1.7 Collocation
The step from the integral equation to the linear system of
equations, as illustrated in Section 1.2, is carried
out by applying an integral equation method to
an equation such as (3) to give an equation like (5).
There are a range
of methods for carrying this out (ref [3])
but the most favoured technique is that of collocation,
because of its inherent simplicity.
Collocation can be applied in a remarkably elementary form,
which is termed the C-1 collocation method in this text
since it is derived by approximating the boundary functions by
a constant on each panel.
In this subsection the C0 collocation method is briefly
outlined.
First the boundary S is assumed to be expressed as a set of
panels; S = åj = 1n DSj.
Often the DSj are referred to as elements
in other texts. However, the term element
refers not only to the geometry of DSj but also
to the method of representing the boundary functions on DSj.
The C0 collocation method involves representing
the boundary function by a constant on each panel.
For example
|
f(p) » fj , v(p) » vj if p Î DSj . |
| (7) |
The substitution of representations of this form for the
boundary functions in the integral equation reduce it to
discrete form. The combination of the representation of the
panels and the approximation of the boundary functions,
as typified by (7), defines the element.
The simplifications allow us to re-write equation (3)
as the approximation
|
|
n
å
j = 1
|
{(M + |
1
2
|
I) e }DSj(p) fj » |
n
å
j = 1
|
{ L e }DSj(p) vj (p Î S) |
|
where e is the unit function (e º 1). The
{ L e }DSj(p), for example, for
a specific point p,
are the numerical values of definite integrals.
The { L e }DSj(p) are independent of the
boundary functions
and are termed the discrete form of the L integral operator.
The
constant approximation is taken to be the value of the boundary
functions at the representative central point (the
collocation point) on each panel.
By finding the discrete forms of the
relevant integral operators for all the collocation points
a system of the form
|
|
n
å
j = 1
|
{(M + |
1
2
|
I) e}DSj(pi) fj » |
n
å
j = 1
|
{ L e }DSj(pi) vj |
| (8) |
for i = 1,2,...,n is obtained by putting p = pi
in the previous approximation.
Note that because of the boundary approximation and approximation
of the boundary functions, the discrete equivalent of
equation (3) is an approximation relating the
exact values of the boundary functions at the collocation points.
This system of approximations (8)
can now be written in the matrix-vector form
with
|
[L]ij = { L e }DSj(pi) , [M]ij = { M e }DSj(pi). |
| (10) |
The vectors f and v
represent the exact values of f and v at the collocation
points. The approximation between the exact values (9)
can be interpreted as an exact relationship between approximate values,
equation (5).
The terms on the right hand side of the equations in (10)
are definite integrals that need to be computed to return their
value on the left hand side of the equations. The process
of computing the integral is termed the discretisation
of the integral operators. In many cases
this is carried out through a straightforward
application of a numerical integration method. However, in the
implementation of boundary element methods it is well-known
that some of the integrals that arise have singular integrands,
requiring special treatment. See Chapter 3
for more details on how this is done in the
accompanying software.
Return to boundary-element-method.com/helmholtz
Return to boundary-element-method.com