3.3  Discretization of the Integral Operators

The computational method is developed in a similar way to that described in reference [15]. In order to derive the discrete forms of the integral operators (12), (13), (14) and (15), G is approximated by a set of n panels G' = ånj = 1 DG'j . The boundary function m is replaced by its equivalent on the approximate boundary G. The function is then replaced by a constant on each panel. Thus for the L operator:

{ L m}G (p) » { L
m'
 
 }G' (p) » n
å
 j = 1 
 ó
õ

DG'j 
 G(p,q)
m'
 
 (pj) dSq = n
å
 j = 1 
 [
m'
 
 (pj) { L
e'
 
 }DG'j (p) ] ,
(27)
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:

{ L
e'
 
 }DG'j(p) = ó
õ

DG'j 
 G (p,q) dSq ,
(28)

{ M
e'
 
 }DG'j(p) = ó
õ

DG'j 
 G
nq
 (p,q) dSq ,
(29)

{ Mt
e'
 
 }DG'j(p; vp) =
vp
 ó
õ

DG'j 
 G (p,q) dSq ,
(30)

{ N
e'
 
 }DG'j(p; vp) =
vp
 ó
õ

DG'j 
 G
nq
 (p,q) dSq ,
(31)

The derivative operator in (30) can always be carried inside the integral. The same is true for the operator in (31) when p does not lie on the element DG'j. Thus we may write:

{ Mt
e'
 
 }DG'j(p; vp) = ó
õ

DG'j 
 G
vp
 (p,q) dSq ,
(32)

{ N
e'
 
 }DG'j(p; vp) = ó
õ

DG'j 
 2 G
vp nq
 (p,q) dSq when  p \not Î D
G'
 
j 
  .
(33)
When p \not Î DG'j the integrals of (28), (29), (30) and (31) will all be regular and hence are amenable to standard quadrature. The same is true for the integrands of (29) and (30) when p Î DG'j (though not on the edge of the element). However, the evaluation of the discrete integral operators (28) and (31) generally require special treatment when p Î DG'j.

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 L and N operators.

Return to boundary-element-method.com/laplace
Return to boundary-element-method.com