# Next generation finite element methods for wave problems

Lead Research Organisation:
University College London

Department Name: Mathematics

### Abstract

Efficient and accurate simulation of wave phenomena is a key enabling technology across science and engineering. Applications span diverse areas, and include the whole of acoustics and noise control, non-destructive testing and ultrasonic and microwave technologies for medical imaging, problems of seismic and radar propagation and imaging, and even quantum scale simulations. But even though the underlying partial differential equations are usually linear and well understood, wave phenomena are complex and hard to simulate whenever the wavelength is small compared to the diameter of the region to be simulated.A main and standard computational tool for simulations of wave problems is the so-called finite element method. The idea of the method is to break up the computational domain into small elements and to approximate the solution on each of them in a simple way, e.g. as a linear variation. However, this gives accurate solutions only if the diameter of each element is small compared to the wavelength. Thus the number of elements needed and the associated computational cost and storage is infeasible if the diameter of the region to be simulated is very large compared to the wavelength, as it is for very many complex problems of wave propagation and scattering, e.g. seismic wave propagation for hydrocarbon exploration.Recently, there has been strong international interest in novel finite element formulations that try to solve this problem by representing the wave field on each element by functions that are themselves waves. This allows much bigger element sizes and so a significant reduction of the computational cost. However, these novel finite element methods are still in their infancy and it is poorly understood how to implement them in an optimal way. For example, one key open problem is the question of which wave functions to use. Another open question is how to achieve numerical stability, i.e. an algorithm whose results are not garbled by effects resulting from the limited accuracy that computers have. These and other questions are particularly unclear for three dimensional problems, although most practical applications are three dimensional.The fellowship addresses this wide open research area. Building upon novel ideas about how to locally model wave phenomena in a stable way it combines fundamental research in diverse areas of applied and computational mathematics in order to develop the next generation of finite element methods for wave problems. These new methods have the potential to be orders of magnitude faster than current methods allowing for numerical simulations of phenomena that are currently out of reach. In close collaboration with partners in science and industry the new methods will be applied to exciting research problems in science and engineering. In particular, a major part of the hydrocarbon exploration business is enabled through the modelling and inversion of large scale 3D seismic and electromagnetic data sets, and Schlumberger Cambridge Research will be a key project partner. Throughout the fellowship annual international workshops on next generation finite element methods for wave problems will be organised, at Reading and Schlumberger. These will bring together leading researchers in the area of numerical wave simulations from academia and industry and will drive this research area forward by intensifying collaborations and developing and exploring application areas for these methods.Numerical wave simulations are an essential technology in science and engineering. Innovations in many areas depend upon the ability to simulate complex wave phenomena. The UK is one of the leading countries for wave-related research. This fellowship will enhance this role by building up an internationally outstanding research group on novel finite element methods for wave problems that will have a strong impact on wave-related research and applications long after the duration of the fellowship.

## People |
## ORCID iD |

Timo Betcke (Principal Investigator) |

### Publications

Betcke T
(2011)

*Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation*in Numerical Methods for Partial Differential Equations
Betcke T
(2014)

*Domain Decomposition Methods in Science and Engineering XXI*
Betcke T
(2011)

*Perturbation, extraction and refinement of invariant pairs for matrix polynomials*in Linear Algebra and its Applications
Betcke T
(2013)

*Spectral decompositions and nonnormality of boundary integral operators in acoustic scattering*in IMA Journal of Numerical Analysis
Chaulet N
(2014)

*The factorization method for three dimensional electrical impedance tomography*in Inverse Problems
Jehl M
(2015)

*A fast parallel solver for the forward problem in electrical impedance tomography.*in IEEE transactions on bio-medical engineering
Malone E
(2014)

*Stroke type differentiation using spectrally constrained multifrequency EIT: evaluation of feasibility in a realistic head model.*in Physiological measurement
Smigaj W
(2015)

*Solving Boundary Integral Problems with BEM++*in ACM Transactions on Mathematical Software### Related Projects

Project Reference | Relationship | Related To | Start | End | Award Value |
---|---|---|---|---|---|

EP/H004009/1 | 01/10/2009 | 01/02/2011 | £795,615 | ||

EP/H004009/2 | Transfer | EP/H004009/1 | 01/02/2011 | 30/09/2014 | £608,157 |

Description | We have investigated numerical solution methods for high-frequency acoustic wave problems. These occur for example in the acoustic simulation of noise from a car or plane. We focused on methods in which analytical information is used to represent the high-frequency behaviour of the solution. We initially investigated methods where the high-frequency oscillations of the solutions are represented by the functions used to approximate the solution. We then focused on so-called boundary integral equation methods, in which the complete solution is represented by its action on the boundary of the scattered object. For both modalities we investigated fast numerical methods and novel theoretical results. For boundary integral equation methods we developed as part of additional EPSRC funding a fast open-source software library that is freely available for researchers and other industry. As part of novel collaborations we later also investigated efficient simulation techniques for Electrical Impedance Tomography, a novel medical imaging methodology and developed a software package for large-scale EIT simulations of complex brain models. |

Exploitation Route | We have developed a boundary element software library called BEM++ as part of an associated EPSRC Software Development Grant. They library is open-source and can be downloaded for free. It has a growing user community outside UCL. |

Sectors | Aerospace, Defence and Marine,Healthcare |

Description | Our fundings have appeared as scientific publications, which attract a growing number of citations and have been implemented as freely available software (BEM++). While we are not tracking user statistics for the software there are several external groups in engineering and mathematics who use this software. |

First Year Of Impact | 2012 |

Sector | Other |