POTENTIAL PROBLEMS 39
EVALUATION OF INTEGRALS
The integrals H
0
and G
u
can be calculated using the simple Gauss
quadrature rule (four points for the two-dimensional case) for all
points, except the one corresponding to the node under consideration.
For this integral it is recommended to use higher-order integration
rules or a special logarithmic formula which will be described later on.
For the particular case of constant elements, however, the H
ti
and G
u
integrals can be easily computed analytically. The H
H
term for
instance is identically zero, for fundamental solutions with no
Γ dependence, i.e.
H
it
=
q*dr =
du*
dr
dr dn*
άΓ = 0
(2.52)
This is because n and r are orthogonal over the element.
The G
u
integral can be calculated analytically as follows,
G,.
=
f
w
*dr =
i- j
In
0 W (2.53)
or using the homogeneous coordinate ξ over a segment (Figure 2.6):
ο
»
=
έίΓ'"0>
Γ
%-Ι>(Ο
αΓ
,251)
Figure 2.6 Constant element
f=-i
Transforming coordinates, r = ξτ
ι
,
where 1^ | ^
j/-
2
1
we have,
G
u
= -
π
(2)
i(0)
'"Oh^KräH'
1
"©«]
(2.55)
Noting that the last integral is equal to 1 we have that,