**Research group leader:** Dr. Paul Harris

**Research staff**:
Dr. Roma Chakrabarti,
Dr. David Henwood,
Mr. Keith Parramore, and Dr. William Wilkinson.

**Research students:** Huan Wang

The Computational Mathematics Group is involved with a number of on-going research projects. The details of some of these are listed below.

### The Boundary Element Method

Previous research has covered a variety of application areas, notably bubble dynamics. However current activity focuses on acoustics.

The group is currently working on the
modelling of the transient sound
field radiated by a loud-speaker cone as part of a long-standing collaboration
between the University of Brighton and B&W Loudspeakers Ltd. Prediction of the
sound field enables the design process to be simplified.

Calculated results
are compared to actual measurements obtained using a laser interferometer and
anechoic chamber at B&W. The animation shows the calculated pressure field
generated on the surface of the loudspeaker
cone by an axisymmetric oscillation of the cone.

The group is also collaborating with the University of Liverpool on precondtioners for iterative solution of boundary element equations.

### Space Plasmas

The group's research interest in space plasmas focuses on the various aspects of collisionless shock waves; quasi-perpendicular and quasi-parallel shock structure; shock thermalization processes; particle dynamics at shocks; upstream and downstream wave phenomena; particle distributions and wave-particle interactions and co-rotating interaction regions in the solar wind. The work involves theoretical and computer simulation studies supported by some analysis of spacecraft measurements.

The supersonic solar wind is deflected past the Earth by a standing shock wave (pink in the picture). This picture is from the ESA/NASA SOHO spacecraft portfolio and is courtesy of NASA.

The plasma is heated as it passes through the shock. Phase space analysis allows the particle distribution to be determined. On the diagram, the dots show the calculated velocity distribution (white dots, one standard deviation; black dots, mode and two standard deviations), and the ellipses indicate a Gaussian approximation to the distribution. The diagram is taken from Ellacott and Wilkinson, 2003 (see publications below).

#### Recent PhDs / MPhils

Philip Newman 2012, D. Chappell 2007

H. Wang, *Boundary integral modelling of transient wave
propagation with application to acoustic radiation from
loudspeakers*, University of Brighton, 2004

### Recent Publications

Mason, P.J.R., Alharbi, T., Regan, P.H., Mărginean, N., Podolyàk, Zs., Simpson, E.C., Alkhomashi, N., Bender, P.C., Bowry, M., Bostan, M., Bucurescu, D., Bruce, A.M., Căta-Danil, G., Căta-Danil, I., Chakrabarti, R., Deleanu, D., Detistov, P., Erduran, M.N., Filipescu, D., Garg, U., Glodariu, A., Ghiţă, D., Ghugre, S.S., Kusoglu, A., Mărginean, R., Mihai, C., Nakhostin, M., Negret, A., Pascu, S., Rodriguez Triguero, C, Sava, T., Sinha, A.K., Stroe, L., Suliman, G. and Zamfir, N.V. (2012) *Half-life of the Iπ=4− intruder state in 34P: M2 transition strengths approaching the island of inversion* Physical Review C, 85 (6). ISSN 0556-2813

Newman, P.R., Ellacott, S.W. and WILKINSON, WILLIAM (2011) *A statistical physics perspective on the evolution of Ion distribution across low-mach number quasi-perpendicular collisionless shocks* Journal of Geophysical Research, 116 (9). ISSN 0148-0227

Harris, P.J. (2011) *On methods for evaluating a class of highly oscillatory integral* In: Proceedings of the 8th UK Conference on Boundary Integral Methods, 4-5 July, 2011, University of Leeds, UK.

Harris, P.J. and Hardwidge, C (2010) *A porous finite element model of the motion of the spinal cord* In: Tenth International Conference on Integral Methods in Science and Engineering, 7-10 July 2008, Santander, Spain.

Harris, Paul J. and Chen, Ke (2009) *An efficient method for evaluating the integral of a class of highly oscillatory functions* Journal of Computational and Applied Mathematics, 230 (2). pp. 433-442. ISSN 0377-0427

Chappell, D. and HARRIS, PAUL (2009) *On the choice of coupling parameter in the time domain Burton and Miller formulation* Quarterly Journal of Mechanics and Applied Mathematics, 62 (4). pp. 431-450. ISSN 0033-5614

Blake, A.L., Winstanley, G. and Wilkinson, W.P. (2009) *Deriving Displacement from 3-Axis Accelerometers* In: Computer Games, Multimedia & Allied Technology, May 2009, Singapore.

Chappell, D. and Harris, P.J. (2009) *A Burton–Miller inverse boundary element method for near-field acoustic holography* The Journal of the Acoustical Society of America, 126 (1). pp. 149-157. ISSN 0001-4966

Chen, K., Cheng, J. and Harris, P.J. (2009) *A new study of the Burton and Miller method for the solution of a 3D Helmholtz problem* IMA Journal of Applied Mathematics, 74 (2). pp. 163-177. ISSN 0272-4960

Chappell, D., Harris, P.J., Henwood, D. and Chakrabarti, R. (2008) *Modelling the transient interaction of a thin elastic shell with an exterior acoustic field* International Journal for Numerical Methods in Engineering, 75 (3). pp. 275-290. ISSN 0029-5981

Chappell, D., Harris, P.J., Henwood, D. and Chakrabarti, R. (2008) *Transient acoustic radiation from a thin spherical elastic shell* In: Constanda, C. and Potapenko, S., eds. Integral Methods in Science and Engineering: Techniques and Applications. Birkhauser, Boston, pp. 47-54. ISBN 9780817646707

Harris, P.J. and Hardwidge, C (2008) *The mathematical modelling of syringomyelia* In: The Ninth International Conference on Integral Methods in Science and Engineering, July 23-27, 2006, Niagara Falls, Ontario, Canada.

Newman, P., Wilkinson, W.P. and Ellacott, S.W. (2007) *Trajectories of ions specularly reflected from non-planar collisionless shocks* Planetary and Space Science, 55 (15). pp. 2295-2298.

Wang, H., Henwood, D., Harris, P. and Chakrabarti, R. (2007) *Concerning the cause of instability in time-stepping boundary element methods applied to the exterior acoustic problem* Journal of Sound and Vibration, 305 (1-2). pp. 289-297. ISSN 0022-460X

Ellacott, S.W. and Wilkinson, W.P. (2007) *Heating of directly transmitted ions at low Mach number oblique collisionless shocks: a statistical physics formulation* Planetary and Space Science, 55 (15). pp. 2251-2256. ISSN 0032-0633

Hawkins, S.C., Chen, K. and Harris, P.J. (2007) *On the influence of the wavenumber on compression in a wavelet boundary element method for the Helmholtz equation* International Journal of Numerical Analysis and Modeling, 4 (1). pp. 48-62. ISSN 1705-5105

Chappell, D., Harris, P., Henwood, D. and Chakrabarti, R. (2006) *A stable boundary element method for modelling transient acoustic radiation* Journal of the Acoustical Society of America, 120 (1). pp. 74-80. ISSN 0001-4966

Hawkins, S.C., Chen, K. and Harris, P.J. (2005) *An operator splitting preconditioner for matrices arising from a wavelet boundary element method for the Helmholtz equation* International Journal of Wavelets, Multiresolution and Information Processing (ijwmip), 3 (4). pp. 601-620. ISSN 0219-6913

Harris, P.J., Wang, H., Chakrabarti, R. and Henwood, D.B. (2005) *Stability analysis of a collocation method for solving the retarded potential integral equation* Journal of Computational Acoustics, 13 (2). pp. 287-299. ISSN 0218-396X

Wilkinson, W. and Ellacott, S.W. (2005) *Irreversibility and entropy issues concerning the heating of directly transmitting ions at low Mach number perpendicular collisionless shocks* Advances in Space Research, 37 (3). pp. 473-482. ISSN 0273-1177

Harris, P., Wang, H., Chakrabarti, R. and Henwood, D. (2004) *A method for modelling non-harmonic periodic acoustic radiation from a loudspeaker* In: Constanda, C., Ahues, M.P. and Largillier, A.R., eds. Integral methods in science and engineering: analytic and numerical techniques. Birkhauser, Boston, USA, pp. 73-78. ISBN 9780817632281

Harris, P.J., Al-Awadi, H. and Soh, W.K (2004) *An investigation into the effects of heat transfer on the motion of a spherical bubble* The ANZIAM Journal, 45 (3). pp. 361-372.

Wilkinson, W.P. (2003) *The earth's quasi-parallel bow-shock: review of observations and perspectives for Cluster* Planetary and Space Science, 51 (11). pp. 629-647. ISSN 0032-0633

Harris, P.J. and Chen, K. (2003) *On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem* Journal of Computational and Applied Mathematics, 156 (2). pp. 303-318. ISSN 0377-0427

Harris, P.J. (2003) *An investigation into the use of the boundary integral method to model the motion of a single gas or vapour bubble in a liquid* Engineering Analysis with Boundary Elements, 28 (4). pp. 325-332. ISSN 0955-7997

Ellacott, S.W. and Wilkinson, W.P. (2003) *Heating of directly transmitted ions at low Mach number perpendicular shocks: new insights from a statistical physics formulation* Journal of Geophysical Research, 108 (A111). p. 1409. ISSN 0148-0227

Harris, P., Wang, H., Chakrabarti, R. and Henwood, D. (2003) *Modelling transient acoustic radiation using the retarded potential integral equation method* In: Fourth UK Conference on Boundary Integral Methods, Salford, 2003.

Harris, P. and Bone, A. (2002) *The experience of external examiners for undergraduate law degrees - a research survey* The Law Teacher, 36 (2). pp. 168-183. ISSN 0306-9400

Wilkinson, W.P. (2001) *What the Rankine-Hugoniot relations tell us about co-rotating interaction regions in the solar wind, and what they don't* In: Proceedings of the Les Woolliscroft Memorial Conference/Sheffield Space Plasma Meeting: Multi-Point Measurements versus Theory, 24-26 April 2001, Sheffield, UK.

Chen, K. and Harris, P. (2001) *Efficient preconditioners for iterative solution of the boundary element equations for the three-dimensional Helmholtz problem* Applied Numerical Mathematics, 36 (4). pp. 475-489. ISSN 0168-9274