
118 Meshless methods and their numerical properties
where Γ
w
, Γ
θ
, Γ
M
and Γ
v
are the boundaries of w, θ, M, and V imposed,
respectively, and t is the time and t
0
is the initial time. The local weak
form of the differential Equation (3.166) over a local support domain Ω
s
bounded by Γ
s
can be constructed by the weighted residual method as
∫
∫
ρ
∂∂
∂∂
∂
∂
+θ
−
Ω=
ρ
∂θ ∂
∂∂
∂θ
∂
+
∂
∂
+θ
Ω=
Ω
Ω
vA
w
tx
GAk
w
x
fd
vI
tx
EI
x
GAk
w
x
d
0
0
s
s
2
2
2
2
s
s
(3.171)
where v is the weight function. Integrating Equation (3.171) by parts, one
can have
∫∫
∫∫
ρ
∂
∂
Ω+
∂
∂
+θ
−
Ω−
∂
∂
+θ
=
ρ
∂θ
∂
Ω+
∂θ
∂
+
∂
∂
+θ
Ω−
∂θ
∂
=