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

2. 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 prototyp
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.

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

4. Boundary Value Problem - Conditions
The BEM solves a PDE with boundary conditions; a boundary value problem
𝜕𝜑 𝜕𝑛 =𝑓
α 𝜑+β 𝜕𝜑 𝜕𝑛 =𝑓
𝜑=𝑓
Dirichlet or Essential
Neuman or flux
Robin eg impedance
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 𝜑 in the domain
To solve the boundary value problem

5. Motivation Finite element method and the finite difference method?
FDM: 27 internal nodes
FEM: 42 elements
BEM: 16 elements
Finer Mesh?
BEM is possibly more efficient to scale
3D?
Only the surface needs to be meshed – benefits of BEM seem more pronounced
Ease of use?
Boundary mesh is ‘easier’ or ‘less bother’ to set up than a domain mesh

6. Motivation Motivation - FEM for exterior problems
Possible Solution – Infinite elements

7. Motivation Motivation - FDM for exterior problems
Possible Solution – Perfectly matched layer

8. Motivation - BEM for exterior problems

9. Initial Summary
In the boundary element method only a boundary mesh is required
Hence possibly a lot less elements or nodes – possible efficiency saving
Possible reduction in run time
Less ‘bother’ in comparison to the other methods
Saves development time for the eventual user – typically an engineer
Raises the profile of the BEM as a teaching package
An important research area in humble SK’s view and one that has been largely overlooked
Requires a reformulation of the PDE as a boundary-only equation
Can be done for some linear elliptic PDEs
Quite a restriction although some progress with DRM/MRM

10. BEM in Practice
Is used in most areas of physics/engineering, but not as widely as FEM
Can be thought of as a competitor to the FEM or FDM
More realistic to see it as a complement to these methods
Could be thought of as an alternative to PML or infinite
elements for exterior problems
In a ‘multiphysics’ situation, the preferred method
could be used in each domain
eg fluid-structure BEM-FEM
BEM
FEM/FDM
BEM
FEM

11. Wot? Still not much maths?
Where are we coming from?
Mathematician or Engineer?
Researcher, Teacher or Student?
How do I learn about the method?
Mathematical basis of BEM is complicated
Traditionally, books on BEM are impenetrable
Educational viewpoint in SK’s humble opinion
Work towards making BEM theory accessible and inclusive
Easy to use the method and so educational possiblities within and outside of BEM world

12. 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