Chapter 16
Boundary Element Method
We will proceed to learn about Object Oriented Programming and Design by look
ing at a specific example that revolves around the solution of the Laplace equation
by the Boundary Element Method (BEM). The benefits of this approach is that
it takes a 2D or 3D PDE and reduces it to an algorithm that involves 1D or 2D
computations only, respectively. For simplicity we will focus on the 2D case only.
16.1
The equations and the boundary conditions
The Laplace equation takes the form
∇
2
φ
=
∇ ·
(
∇
φ
) = 0
(16.1)
subject to boundary conditions of the type
1.
Dirichlet
where the unknown function is imposed on a portion of the bound
ary, i.e.
φ
(
x
) =
a
(
s
),
s
being a coordinate system along the boundary.
2.
Neumann
where the normal derivative of the function is known
∇
φ
·
n
=
q
(
s
)
where
n
is the outward unit normal.
3.
Robin
where a relationship of the form
αφ
+
β
∇
φ
·
n
=
r
(
s
). A wellposed
problem requires the specification of only one type of boundary conditions
on any portion of the boundary, that is either the function value is specified,
or the normal derivative but not both.
The data imposed on the boundary form are part of the problem’s specifi
cation and can be quite arbitrary except for some minor constraints.
For
example, if Neumann conditions are applied over the entire boundary, then
the solution can be determined only up to an additive constant, that is if
φ
is a solution of
∇
2
φ
= 0 with
∇
φ
·
n
=
q
, then so is
φ
+
C
where
C
is an
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CHAPTER 16. BOUNDARY ELEMENT METHOD
arbitrary constant. In that case the flux have to satisfy a compatability con
dition that can be derived by applying the divergence theorem to equation
16.1:
integraldisplay
Ω
∇
2
φ
d
A
=
integraldisplay
∂
Ω
∇
φ
·
n
d
s
=
integraldisplay
∂
Ω
q
d
s
= 0
(16.2)
16.2
From PDE to Integral Equation
Consider a function
G
, solution to a Poisson equation of the form:
∇
2
G
=
Q
(
x
−
x
i
)
(16.3)
where
Q
(
x
−
x
i
) is a function that depends on the space variable
x
and on the
source
point
x
i
where an impulse forcing is applied. I will keep these terms vague
for the time being, and will delay specifying them until later. Multiplying equation
16.3 by
φ
and 16.1 by
G
and substracting the resulting expressions we get:
φ
∇
2
G
−
G
∇
2
φ
=
φG
(
x
−
x
i
)
(16.4)
Integrating the above equation over the area of the domain Ω we get:
integraldisplay
Ω
(
φ
∇
2
G
−
G
∇
2
φ
) d
A
=
integraldisplay
Ω
φG
(
x
−
x
i
) d
A
(16.5)
The chain rule of differentiation and the Gauss divergence theorem can be used
to turn the left hand side into a boundary integral only.
Indeed since for any
differentiable functions
a
and
b
:
∇ ·
(
a
∇
b
) =
∇
a
· ∇
b
+
a
∇
2
b
, we can then write:
φ
∇
2
G
−
G
∇
2
φ
=
∇ ·
(
φ
∇
G
)
− ∇
φ
· ∇
G
− ∇ ·
(
G
∇
φ
) +
∇
φ
· ∇
G
(16.6)
=
∇ ·
(
φ
∇
G
)
− ∇ ·
(
G
∇
φ
) =
∇ ·
(
φ
∇
G
−
G
∇
φ
)
(16.7)
Furthermore the divergence form of the last equations permits the application of
the Gauss divergence theorm to get:
integraldisplay
Ω
(
φ
∇
2
G
−
G
∇
2
φ
) d
A
=
integraldisplay
Ω
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 Fall '08
 Staff
 Derivative, Normal mode, Green's function, integral equation

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