Mark Walkley

General interests

Numerical methods, engineering applications, the finite element method, adaptivity

My main interests lie in the development and application of numerical methods for engineering problems. In particular many 2d and 3d problems can be modelled with unstructured finite element methods using adaptivity in both space and time. I am also interested in techniques for mesh generation and adaptivity in 2d and 3d and visualisation of scientific data on unstructured grids.

Relevant publications are listed with each project below. Most can be downloaded as pdf. A full list of my publications is available here.

Please contact me if you would like to discuss any of this work.

If you are interested in PhD or Masters research relating to any of these topics please get in touch and we can discuss details and options for funding. Further opportunities in the Scientific Computation research group are listed on the research group pages.

Projects

PhD, post-doctoral, CPDE Unit, supervisory, collaborations

Finite element models for polymer processing

Post-doctoral, 2005-date, EPSRC grant GR/T11807/01

The MuPP2 project, managed by Tom McLeish.

Our role is to implement finite element software for 3d viscoelastic flows using a range of state-of-the-art constitutive models. Detailed comparisons with experiments are possible to validate the models.

Publications

R. Tenchev, O. Harlen, P.K. Jimack and M.A.Walkley. Finite element modelling of two- and three-dimensional viscoelastic polymer flows. In: B.H.V. Topping and M.Papadrakakis (editors). Trends in Engineering Computational Technology, (Saxe-Coburg), to appear, 2008.
R. Tenchev, O. Harlen, P.K. Jimack and M.A.Walkley. Finite element analysis of 3-d polymer melts. Part I: Methods and validation. in preparation, 2008.
R. Tenchev, T, Gough, O. Harlen, D.H. Klein, P.K. Jimack and M.A.Walkley. Finite element analysis of 3-d polymer melts. Part II: Comparison with experiments. in preparation, 2008.

An AMR finite element model of polymeric fluids

Supervision of PhD research student Sung-Ho Yoon, 2007-date

Co-supervised by Oliver Harlen.

An immersed-interface approach to approximating particle suspensions in polymeric fluids. An adaptive finite element model in 2d and 3d is developed using deal.II.

Finite element models of Boussinesq equations

Collaboration with LNEC, 2006-date, funded by GRICES

Work with Juana Fortes and Liliana Pinheiro at LNEC extending my PhD work to solve more realistic near-shore problems with more general and physical boundary conditions.

Publications

L. Pinheiro, C.J. Fortes, and M.A. Walkley. Random waves in Sines harbour using a finite element Boussinesq model. In: Proceedings of ICCE 2008.
L. Pinheiro, A. Palha, C.J. Fortes, and M.A. Walkley. Internal wave generation on finite element model BOUSS1D_iw: comparison with experimental data In: Chung, J S, Kashiwagi, M, Losada, I J & Chien, L-K (editors). The Proceedings of The Seventeenth International Offshore and Polar Engineering Conference (ISOPE) Volume III, pp. 2309-2316. 2007.

Computational analysis of integral and differential formulations of the EHL film thickness equation

Supervision of MRes research student Elham Afandizadeh Zargari, 2006-2007

Co-supervised by Peter Jimack.

An investigation of two alternative formulations of the EHL problem and the implications for a numerical approximation.

Publications

E. Afandizadeh Zargari, Computational Analysis of Integral And Differential Formulations of the Elastohydrodynamic Lubrication Film Thickness Equation, MRes thesis, School of Computing, University of Leeds, 2007.
E. Afandizadeh Zargari, P.K. Jimack and M.A. Walkley, An investigation of the film thickness calculation for elastohydrodynamic lubrication problems, In: Proceedings of the 9th Conference on Numerical Methods for Fluid Dynamics, ed. M. Arthur et al., ICFD, Reading, 2007.

Elastohydrodynamic lubrication

CPDE Unit contract, 2004-date, funded by Shell Global Solutions

Maintenance and development of the EHL software suite designed in the School of Computing at Leeds.

The elastohydrodynamic lubrication (EHL) model is a thin-film approximation typically governing the fluid dynamics of oils in gears and bearings. The high pressures created in these situations cause large local changes in the fluid density and viscosity leading to a highly nonlinear discrete equation system. Work at Leeds has developed efficient and scalable software tools for this problem based on finite difference multigrid methods and multi-level numerical integration techniques.

Publications

H. Lu, M. Berzins, C.E. Goodyer, P.K. Jimack and M.A. Walkley, Adaptive High-Order Finite Element Solution of Transient Elastohydrodynamic Lubrication Problems, Proc. IMechE Part J: J. Engrg. Tribology, 220:215-225, 2006.

A collaborative problem-solving environment

Collaborative work with the Visualization Research Group, 2004-2005, funded by the DTI

Managed by Ken Brodlie, and collaborating with Jason Wood, Martin Thompson and Sally Mason.

An e-Science Demonstrator project integrating numerical simulation software with the IRIS Explorer visualisation system in a collaborative problem solving environment. A computational steering environment is developed by allowing parameters that control various aspects of the numerical simulation to be interactively modified from within the visualisation environment. Numerical solution and visualisation proceed simultaneously within IRIS Explorer. A collaborative environment is developed by making use of the COVISA modules allowing geographically separate users to share some or all of the dataflow pipeline. The sharing of data can be at several levels, including dial parameters, rendered images and the full solution data produced by the numerical software, allowing for a wide range of collaborative scenarios. In a further enhancement the numerically-intensive simulation can be distributed across a computational Grid allowing a light-weight collaborative interface that controls and visualises the resulting solution data.

Publications

K. Brodlie, S. Mason, M. Thompson, M. Walkley and J. Wood. Evolving dataflow visualization environments to grid computing In: Bonneau, G-P, Ertl, T & Nielson, G M (editors) Scientific Visualization: The Visual Extraction of Knowledge, pp. 395-408 Springer. 2005.
M. Walkley, J. Wood and K. Brodlie. A distributed co-operative problem solving environment In: Sloot, P M A, Tan, C J K, Dongarra, J J & Hoekstra, A G (editors) Computational Science - ICCS 2002, Proceedings, Part I, pp. 853-861 Springer. 2002.
K. Brodlie, S. Mason, M. Thompson, M. Walkley and J. Wood. Reacting to a crisis: benefits of collaborative visualization and computational steering in a grid environment. In: Proceedings of the UK e-Science All Hands Conference EPSRC. 2002.

A finite element model for surface-tension-dominated flow

Post-doctoral, 2001-2004, EPSRC grant GR/R25453/01

The Principal Investigator on this project was Peter Jimack from the School of Computing, and Co-Investigators were Prof Phil Gaskell and Dr Jon Summers from the Engineering Fluid Mechanics Group, in the School of Mechanical Engineering, and Dr Mark Kelmanson from the Dept. of Applied Maths.

The purpose of this work is to develop efficient and reliable adaptive finite element techniques for the solution of quite general transient three dimensional free surface flow problems, and to apply these techniques to the solution of a number of fundamentally important, surface tension dominated, engineering flow problems. The work will build upon the diverse and compatible expertise of the investigators in both fluid mechanics and computation algorithms to develop a new finite element solver which follows the evolution of the fluid using an arbitrary Lagrangian-Eulerian (ALE) approach. This will use both continuous and discrete remeshing involving both local and global adaptivity. A highly accurate representation of the domain boundary will be maintained in order to ensure the accurate implementation of the the free-surface boundary conditions which describe the critical infuence of surface tension. A series of test simulations will be undertaken to validate the ongoing work against experimental, analytical and other numerical results (the latter two being based upon lubrication apporximations in the thin-film limit). This will lead onto the simulation of more demanding flow problems which will require the full power of our three-dimensional Navier-Stokes solver. These problems will include spreading liquids over non-smooth surfaces and micro-fluidic devices.

Publications

M.A. Walkley, P.H. Gaskell, P.K. Jimack, M.A. Kelmanson and J.L. Summers, Finite Element Simulation of Three-Dimensional Free-Surface Flow Problems, J. Sci. Comput., 24:147-162, 2005.
M.A. Walkley, P.H. Gaskell, P.K. Jimack, M.A. Kelmanson and J.L. Summers, Adaptive Finite Element Simulation of Three-Dimensional Surface-Tension Dominated Free-Surface Flow Problems, ANZIAM J. (electronic supplement), 46(E):C558-C571, 2005.
M.A. Walkley, P.H. Gaskell, P.K. Jimack, M.A. Kelmanson and J.L. Summers, Finite element simulation of three-dimensional free-surface flow problems with dynamic contact lines, International Journal for Numerical Methods in Fluids, 47(10-11):1353-1359, 2005.
M.A. Walkley, P.H. Gaskell, P.K. Jimack, M.A. Kelmanson, J.L. Summers and M.C.T. Wilson, On the Calculations of Normals in Free-Surface Flow Problems, Comm. Num. Meth. Eng., 20:343-351, 2004.

Anisotropic adaptivity for 3d convection-dominated probelms

Post-doctoral, 1999-2001, EPSRC grant GR/M00077/01

The Principal Investigator on this project was Peter Jimack with Co-Investigator Martin Berzins.

Solutions to three-dimensional convection-dominated problems often contain lower dimensional features, the resolution of which is critical to the accuracy of the method, eg. a plane shock wave, or a boundary layer. Resolving this feature numerically is computationally expensive if an isotropic structured or unstructured mesh is used. It has become common to use anisotropic meshes for these problems, however a rigorous justification of the appropriate mesh, or a reliable measure of the numerical error on such a mesh is not yet apparent. This project investigated the application of finite element methods on 3d tetrahedral grids for such problems and studied the process of error estimation on anisotropic grids and the use of error estimates in producing anisotropic mesh adaption strategies.

Publications

P.K. Jimack, R. Mahmood, M.A. Walkley and M. Berzins, A Multilevel Approach for Obtaining Locally Optimal Finite Element Meshes, Advances in Engineering Software, 33:403-415, 2002.
M.A. Walkley, P.K. Jimack and M. Berzins, Anisotropic adaptivity for the finite element solutions of three-dimensional convection-dominated problems, International Journal for Numerical Methods in Fluids, 40(3-4):551-559, 2002.
M.A. Walkley, P.K. Jimack, and M. Berzins, Mesh Quality for Three-dimensional Finite Element Solutions on Anisotropic Meshes, In: Proceedings of FEM3D, GAKUTO International Series, Mathematical Sciences and Applications, 15:310-321, 2001.
T. Apel, M. Berzins, P.K. Jimack, G. Kunert, A. Plaks, I. Tsukerman, and M.A. Walkley, Mesh Shape and Anistropic Elements: Theory and Practice, In: The Mathematics of Finite Elements and Applications X, ed. J.R. Whiteman (Elsevier), pp.367--376, 2000.
M. Berzins, P.K. Jimack, M.A. Walkley and L.J.K. Durbeck, Mesh Quality and Moving Meshes for 2D and 3D Unstructured Mesh Flow Solvers, In: VKI Lecture Series 2000-05, 31st Computational Fluid Dynamics, ed. N.P. Weatherill and H. Deconinck (von Karman Institute ISSN0377-8312), 2000.

A finite element model for extended Boussinesq equations

PhD, 1995-1999, EPSRC CASE funded with HR Wallingford

Supervised by Martin Berzins. My thesis can be downloaded as pdf.

Boussinesq equations include the effects of weak dispersion and nonlinearity in a shallow water framework and allow accurate nearshore simulation of wave transformation processes. Extended Boussinesq equation systems allow the models to be applied in deeper water and so extend the range of usefulness of the models, as well as increasing the accuracy of the linear dispersion characteristics of the model. I chose to solve Nwogu's extended Boussinesq system, using a method of lines approach with a finite element spatial discretisation combined with an adaptive time integration strategy. The time integration uses the SPRINT software, which was developed within this department, for the 1D system and DASPK for the 2D system. Both solve differential-algebraic equation systems with adaptive order, adaptive step size BDF time integration based on sophisticated error control.

Publications

M.A. Walkley, M. Berzins, A finite element method for the two-dimensional extended Boussinesq equations, International Journal for Numerical Methods in Fluids, 39(10):865-885, 2002.
M.A. Walkley, Numerical solution of the Boussinesq-type shallow water wave equations, PhD thesis, School of Computing, University of Leeds, 2000.
M.A. Walkley, M. Berzins, A finite element method for the one-dimensional extended Boussinesq equations, International Journal for Numerical Methods in Fluids, 29(2):143-157, 1999.