6
1 Idea and Derivation of the Method
Fig. 1.3 Typical cell for the cell collocation method and appropriate weight function
N i−1 =
1
2
ξ(ξ − 1) , N i = (1 − ξ)(1 + ξ) , N i+1 =
1
2
ξ(1 + ξ) .
(1.16)
The second-order derivative of the approximate function as given in Eq. (1.15) can
be written under consideration of
dξ
dX
=
1
as:
d
2 u
dX 2 =
1
X 2
d
2 N i−1
dξ 2 u i−1 +
d
2 N i
dξ 2 u i +
d
2 N i+1
dξ 2 u i+1
=
1
X 2 (1u i−1 − 2u i + 1u i+1 ) .
(1.17)
Thus, the residual
r =
1
X 2 (1u i−1 − 2u i + 1u i+1 ) = 0
(1.18)
can be used to formulate the collocation statement at node i as:
i+1
i−1
r W dX =
i+1
i−1
1
2 (1u i−1 − 2u i + 1u i+1 ) δ(X − X i )
!
= 0 .
(1.19)
Under consideration of the properties of the Dirac delta function [8], i.e.,