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

Lead Research Organisation: Queen Mary University of London
Department Name: Sch of Mathematical Sciences

Abstract

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.

Planned Impact

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.

Publications

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Bandtlow O (2021) New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces in Journal of the European Mathematical Society

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Chappell D (2021) A Direction Preserving Discretization for Computing Phase-Space Densities in SIAM Journal on Scientific Computing

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Güven Sarihan A (2021) Quantitative spectral perturbation theory for compact operators on a Hilbert space in Linear Algebra and its Applications

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Just W (2023) Synchronization of non-identical systems by non-invasive mutual time-delayed feedback in Chaos: An Interdisciplinary Journal of Nonlinear Science

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Otto A (2019) Nonlinear dynamics of delay systems: an overview. in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences

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Panagiotopoulos I (2022) Control of collective human behavior: Social dynamics beyond modeling in Physical Review Research

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Panagiotopoulos I (2023) Continuation with Noninvasive Control Schemes: Revealing Unstable States in a Pedestrian Evacuation Scenario in SIAM Journal on Applied Dynamical Systems

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Richter M (2021) Solving linear systems from dynamical energy analysis - using and reusing preconditioners in INTER-NOISE and NOISE-CON Congress and Conference Proceedings

 
Description The goal of this award was a rigorous mathematical and numerical analysis of a method called Dynamical Energy Analysis (DEA), developed for modelling noise and vibrations at high frequencies in complex built-up structures. This numerical method is formulated in terms of the approximation of a so-called transfer operator for the billiard map modelling the underlying ray dynamics. The main objective was to establish rigorous convergence results and error bounds, and improve existing computational approximation schemes accordingly. The work was structured into the following objectives:

O1 Establishing common ground: constructing and testing explicitly solvable models.
O2 Determining function spaces for the DEA transfer operator: laying the foundations for convergence estimates and efficient implementation.
O3 Providing an optimal preconditioning solution for the DEA matrix solver.
O4 Establishing convergence results: giving explicitly accessible error estimates.
O5 Providing a validated software package with enhanced efficiency and error control capability.

The first rigorously studied explicitly solvable model case (O1) is the circular billiard geometry [Ref 1]. In this case, using finite-rank Fourier basis approximations of a transfer operator on a suitably chosen anisotropic function space (O2), we proved strong convergence rate bounds, establishing a power law bound with the power determined by regularity properties of the system's emitting energy source (O4). Numerical experiments verified that these bounds are not significantly affected for certain more complex geometries (deformed circular billiards) (O4).

The application-relevant case of more complex geometrical structures described by meshes (typically consisting of convex polygonal mesh cells) (O2, O4) rests on analogous approximation results for polygonal geometries. This case (exemplified by the triangular billiard) requires an adaptation of the function spaces and approximation methods (namely spaces of generalised bounded variation and Ulam approximation methods). This is currently a work in progress, and provides a fertile ground for further research.

On the computational side, we investigated a range of numerical approaches for solving large, sparse, non-symmetric linear systems as required for the DEA approximation schemes, identifying a well-suited choice (optimized for run time) for the given problem (O3), published in [Ref 2]. Deploying this method alongside an existing software package developed at the University of Nottingham results in a high-efficiency computational approach for predicting vibrational energy density within complex structures (O5). Error control capabilities for this depend directly on corresponding convergence rate bounds referenced above (available for simpler, e.g. circular, geometries, and in progress for more general polygonal meshes).

[Ref 1] Transfer operator approach to ray-tracing in circular domains. Slipantschuk et al., Nonlinearity, 2020.
https://doi.org/10.1088/1361-6544/ab9dca
[Ref 2] Solving linear systems from dynamical energy analysis - using and reusing preconditioners. Richter et al., Inter-Noise, 2021.
https://uniofnottm-my.sharepoint.com/:b:/g/personal/martin_richter_nottingham_ac_uk/EaCoc-cjBppOrZd5WmHAmqIBnVM5__tsKwhjSjXqaGlGgw?e=XUCF4d
Exploitation Route Our work on the functional analytic properties of transfer operators and the construction of approximation procedures for systems arising in engineering applications has, so far, been well-received by researchers in mathematical analysis, in particular those in dynamical systems, ergodic theory and operator theory, as well as the numerical analysis community. In the longer run, we expect that our exploitation of dynamical systems tools to model mechanical structures will provide new stimuli to researchers in industrial and applied mathematics, whilst our concrete example-driven modelling of the mid- and high-frequency response of built-up structures will be taken up in the mechanical engineering community. In fact, being able to determine reliable a-priori error bounds is crucial for the application of our methods in real world scenarios. During the course of the project, contact to engineering groups was established, who are now using the software in their research. This has lead to joint publications with engineers from
Germany [Ref 3], France [Ref 4], and a collaboration with a French ship-builder as part of a summer student project. The latter connection will lead to a joint PhD program to further strengthen the impact of the research.

[Ref 3] Finite element method and dynamical energy analysis in vibro-acoustics - A comparative study, Zettel et al, INTER-NOISE 2021,
https://doi.org/10.3397/IN-2021-1906
[Ref 4] Convergence of Ray-Density Methods Using Transfer Operators in Different Bases, Richter et al, Forum Acusticum, 2020, pp.231-237,
https://doi.org/10.48465/fa.2020.0738
Sectors Manufacturing, including Industrial Biotechology

URL https://doi.org/10.1088/1361-6544/ab9dca
 
Description The techniques developed in the grant have had important implications for the automotive industry, where the modelling of high-frequency wave fields was used to analyse the performance and safety of car components. The development of more efficient and accurate modelling techniques resulted in advancements in design and manufacturing processes, ultimately leading to safer and more reliable transportation.
First Year Of Impact 2021
Sector Manufacturing, including Industrial Biotechology
 
Description Faculty of Science and Engineering Research Support Fund
Amount £60,000 (GBP)
Organisation Queen Mary University of London 
Sector Academic/University
Country United Kingdom
Start 09/2018 
End 08/2021
 
Description Effective computation of resonances of Laplace operators 
Organisation Max Planck Society
Department Max Planck Institute for Mathematics
Country Germany 
Sector Academic/University 
PI Contribution Provided idea for proof of convergence of algorithm as well as error estimates.
Collaborator Contribution Partners in Bremen and at Max Planck Society finessed proof of convergence and provided effective implementation of algorithm.
Impact Preprint: https://arxiv.org/abs/2002.03334
Start Year 2018
 
Description Effective computation of resonances of Laplace operators 
Organisation University of Bremen
Country Germany 
Sector Academic/University 
PI Contribution Provided idea for proof of convergence of algorithm as well as error estimates.
Collaborator Contribution Partners in Bremen and at Max Planck Society finessed proof of convergence and provided effective implementation of algorithm.
Impact Preprint: https://arxiv.org/abs/2002.03334
Start Year 2018
 
Description PACSYS Ltd 
Organisation PACSYS Ltd
Country United Kingdom 
Sector Private 
PI Contribution PACSYS Ltd is an SME specialising in software solutions, primarily focussing on finite and boundary element methods for vibroacoustics applications. The research carried out for the project enhances reliability of vibroacoustic simulations based on transfer operator methods.
Collaborator Contribution PACSYS Ltd provides expertise in engineering analysis as well as complimentary software licences.
Impact No outputs yet
Start Year 2018
 
Description Romax 
Organisation Romax Technology
Country United Kingdom 
Sector Private 
PI Contribution Romax Technology is a global company providing software and services for gearbox, bearings and driveline systems. The research carried out in the project assists in the implementation of a transfer operator based vibroacoustic solver for gearbox modelling.
Collaborator Contribution Romax supports the project through the provision of FEM meshes as well as a software license for RomaxDesigner.
Impact No output yet
Start Year 2018
 
Description Public outreach event within the Nottingham Festival Science and Curiosity 
Form Of Engagement Activity Participation in an open day or visit at my research institution
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Public/other audiences
Results and Impact Demonstration and explanation of sound and vibrations to families and students. This included a small hands-on activity where visitors could try doing Chladni Figures with their own voice.
Year(s) Of Engagement Activity 2020
URL http://nottsfosac.co.uk/
 
Description Public outreach event within the Nottingham Festival Science and Curiosity 
Form Of Engagement Activity Participation in an open day or visit at my research institution
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Public/other audiences
Results and Impact Demonstration and explanation of sound and vibrations to families and students.

This included a small hands-on activity where visitors could try doing Chladni Figures with their own voice.
Year(s) Of Engagement Activity 2018
URL http://nottsfosac.co.uk/