3.3 Discretisation
In order to derive the discrete forms of the integral operators
(3.1), (3.2), (3.3) and (3.4),
G is approximated by a set of n panels
G* = ånj = 1 DG*j . The boundary function z
is replaced by its equivalent on the approximate boundary
G*. The function is then replaced by a constant on
each panel. Thus for the Lk integral operator:
|
{ Lk z}G (p) » { Lk |
~
z
|
}G* (p) » |
n
å
j = 1
|
|
ó
õ
|
DG*j
|
Gk(p,q) |
~
z
|
(pj) dSq » |
n
å
j = 1
|
[ |
~
z
|
(pj) { Lk |
~
e
|
}DGj (p) ] , |
| (3.29) |
where [e*] is the unit function.
The other integral operators may be discretised in a similar way.
The discrete forms are thus
defined as follows:
|
{ Lk |
~
e
|
}DG*j(p) = |
ó
õ
|
DG*j
|
Gk (p,q) dSq , |
| (3.30) |
|
{ Mk |
~
e
|
}DG*j(p) = |
ó
õ
|
DG*j
|
|
¶Gk
¶nq
|
(p,q) dSq , |
| (3.31) |
|
{ Mkt |
~
e
|
}DG*j(p; up) = |
¶
¶up
|
|
ó
õ
|
DG*j
|
Gk (p,q) dSq and |
| (3.32) |
|
{ Nk |
~
e
|
}DG*j(p; up) = |
¶
¶up
|
|
ó
õ
|
DG*j
|
|
¶Gk
¶nq
|
(p,q) dSq . |
| (3.33) |
The derivative operator in (3.32) can always be
carried inside the integral.
The same is true for the operator in (3.33) when p does not lie
on the panel DG*j. Thus we may write:
|
{ Mkt |
~
e
|
}DG*j(p; up) = |
ó
õ
|
DG*j
|
|
¶Gk
¶up
|
(p,q) dSq , |
| (3.34) |
|
{ Nk |
~
e
|
}DG*j(p; up) = |
ó
õ
|
DG*j
|
|
¶2 Gk
¶up ¶nq
|
(p,q) dSq when p \not Î D |
~
G
|
j
|
. |
| (3.35) |
When p \not Î DG*j the integrals of
(3.30), (3.31), (3.32) and (3.33) will all
be regular and hence are amenable to standard quadrature.
The same is true for the integrands of (3.31) and (3.32)
when
p Î DG*j (although not on the edge of the panel). However, the
evaluation of the discrete integral operators (3.30) and
(3.33) generally require special
treatment when p Î DG*j.
The special techniques applied here involve `subtracting out'
the singularity and evaluating the singular part and remaining regular
part separately. The following results are immediate from the
asymptotic properties of the kernel functions (3.25) and
(3.36):
|
{ Lk |
~
e
|
}DG*j (p) = { L0 |
~
e
|
}DG*j (p) + |
ó
õ
|
DG*j
|
( Gk(p,q) - G0(p,q) ) dSq , |
| (3.36) |
|
{ Nk |
~
e
|
}DG*j (p; up) = { N0 |
~
e
|
}DG*j (p; up) - |
1
2
|
k2 { L0 |
~
e
|
}DG*j (p) + |
|
|
|
ó
õ
|
DG*j
|
( |
¶2 Gk
¶up ¶nq
|
(p,q) - |
¶2 G0
¶up ¶nq
|
(p,q) + |
1
2
|
k2 G0(p,q) ) dSq , |
| (3.37) |
where in each of (3.36) and (3.37) the explicit integral is non-singular.
Evaluation in this way requires the computation of the regular integral
(amenable to standard quadrature) and the determination of the
subtracted out part.
In summary, the evaluation of the integral operators requires a summation
of a set of integrand values multiplied by quadrature weights. In the
case when p Î DG*j the evaluation of the
subtracted out part is also required for the Lk and Nk operators.
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