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.

Many real-world problems in homogeneous media, such as electromagnetic scattering from an airplane or the noise radiation from a car, can be formulated as integral equations over the boundary of the domain. The advantage of such formulations is that only the boundary and not a (possibly infinite) medium needs to be discretised. Computing solutions of such a boundary integral equation is the task of a boundary element method (BEM). We have developed BEM++ (www.bempp.org), a modern open source boundary element software library, which can solve a wide range of problems in electrostatics, diffuse optics and acoustics. Support for electromagnetic and elasticity problems is coming soon. The library can be used either from C++ or via a convenient scripting interface directly from Python. The current version of BEM++ already parallelises on modern multi-core workstations and has been deployed successfully for the computation of light diffusion in optical tomography applications. In this Stage 2 proposal we build on the successful core library and extend it to solve challenging application problems on modern heterogeneous computing architectures. To achieve this goal the following software library components will be developed as part of the project. 1.) We will implement hierarchical matrix and fast multipole methods on distributed CPU/GPU clusters. These methods make possible the fast solution of boundary integral equation problems with millions of unknowns and are essential for very large application problems. 2.) We will interface BEM++ with Dune, a well known high-performance library for large scale grid based applications such as finite element and finite volume methods. While boundary element formulations are suitable for homogeneous media problems they do not work for heterogeneous media, where finite element methods (FEM) are frequently being used. For coupled problems, where parts of the medium are homogeneous and others are heterogeneous it is often beneficial to couple FEM and BEM methods. The interface to Dune will allow us to do this, and to solve complicated coupled problems on large parallel computing architectures. These extensions will allow us to solve large scale real world application problems. The first medical application that we will study is High Intensity focused ultrasound (HIFU) treatment. The principle here is to use focused ultrasound targeted at a tumor to destroy the cancerous tissue by localized thermal expansion. However, to properly plan HIFU treatment, careful numerical simulations are necessary and we will build a HIFU toolbox based on BEM++ and Dune for this treatment planning simulation. This is work in collaboration with the National Physics Lab and is of large industrial interest for the development of novel cancer therapies. The second medical application is in functional neuroimaging. An example is diffuse optical cortical mapping, where we use the library to simulate the diffuse scattering of light in realistic head geometries. These techniques are useful to map the brain activities of patients and can have potential applications for example in the treatment of sufferers of locked-in syndrome. The other large application area studied in this proposal is atmospheric sciences in collaboration with the Met Office. The interest here is to accurately simulate the electromagnetic properties of atmospheric particles such as ice crystals. Remote sensing of cloud properties relies heavily on electromagnetic scattering models and questions regarding the accuracy of standard approaches have potentially profound implications for our understanding of how the water cycle and clouds influence the radiation balance of the earth. BEM++ computations of scattering by these atmospheric particles will be compared with results from previously used asymptotic methods, and the implications for remote-sensing and radiative transfer in the atmosphere investigated.

The rapid growth of computing power during the last 50 years has given rise to a whole simulation industry serving the needs of the manufacturers looking to design products in an optimal manner, without the time and costs associated with building a series of physical prototypes. Design and construction decisions are increasingly made by means of virtual prototyping as part of Computer Aided Engineering (CAE), and efficient simulation tools in all areas of engineering are sought after. Noise and vibration are particularly important performance aspects in the design of many mechanical systems. High noise and vibration levels can be damaging to structures and to their users (potentially causing hearing loss, for example). Developing computational techniques to improve our understanding of the vibration and acoustics of complex built-up structures can enhance performance, speed up the design cycle and ultimately result in safer and less noisy products. Methodologies have long been sought after for modelling large-scale complex structures such as aircraft, trains and cars. The sheer size of these structures makes building full-scale physical prototypes expensive, and often infeasible. It also poses problems for simulation methods and limits many CAE products to low frequencies, where computational run times are relatively low and uncertainties have little influence on the vibrational behaviour. Uncertainties arising during the manufacturing process (for example, in material properties or physical dimensions) can lead to large variations in the levels of noise and vibration of a structure at high frequencies, and so mechanical engineers have turned to statistical methods to instead predict averages of these noise and vibration levels. Unfortunately, these statistical methods are based on a set of assumptions that are hard to control and generally only fulfilled for more traditional structural designs. They are not fulfilled for the large curved and moulded components used today. Therefore the CAE tools available at present for simulating mid- and high- frequency noise and vibration do not meet the needs of engineers in the transport sector. As a result of the 2008 climate change act in the UK and similar initiatives around the globe, transport industries are undergoing a period of great change. Alternative fuel sources and lightweight materials are two of the major areas of development. An increasing number of hybrid and electric powered vehicles are appearing on the market and the use of lightweight and composite materials is increasing across the sector. Engineers were already in need of new and more versatile simulation methods at mid-to-high frequencies, but the increasing popularity of lightweight materials and electric power sources has compounded this situation for three main reasons: - only estimates of the material properties for newly manufactured lightweight and composite materials are available introducing considerable uncertainty into the model; - lightweight and composite materials typically emit noise at higher frequencies than more traditional steel or aluminium based structures; - sources of noise and vibration (eg. electric motors, air resistance etc.) will mostly be at high frequencies. In this proposal, random (or stochastic) transfer operator methods will be developed for modelling mid-to-high frequency structural vibrations in large complex structures. These methods will have the advantages of the current statistical approaches in terms of being able to model uncertainties in the structural design and materials, but crucially will be applicable to a far wider range of structures, including large moulded components and novel lightweight materials. The approach to be developed therefore has the potential to provide a black-box design tool for mechanical engineers looking to develop the next generation of green and lightweight transport structures.