear and (3) a continuous chemical reactor. It will demonstrate the next generation of multi-phase high-fidelity flow simulations. We will exploit novel computing hardware through the extension and use of a state of the art fully implicit parallel library developed at the University of Oxford. The library, which enables 'future proofing' of CFD codes for modern hardware architectures, has been shown to give a 27x speedup on a GPU compared with the Intel Math Kernel Library tri-diagonal solver on a CPU. The research will be led by Dr. Richard Jefferson-Loveday, Assistant Professor in the department of Engineering at Nottingham University. It will be undertaken in collaboration with industrial partners MAHLE Powertrain, Rolls-Royce, ROMAX and GSK.

Vibration absorbers are commonly used in infrastructure assets (e.g. wind turbines, buildings, bridges) and in the dynamic systems which operate on them (e.g. railway and road vehicles). To achieve more structurally resilient, low carbon and lifetime cost efficient infrastructure assets, a step change in the performance of vibration absorbers is urgently needed. There are numerous absorber design possibilities considering components from multiple domains (mechanical, hydraulic, pneumatic and electrical). However, because there is no systematic approach available, only an extremely limited number of designs have been studied to date. This fellowship will establish an optimal multidomain vibration-absorber synthesis tool, which will fully unlock the significant potential of vibration absorber designs. The superiority of the proposed synthesis tool, and the subsequent design improvements, will be demonstrated using industrially driven and supported case studies in three infrastructure sectors. These include the alleviation of wind- and wave-induced loads to wind turbines (wind energy sector); the mitigation of environmental- and human-induced oscillations in buildings and bridges (civil structure sector); the enhancement of vehicle-track and pantograph-catenary interactions (rail sector). The developed absorber synthesis tool will be applicable to solving the dynamic performance challenges in a wide range of mechanical structures, for example, minimising road damage produced by heavy duty vehicles, vibration mitigation of hydraulic and pneumatic pipelines, and dynamic performance enhancement for robotics and autonomous vehicles. These present a significant opportunity for the PI, UK Academia and UK Industry to establish a world leading capability in this challenging field with unique expertise.

The aim of this proposal is to transform the design and manufacture of structural systems by relieving the bottleneck caused by the current practice of restricting designs to a linear dynamic regime. Our ambition is to not only address the challenge of dealing with nonlinearity, but to unlock the huge potential which can be gained from exploiting its positive attributes. The outputs will be a suite of novel modelling and control techniques which can be used directly in the design processes for structural systems, which we will demonstrate on a series of industry based experimental demonstrators. These design tools will enable a transformation in the performance of engineering structural systems which are under rapidly increasing demands from technological, economic and environmental pressures. The performance of engineering structures and systems is governed by how well they behave in their operating environment. For a significant number of engineering sectors, such as wind power generation, automotive, medical robotics, aerospace and large civil infrastructure, dynamic effects dominate the operational regime. As a result, understanding structural dynamics is crucial for ensuring that we have safe, reliable and efficient structures. In fact, the related mathematical problems extend to other modelling problems encountered in other important research areas such as systems biology, physiological modelling and information technology. So what exactly is the problem we are seeking to address in this proposal? Typically, when the behaviour of an engineering system is linear, computer simulations can be used to make very accurate predictions of its dynamic behaviour. The concept of end-to-end simulation and virtual prototyping, verification and testing has become a key paradigm across many sectors. The problem with this simulation based approach is that it is built on implicit assumptions of repeatability and linearity. For example, many structural analysis methods are based on the concept of a frequency domain charaterisation, which assumes that response of the system can be characterised by linear superposition of the response to each frequency seperately. But, the response of nonlinear systems is known to display amplitude dependence, sensitivity to transient effects in the forcing, and potential bistability or multiplicity of outcome for the same input frequency. As a result, when the system is nonlinear (which is nearly always the case for a large number of important industrial problems) it is almost impossible to make dynamic predictions without introducing very limiting approximations and simplifications. For example, throughout recent history, there have been many examples of unwanted vibrations; Failure of the Tacoma Narrows bridge (1940); cable-deck coupled vibrations on the DongTing Lake Bridge (1999); human induced vibration on the Millennium Bridge (2000); NASA Helios failure (2003); Coupling between thrusters and natural frequencies of the flexible structure on the International Space Station (2009); Landing gear shimmy. In many cases, the complexity of modern designs has outstripped our ability to understand their dynamic behaviour in detail. Even with the benefit of high power computing, which has enabled engineers to carry out detailed simulations, interpreting results from these simulations is a fundamental bottleneck, and it would seem that our ability to match experimental results is not improving, due primarily to the combination of random and uncertain effects and the failure of the linear superposition approach. As a result a new type of structural dynamics, which fully embraces nonlinearity, is urgently needed to enable the most efficient design and manufacture of the next generation of engineering structures.

The aim of this proposal is to transform the design and manufacture of structural systems by relieving the bottleneck caused by the current practice of restricting designs to a linear dynamic regime. Our ambition is to not only address the challenge of dealing with nonlinearity, but to unlock the huge potential which can be gained from exploiting its positive attributes. The outputs will be a suite of novel modelling and control techniques which can be used directly in the design processes for structural systems, which we will demonstrate on a series of industry based experimental demonstrators. These design tools will enable a transformation in the performance of engineering structural systems which are under rapidly increasing demands from technological, economic and environmental pressures. The performance of engineering structures and systems is governed by how well they behave in their operating environment. For a significant number of engineering sectors, such as wind power generation, automotive, medical robotics, aerospace and large civil infrastructure, dynamic effects dominate the operational regime. As a result, understanding structural dynamics is crucial for ensuring that we have safe, reliable and efficient structures. In fact, the related mathematical problems extend to other modelling problems encountered in other important research areas such as systems biology, physiological modelling and information technology. So what exactly is the problem we are seeking to address in this proposal? Typically, when the behaviour of an engineering system is linear, computer simulations can be used to make very accurate predictions of its dynamic behaviour. The concept of end-to-end simulation and virtual prototyping, verification and testing has become a key paradigm across many sectors. The problem with this simulation based approach is that it is built on implicit assumptions of repeatability and linearity. For example, many structural analysis methods are based on the concept of a frequency domain charaterisation, which assumes that response of the system can be characterised by linear superposition of the response to each frequency seperately. But, the response of nonlinear systems is known to display amplitude dependence, sensitivity to transient effects in the forcing, and potential bistability or multiplicity of outcome for the same input frequency. As a result, when the system is nonlinear (which is nearly always the case for a large number of important industrial problems) it is almost impossible to make dynamic predictions without introducing very limiting approximations and simplifications. For example, throughout recent history, there have been many examples of unwanted vibrations; Failure of the Tacoma Narrows bridge (1940); cable-deck coupled vibrations on the DongTing Lake Bridge (1999); human induced vibration on the Millennium Bridge (2000); NASA Helios failure (2003); Coupling between thrusters and natural frequencies of the flexible structure on the International Space Station (2009); Landing gear shimmy. In many cases, the complexity of modern designs has outstripped our ability to understand their dynamic behaviour in detail. Even with the benefit of high power computing, which has enabled engineers to carry out detailed simulations, interpreting results from these simulations is a fundamental bottleneck, and it would seem that our ability to match experimental results is not improving, due primarily to the combination of random and uncertain effects and the failure of the linear superposition approach. As a result a new type of structural dynamics, which fully embraces nonlinearity, is urgently needed to enable the most efficient design and manufacture of the next generation of engineering structures.