
Recent developments of meshless methods 133
The approximate derivatives of the rst-order for a 2-D problem are simi-
larly given by
∑
=
=
fxyafx y(,
xij
ik
x
k
j
k
N
(1)
1
x
(3.210)
∑
=
=
fxyafx y(,
yij
jk
y
i
k
k
N
(1)
1
y
(3.211)
where a
ik
x
and a
jk
y
are the DQ weighting coefcients in the x and y direc-
tion, respectively. They are computed by Equation (3.209), and N
x
and
N
y
are the total virtual nodes in the local DQ domain of the concerned
virtual node
in the x and y directions, respectively. Similarly, the
approximate derivatives of the second order are as
∑
=
=
dfxy
dx
bf
(,)
(,)
ij
ik
h
k
j
k
N
2
2
1
x
(3.212)
∑
=
=
dfxy
dy
bf
(, )
(, )
ij
jk
h
i
k
k
N
2
2
1
y
(3.213)