1.2 Outline of The Boundary Element Method
In this Section the overall strategy in using the BEM is
briefly outlined. For a more thorough introduction,
the reader is advised to consult the textbooks exclusively
devoted to the boundary element method, such as
Jaswon and Symm [14],
Brebbia [3], Banerjee and Butterfield [2],
Chen and Zhou [6] and Hall [11]. Books on the boundary element
method are listed on
www.science-books.net .
In the derivation of the BEM,
the underlying objective is to replace the partial differential
equation that governs the solution in a domain by an equation
that governs the solution on the boundary alone. For example
the Laplace equation
|
|
N
å
i = 1
|
|
¶2 f(p)
¶pi2
|
= 0 |
|
where N is the dimension of the space, or more concisely,
governing the interior to a domain D bounded by a surface S
can be replaced by an integral equation of the form
|
|
ó
õ
|
S
|
|
¶G
¶nq
|
(p,q) f(q) dSq+ |
1
2
|
f(q) = |
ó
õ
|
S
|
G(p,q) |
¶f
¶nq
|
dSq . |
| (1) |
The function G is known as a Green's function
|
G(p, q) = - |
1
2 p
|
logr in two dimensions, |
|
|
G(p, q) = |
1
4 p
|
|
1
r
|
in three dimensions. |
|
Physically,
G(p,q) represents the effect observed
at a point p of a unit source at the point q.
The terminology ¶*/ ¶nq represents the
partial derivative of the function * with respect to the outward normal
at the point q on the boundary.
The integral equation can be derived from the
Laplace equation by applying Green's second theorem.
The power of the formulation (1) lies in the fact that
it relates the potential f and its derivative on
the boundary alone; no reference is made to f at points in the
domain. In a typical boundary-value problem we may be given
f(q), ¶f(q)/ ¶nq
or a combination of such data on S: equation (1)
is a means of determining the unknown boundary function(s)
from given boundary data.
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