Transfer operator methods for modelling high-frequency wave fields - advancements through modern functional and numerical analysis

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 proposed research programme has a broad range of future impacts ranging from academia across to industry and the general public. Impact will be achieved by channelling knowledge of solving transfer operator problems efficiently and reliably into developing software for better noise and vibration modelling of complex mechanical structures. In particular, engineers in academia and industry will benefit from robust and readily usable methods for high frequency noise modelling, leading to better control of structure-borne noise already at the design stage.

More specifically, with the automotive industry moving progressively towards virtual design, modelling high-frequency vibrations and associated noise radiating from gearbox housings is becoming a key area of research for one of the industrial partners, Romax Technology Ltd, a global leader in software and services for gearbox, bearings and driveline systems. PACSYS Ltd, the other industrial partner, a vibro-acoustic wave analysis software specialist, is interested in the project with a view of enhancing their software product line and opening up new business areas. Both industrial partners will therefore directly benefit from the results of the proposed research. The transfer of knowledge, the development of new paradigms for software development, improved algorithms and methodologies, will thus improve the position of both partners in a competitive worldwide market for software design.

At the same time, the collaboration with the two industrial partners as well as existing contacts at the University of Nottingham and Nottingham Trent University will serve as an entry point for establishing contact with other interested parties, in particular the software and simulation industry as well as end users, mainly in the transport sector such as the automotive, aerospace or railway industries. Prime targets are existing contacts such as CDH AG (noise and vibration modelling), inuTech GmbH (numerical software solutions), CST AG (EM wave field modelling) as well as OEMs such as Jaguar Land Rover Ltd (automotive), Airbus (aerospace), Yanmar Co Ltd (agricultural machinery) and Stadler Rail (railways) who are all potential customers for PACSYS and Romax.

The proposed research will also create a wealth of impact in the academic sector. The investigators, research assistants and PhD student will gain a fuller understanding of the use of transfer operators in the context of high frequency wave models, developed in tandem from both a theoretical and a practical viewpoint thus adding considerable value to an interdisciplinary agenda. At the end of the project the research assistants and PhD student will have become leading experts in the newly developed theory and methodology, helping to enrich the UK skills base. The success of the project in building links between pure and applied mathematics, as well as between academia and industry, will form a solid basis for the career of the research assistants as highly skilled intra- and interdisciplinary researchers.

Finally, high frequency wave modelling has the potential to be a topic of great interest to the public due to applications in reducing noise and vibration in transport structures as well as improving room acoustics in architectural design. A carefully planned series of events will greatly enhance visibility of science and mathematics and hence contributing to the STEM strategy by helping to empower future generations through science, technology, engineering and mathematics to grow a dynamic, innovative economy.