Contact Details
| Room No. | L937 |
|---|---|
| Telephone | +44 (0)141 548 3416 |
| Fax | +44 (0)141 548 3345 |
| penny.davies@strath.ac.uk |
Research Interests
Numerical methods for time-dependent boundary integral equations
Linear wave propagation and scattering is an important area with wide-ranging applications, which, despite its long history, still provides significant challenges in analysis and computation. Wave scattering problems can be formulated in either the frequency or time domains. Problems in the frequency domain involve time harmonic fields and give rise to Helmholtz-type elliptic partial differential equations (PDEs) for the spatial behaviour of the field amplitudes, whilst time domain problems are used for transient wave fields (often short pulses) and give rise to systems of hyperbolic PDEs. In both cases scattering problems can be solved by discretising the PDEs directly, or can often also be reformulated as boundary integral equations (BIEs) posed on the surface of the scatterer.
The numerical analysis of time dependent BIEs is very complicated, and the methods used often have stability problems and a low order of accuracy. My current research in this area is in developing stable methods which are high order (accurate). Improving spatial accuracy is reasonably straightforward since stability appears to be relatively insensitive to the spatial approximation used. But since the stability of an algorithm depends critically on the underlying temporal approximation, it is a challenging problem to obtain stable schemes which use high order approximations in time.
Recent Publications
- Discontinuous Galerkin approximations for Volterra integral equations of the first kind
- Brunner H, Davies PJ and Duncan DB
University of Strathclyde Mathematics Research Report, 2007, #5 - The stability of numerical approximations of the time domain current induced on thin wire and strip antennas
- Davies PJ, Duncan DB and Zubik-Kowal B
Appl. Numer. Math., 2005, 55, 48-68. - Convergence of collocation methods for time domain boundary integral equations; in Computational Electromagnetism; eds. R Hiptmair; R H W Hoppe and U Langer
- Davies PJ and Duncan DB
Oberwolfach Reports, 2004, V1, 579-581. - Fourier stability analysis of a numerical method for time domain electromagnetic scattering from a thin wire
- Zubik-Kowal B and Davies PJ
Numer. Algorithms, 2004, 35(1), 121-130. - Stability and convergence of collocation schemes for retarded potential integral equations
- Davies PJ and Duncan DB
SIAM J. Numer. Anal., 2004, 42, 1167-1188.
Vacancies & Studentships
Modelling and numerics for keyhole surgery simulations
Reference No.: PJD1
Supervisor: Dr Penny Davies
Date Advertised: 6th February 2007
Please contact Dr Penny Davies for further information.
Laparoscopic or "keyhole" surgery is becoming increasingly popular because it is both beneficial to patients and cost-effective. It differs from conventional surgery in that the surgeon cannot see the tissue being operated on directly but instead views its image on a high quality video monitor.
The surgical instruments are inserted into the patient via tubes through small wounds, and are necessarily both slender (5mm) and long (30cm). The main additional difficulties in this type of surgery are that the surgeon operates from a two dimensional image with a limited field of view and has much less tactile feedback.
The surgical techniques needed to perform this type of operation are obviously very different from those needed for conventional surgery, and it is essential that keyhole surgeons are properly trained.
There is growing interest in the potential for computer simulations to enhance training, in a similar way that flight simulators are used by pilots.
My interests in this field are to develop good mathematical models to capture the mechanical properties of intestinal tissues, and to develop new numerical algorithms for solving the resultant complicated system of PDEs.
The modelling has to be done very carefully to provide realistic results and it is essential that the models are based on sensible physical assumptions and clinical measurements.
Nonlinear elasticity seems to provide a good framework for modelling, and gives a good fit to data for spleen tissue.
The aim is to try to improve the modelling by adding in other physically realistic behaviour.
Once a good mathematical model has been derived and verified, the object is to use it to compute the tissue's behaviour when it is indented with surgical probes.
This involves solving a complicated system of partial differential equations numerically (there aren't "exact" solutions in general). The aim here is to develop efficient numerical methods.