# BOUNDARY INTEGRAL EQUATION METHODS FOR HIGH FREQUENCY SCATTERING PROBLEMS

Lead Research Organisation:
University of Bath

Department Name: Mathematical Sciences

### Abstract

This project concerns the invention, analysis and implementation of new numerical methods for computationally simulating high frequency wave scattering problems. These problems have diverse applications, for example in modelling radar, sonar, acoustic noise barriers, medical ultrasound, and scattering of radiation by atmospheric particles. Our research is supported by four industrial/research organisations who comprise a steering committee and will provide motivating physical applications for the project.The chief technological difficulty which we face in the project is that of computing accurately wave solutions which are highly oscillatory (i.e. varying very quickly). Standard approximation techniques usually break the domain of the problem up into small ''elements'', and use simple (e.g. polynomial) approximations on each element. Then it is known that about 5-10 elements are required in each wavelength to achieve acceptable accuracy, and so the computational work required grows at least in proportion to the frequency of the wave (and often faster than this). In this sense the methods are termed ''not robust'' as frequency increases.We are going to devise, analyse and implement new robust methods for which the cost to obtain a fixed accuracy is bounded (or at worst grows very slowly) as the frequency increases. The programme involves solving problems not only of approximation of highly oscillatory solutions, but also (and this is often harder) analysing the stability and conditioning (i.e. sensitivity ) of the equations which govern them.The chief device which we will use to achieve our objective is the great body of asymptotic techniques for high frequency wave phenomena, some of which which are well-known in the mathematics and physics communities but which have so far been very little used in numerical computation. Part of our project will involve the derivation of new asymptotic analyses and putting them in a form suitable for use in numerical analysis. Scattering problems will be reformulated in such a way that high frequency parts of the solution are handled explicitly (and thus exactly), leaving only the approximation of low frequency components which can be done with low cost. This approach leads to new, challenging and deep problems in consistency, stability, conditioning and numerical integration which must be solved before the robustness of the methods can be rigorously determined. Some of the problems which we face require applying technology which we know will work because of our earlier studies; others require a significant element of risk and a spirit of adventure.The project will involve four investigators and two PDRAs, one involved primarily in analysis and one primarily in scientific computation. Both will also work on applications of relevance to our collaborators.

### Organisations

- University of Bath, United Kingdom (Lead Research Organisation)
- Schlumberger Limited (Collaboration)
- BAE Systems Advanced Technology Centre (Project Partner)
- Met Office, United Kingdom (Project Partner)
- Institute of Cancer Research (Project Partner)
- Schlumberger Cambridge Research Ltd, United Kingdom (Project Partner)

### Publications

Betcke T
(2011)

*Numerical Estimation of Coercivity Constants for Boundary Integral Operators in Acoustic Scattering*in SIAM Journal on Numerical Analysis
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
Buckeridge J
(2015)

*Determination of the nitrogen vacancy as a shallow compensating center in GaN doped with divalent metals.*in Physical review letters
Chandler-Wilde S
(2012)

*Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering*in Acta Numerica
Chandler-Wilde S
(2009)

*Condition number estimates for combined potential boundary integral operators in acoustic scattering*in Journal of Integral Equations and Applications
Dominguez V
(2011)

*Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals*in IMA Journal of Numerical Analysis
Graham I
(2014)

*When is the error in the $$h$$ h -BEM for solving the Helmholtz equation bounded independently of $$k$$ k ?*in BIT Numerical Mathematics
Graham, I.G.
(2013)

*Direct and Inverse Problems in Wave Propagation and Applications*
Lyalinov M
(2010)

*Scattering of a plane electromagnetic wave by a hollow circular cone with thin semi-transparent walls*in IMA Journal of Applied MathematicsDescription | The project was concerned with the invention, analysis and implementation of new numerical methods for computer simulation of high frequency wave scattering problems. These problems have diverse applications, for example in modelling radar, sonar, acoustic noise barriers, medical ultrasound, and scattering of radiation by atmospheric particles. The chief technological difficulty which was tackled in the project was that of computing accurately wave solutions that are highly oscillatory. This project was tackled by researchers in the mathematical sciences departments at the Universities of Reading and Bath, working with project partners at the Met Office, the Institute of Cancer Research, Schlumberger Cambridge Research, and BAE Systems. Conventional numerical schemes (finite element methods, finite difference methods, conventional boundary integral equation methods) are very expensive when the frequency is high. For example, computing scattering by an obstacle that is several hundred wavelengths long using a conventional boundary integral equation method (and this is the most efficient of the methods) requires many hours on a top-of-the-range parallel computer. Further, each halving of the wavelength (corresponding to doubling the frequency) increases the cost by at least a multiple of 4. The point of the project was to investigate, by a combination of computational experiment and detailed mathematical analysis and design, whether it is possible to devise numerical schemes for scattering problems (based on boundary integral equation methods) that have hugely lower cost, in particular that require no or negligible increase in computation time as the frequency increases. To a very significant extent the project was very successful. In particular, for large classes of scattering obstacles that are two-dimensional, in the sense that they have a cross-section that is invariant in some given direction, we have devised numerical schemes that achieve this goal. Precisely, we have shown that this is possible for cross-sections that are: convex smooth obstacles, convex polygons, convex polygons with curved sides, and certain classes of non-convex polygons. These results have been exhibited in the various papers arising from the project in careful numerical experiments. Of interest to the more technically minded, we have also been able to prove mathematically that the algorithms will perform as we predict, this via careful mathematical argument, using results from the areas of mathematics that are numerical analysis, applied analysis, and asymptotic analysis. The project, while to a large extent a proof of concept, has been successful to the extent that there is substantial follow-up funding. Notably, two of the project partners (Schlumberger Gould Research and the Met Office) are funding between them three PhD Students through the Research Councils CASE arrangements, two at Reading, one at Bath, working to move results from the project specifically into application and/or to develop similar methods in a finite element context. The project has also been successful in developing the careers of outstanding young researchers. Notably, working with our support, the first two postdocs on the project have won their own EPSRC funding for related follow-up work, through a personal Postdoctoral Fellowship and a Career Acceleration Fellowship. |

Exploitation Route | In follow-up work, in a new funded project with the Met Office, we are exploring whether the methods we have developed can, after further adaptation, be made to work to compute how important classes of atmospheric particles scatter incoming light and other electromagnetic radiation. An ability to do these calculations accurately is a crucial component feeding into computational models of climate change, as scattering by aerosols and ice particles in the atmosphere is a crucial part of the earth's energy balance. In other follow-up work we are exploring, with BAE Systems, whether our methods can be used to increase the accuiracy of and speed up computations they make of electromagnetic scattering problems on ships and aircraft, for example to predict the extent of interference between antennae on different parts of a ship/aircraft. Finally other follow-up work is underway with Schlumberger Gould Research, funded by a CASE award, which has so far reesulted in new solvers for large linear system sets arising in seismic imaging, and a patent has been provisionally filed with the US patent office on this topic. |

Sectors | Aerospace/ Defence and Marine,Energy,Environment,Healthcare |

URL | http://people.bath.ac.uk/eas25/HF/ |

Description | The work forms a key part of an ongoing collaboration with Schlumberger Gould Research Cambridge in their seismic inversion activity. To CASE students have been sponsored by SGR on this field. |

First Year Of Impact | 2007 |

Sector | Aerospace, Defence and Marine,Energy,Environment |

Impact Types | Economic |

Description | EPSRC |

Amount | £206,756 (GBP) |

Funding ID | EP/I025995/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 04/2011 |

End | 04/2014 |

Description | EPSRC |

Amount | £409,428 (GBP) |

Funding ID | EP/I030042/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 10/2011 |

End | 09/2013 |

Description | EPSRC |

Amount | £66,465 (GBP) |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 10/2010 |

End | 03/2014 |

Description | Schlumberger |

Amount | £15,000 (GBP) |

Organisation | Schlumberger Limited |

Department | Schlumberger (France) |

Sector | Private |

Country | France |

Start | 10/2010 |

End | 03/2014 |

Description | Schlumberger |

Amount | £15,000 (GBP) |

Organisation | Schlumberger Limited |

Department | Schlumberger Cambridge Research |

Sector | Academic/University |

Country | United Kingdom |

Start | 10/2010 |

End | 03/2014 |

Description | Visiting Professorship |

Amount | SFr. 7,000 (CHF) |

Organisation | University of Zurich |

Sector | Academic/University |

Country | Switzerland |

Start | 08/2016 |

End | 09/2016 |

Description | Visiting Professorship |

Amount | HK$120,000 (HKD) |

Organisation | Chinese University of Hong Kong |

Sector | Academic/University |

Country | Hong Kong |

Start | 01/2016 |

End | 07/2016 |

Description | Frequency domain helmholtz solvers, waves in random media |

Organisation | Schlumberger Limited |

Department | Schlumberger Cambridge Research |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Schlumberger Gould Research, Cambridge UK |

Collaborator Contribution | We developed new iterative solvers for frequency domain wave propagation and scattering problems relevant to seismic inversion problems. We also are developing new theory for heterogeneous scattering problems. |

Impact | Multidisciplinary: Numerical analysis, signal processing, petroleum engineering |

Start Year | 2009 |