Boundary Element Method in Engineering: Numerical Solutions for Partial Differential Equations

1. Introduction Stephen Kirkup School of Engineering, University of Central Lancashire, England

2. UK to India image by dentistindelhi.co.in

3. UK and me • Grew up on a farm in Co. Durham • Studied BSc Mathematics at the University of Newcastle • Studied MSc Numerical Analysis and Programming at the University of Dundee • Studied PhD “Solution of Exterior Acoustic problems by the Boundary Element Method” at the University of Brighton • Mostly worked in universities in the North West of the England (Historically Lancashire). Universities of Salford, Liverpool, Blackburn and UCLan http://www.map-of-uk.com/

4. Around UCLan, Preston

5. Lancashire and the Industrial Revolution

6. Lancashire and the Industrial Revolution C N Trueman "Lancashire and the Industrial Revolution"historylearningsite.co.uk. The History Learning Site, 31 Mar 2015. 16 Aug 2016. C N Trueman "Lancashire and the Industrial Revolution"historylearningsite.co.uk. The History Learning Site, 31 Mar 2015. 16 Aug 2016. C N Trueman "Lancashire and the Industrial Revolution"historylearningsite.co.uk. The History Learning Site, 31 Mar 2015. 16 Aug 2016. • Lancashire Today • Manufacturing hub of UK • Engineering – Particularly Aerospace

7. UCLan – University of Central Lancashire • About UCLan • Engineering Innovation Centre • Engineering Degrees BEng and MEng • Aerospace Engineering • Mechanical Engineering • Energy Engineering • Motorsports Engineering • Computer-Aided Engineering • Postgraduate degrees and research • See website for more information www.uclan.ac.uk

8. Teaching • My teaching is currently focussed on mathematics for engineers, • Year 1 to year 3 of engineering degrees. • Year 2 and year 3 modules are rather “mathematics and simulation” modules • In the past I have also taught • Computer Science (Year 1 to Year 3 of BSc course) • Programming and object-oriented software engineering (Java) • Formal Computer Science • Analytic Engineering • Dynamics • Control • Sustainablity in construction and nuclear industry • Current Focus • Use existing and develop on-line resources for teaching maths • Handouts (egwww.mathcentre.ac.uk) • Software (eg spreadsheets, matlab codes (freemat/octave,Simulink)) • Develop students towards technological solutions to maths problems to support pen-and-paper solutions (egmatlab-mupad, mathematica, maxsyma (free)) • Accessibilty and empowerment (VLE, available technologies) (Example – inverting a matrix)

9. Brief Introduction on BEM from Speaker • Worked in the BEM – amongst many other things – for 30 years • Started as a mathematician but became also interested in programming (software design), engineering and education • Book “The Boundary Element Method in Acoustics” and accompanying Fortran library 1998/2007 • Mainly worked in Helmholtz Equations (Acoustics) and Laplace Equation • Website : www.boundary-element-method.com which has published ‘open source’ software and educational materials • This presentation can be downloaded from www.kirkup.info/LDCE • Most of my publications and projects can be found on Researchgate • Member of LinkedIn • User groups : Boundary Element Method, Computational Mechanics, Applied Mathematical Modelling

10. Purpose • The Boundary Element Method is a Numerical Method • From a mathematical viewpoint the BEM finds an approximate solution to a partial differential equation (PDE) governing a domain • For an engineer the BEM can be used to simulate the properties of a design before committing the building the prototype • Domain of ‘just about any reasonable’ shape and general conditions • Analytic methods only work for simple separable geometries - eg circles, squares, spheres and cylinders. Useful for generating test problems.

11. Boundary Value Problem - Boundary • The BEM solves a PDE a boundary value problem • Domain is OR

12. Boundary Value Problem - Conditions • The BEM solves a PDE with boundary conditions; a boundary value problem α Robin eg impedance Dirichlet or Essential Neuman or flux • Note: Exterior problems also have a condition at ∞, that is in the far-field • Game is : Given the PDE, the boundary and the boundary condition, we need to find in the domain • To solve the boundary value problem

13. Summary of my BEM work • Practical Problems • Computer-Aided Analysis of Engine Noise (1992) • Computational methods for the acoustic modal analysis of an enclosed fluid with application to a loudspeaker cabinet (1997) • DC capacitor simulation by the boundary element method (2006) • Simulation of the Acoustic Field of a Horn Loudspeaker by the Boundary Element–Rayleigh Integral Method (2013) • General method development and analysis • Modal analysis of acoustically-loaded structures via integral equation methods (1991) • The computational modelling of acoustic shields by the boundary and shell element method (1992) • Computational solution of acoustic radiation problems by Kussmaul's boundary element method (1992) • Methods for Speeding Up the Boundary Element Solution of Acoustic Radiation Problems (1992) • The influence of the weighting parameter on the improved boundary element solution of the exterior Helmholtz equation (1992) • Solution of the Helmholtz eigenvalue problem via the boundary element method (1993)

14. Summary of my BEM work (continued) • General method development and analysis (continued) • Computational solution of the acoustic field surrounding a baffled panel by the Rayleigh integral method (1994) • An empirical analysis of the boundary element method applied to Laplace's equation (1994) • Solution of Helmholtz Equation in the Exterior Domain by Elementary Boundary Integral Methods (1995) • Solution of discontinuous interior Helmholtz problems by the boundary and shell element method (1997) • Software • Fortran codes for computing the discrete Helmholtz integral operators (1998) • Educational/Developmental • A Gentle Introduction to the Boundary Element Method in Matlab/Freemat(2008)

15. SIMS analysis and Inverse Problems • SIMS experiment • Concentration-depth profiles of the original material structure are given in the top diagram • Material (semiconductor) is bombarded with ions. • Simulation of outcome is given in lower figure • Surface recedes • Atoms are mixed within the material structure • Sputtering is related to the surface concentrations

16. SIMS analysis and Inverse Problems (continued) • Simulation of the ‘forward problem’ • Convection-diffusion equations • Finite difference methods • Computational solution of the atomic mixing equations (1998) • Computational solution of the atomic mixing equations: Special methods and algorithm of IMPETUS II (1998) • Quantification Problem • Finding the initial material structure from the SIMS profiles • Inverse / ill-posed problem • Inverse Diffusion problem : Classically the Backward Heat conduction problem

17. FDTD Method for Electromagnetic Simulation in Java www.kirkup.info/FDTD

18. This week • Develop the BEM for the solution of Laplace’s equation • Simplest equation to which the BEM can be applied • Is widely applicable in physics and engineering: ‘potential’ problems • The model equation for the BEM • Most problems you find with the application of the BEM to any other problem area will be found in this one • It is the best equation with which we learn and teach the BEM and …. • Develop the BEM for the solution of the Helmholtz Equation • Acoustics • Eigenvalue Problems • Fluid-Structure Interaction • Other methods, depending on time etc • Tutorial Today • Download Excel spreadsheet and User Guide