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

2. Where did it start?
The mathematical basis of the BEM is in the reformulation of the BEM as a boundary integral equation
Eg Kellog’s book Foundations of Potential Theory 1929
Availability of computers and the Fortran programming language led to the initial solutions of boundary integral equations in the 1960s
In the 1970s the FEM was beginning to be used in practical engineering.
By the 1980s the boundary integral equation method was being applied in practical engineering and given the name ‘Boundary Element Method’

3. Where do we start? Mathematics Algorithm Numerical Analysis
PDE and its reformulation as a boundary integral equation
Algorithm
Elements, discretization, solution method
Numerical Analysis
Analysis of algorithm, accuracy, efficiency, reflection. Difficult area for this method.
Programming
Language, facilities of language, facilities available from others
Software Design
Abstraction (Generalisation), Components, Hiding Complexity, Reliability
Engineering
Availability, Applicability, Ease of Use, Efficiency
Teaching
Reliability, Ease of use, Simplicity (Hiding Complexity,), Transparency
Me : Experienced in some of these areas.
Areas in red are the neglected areas in humble SK’s view and areas where he is most interested to develop.
Try to be ‘holistic’ : Diferent shills, different audiences : Few becomes many : Educational tool

4. Laplace’s Equation 𝛻 2 𝜑=0
2D : 𝜕 2 φ 𝜕 x 𝜕 2 φ 𝜕 y 2 =0
3D : 𝜕 2 φ 𝜕 x 𝜕 2 φ 𝜕 y 𝜕 2 φ 𝜕 z 2 =0
Has lots of analytic solutions that can be used as test problems
Simple ones
like 𝜑=𝑥 , 𝜑=𝑦 , 𝜑=𝑥+𝑦
More comlicated ones
2D 𝜑=𝑥𝑦 , 𝜑= 𝑥 2 − 𝑦 2
3D 𝜑=𝑥𝑦𝑧 , 𝜑= 2𝑥 2 − 𝑦 2 − 𝑧 2
Useful for test problems
Applications : heat conduction, electro-statics, groundwater flow, gravitation and ideal fluid flow

5. Heat Conduction Modelling Using Partial Differential Equations

6. Green’s Functions
A Green’s function models the effect of a unit point source
Simple examples are a planet is a gravitational point source in the gravitational field of the universe, an electron is a unit point source of charge.
𝛻 2 𝐺 𝒑,𝒒 =δ(𝒒−𝒑)
Influence Function
Linearity - Superposition – Summation – Integration
PDE to BIE
For Laplace’s equation the Green’s functions are
𝐺 𝒑,𝒒 =− 1 2𝜋 ln 𝑟 in two dimensions,
𝐺 𝒑,𝒒 = 1 4𝜋𝑟 in three dimensions
where r = |r|, r = p-q.
Green’s functions are singular

7. Direct BEM E S D Green’s Second Theorem
𝐷 (𝜓 𝛻 2 φ−φ 𝛻 2 𝜓 )dD= 𝑆 ( 𝜓𝛻φ−φ𝛻𝜓)dS
Let 𝜓=𝐺 𝒑,𝒒 and noting that 𝛻 2 φ=0
Hence − 𝐷 𝜑𝛿 𝒒−𝒑 d 𝐷 𝑞 = 𝑆 ( 𝐺 𝒑,𝒒 𝛻φ−φ𝛻𝐺 𝒑,𝒒 )d 𝑆 𝑞
Since 𝐷 φδ(𝒒−𝒑) d 𝐷 𝑞 = φ 𝒑 if 𝒑∈𝐷 0 if 𝒑∈𝐸 1 2 φ 𝒑 if 𝒑∈𝑆
and 𝛻∗𝑑 𝑆 𝑞 =𝛻∗. 𝒏 𝑞 d 𝑆 𝑞 = 𝜕∗ 𝜕𝑛 d 𝑆 𝑞 then
𝑆 ( 𝐺 𝒑,𝒒 𝜕φ 𝜕 𝑛 𝑞 −φ 𝜕𝐺 𝒑,𝒒 𝜕 𝑛 𝑞 )d 𝑆 𝑞 = −φ 𝒑 if 𝒑∈𝐷 0 if 𝒑∈𝐸 − 1 2 φ 𝒑 if 𝒑∈𝑆 E S D

8. With the normal pointing into S. Let’s reverse the normal
𝑆 ( 𝐺 𝒑,𝒒 𝜕φ 𝜕 𝑛 𝑞 −φ 𝜕𝐺 𝒑,𝒒 𝜕 𝑛 𝑞 )d 𝑆 𝑞 = −φ 𝒑 if 𝒑∈𝐷 0 if 𝒑∈𝐸 − 1 2 φ 𝒑 if 𝒑∈𝑆
With the normal pointing into S. Let’s reverse the normal
𝑆 ( −𝐺 𝒑,𝒒 𝜕φ 𝜕 𝑛 𝑞 +φ 𝜕𝐺 𝒑,𝒒 𝜕 𝑛 𝑞 )d 𝑆 𝑞 = −φ 𝒑 if 𝒑∈𝐷 0 if 𝒑∈𝐸 − 1 2 φ 𝒑 if 𝒑∈𝑆
This equation is the basis of the direct boundary element method.
The 𝒑∈𝑆 equation is the key that unlocks the BEM
Everything is only defined on the boundary; no reference to a domain point
A boundary integral equation (BIE)
Only a surface mesh is required
Once solved, we will know φ and 𝜕φ 𝜕𝑛 on S
The 𝒑∈𝐷 equation can then be used to find φ anywhere in the domain

9. Indirect BEM
Let σ be an unknown function defined only on the boundary, having no physical meaning.
Let φ(𝒑)= 𝑺 G(p,q)σ(q)d 𝑆 𝑞 , this is called a single layer potential 𝒑∈𝐷∪𝑆
Let 𝜕φ 𝜕 𝑛 𝑝 (𝒑)= 𝑆 𝜕𝐺 𝒑,𝒒 𝜕 𝑛 𝑝 σ 𝒒 𝑑 𝑆 𝑞 , 𝒑∈𝐷
This gives 𝜕φ 𝜕 𝑛 𝑝 (𝒑)= 𝑆 𝜕𝐺 𝒑,𝒒 𝜕 𝑛 𝑝 σ 𝒒 𝑑 𝑆 𝑞 σ for, 𝒑∈𝑆
because there is a jump dicontinuity in the operator at 𝑆.
Again the equations on the boundary can be used to find σ
Once σ is found the domain equations can be used to find φ

10. Direct and Indirect BEM
All boundary value problems can reformulated as
a direct integral equation, based on Green’s second theorem
an indirect integral equation, based on layer potentials
In discrete form these give rise to the direct and indirect BEM
The methods have a similar efficiency in terms of memory and processing time
Slight methodical advantage with the indirect method for problems with a generalised Robin boundary condition
Most contributors implement one or the other
direct more popular
no comparative study in terms of relative accuracy, theoretical or anecdotal (that I know of).
SK prefers the direct BEM (as in book) but uses both (teaching, generalisation)

11. Fredholm Boundary Integral Equation
The boundary integral equations are Fredholm Integral Equations
Usually the Fredholm integral equations are second kind but can be first kind
At the core of the BEM we are in the realm of methods for solving integral equations
There are many methods for solving Fredholm integral equations
In the BEM context only two are used – the Galerkin method and that of collocation
Collocation is easier to apply
Background Solution of Fredholm Integral Equations by Collocation