
Formulation of classical meshless methods 45
Matrix G is symmetric. The solution of Equation (2.115) leads to
aGU
0
1
S
=
−
(2.118)
Substituting Equation (2.118) into Equation (2.108) results in
uxx
h
Q
TT
S
(, )R(x)p(x)GU
1
=
−
(2.119)
uxx
h
Q
T
S
T
TT
(, )(x) U,where(x)
R(x) p(x)
G
1
ΦΦ=
−
(2.120)
Equation (2.120) can be rewritten as
ux
h
Q
T
i
i
n
(, )(x) U
1
∑
=Φ
=
(2.121)
Equation (2.121) is the nal expression of the RBF shape functions. The
approximate derivatives of the eld variables are computed by simply dif-
ferentiating Equation (2.121) as
u
d
l
d
T
S
(x
x) U
=Φ
(2.122)
where,
denotes the derivative in either the x or y coordinate direction.
The working ...