96
5 Finite Element Formulations
Substituting the above assumptions into the dynamic equation one obtains the displacement vector at time t + Δt as
q
(t+Δt)
=
1
(Δt) 2 M
(t)
uu +
1
2(Δt)
C
(t)
uu
−1
F
(t)
Residual
(5.102)
where
F
(t)
Residual = F
(t)
ue − F
(t)
ui +
1
(Δt) 2 M
(t)
uu
2q
(t)
− q
(t−Δt)
+
1
2(Δt)
C
(t)
uu q
(t−Δt)
+ K
(t)
uu
q
(t−Δt)
− q
(t)
.
(5.103)
For the first step of CDA, the displacement at time t − Δt is needed, which can be
built as
q
(t−Δt)
= q
(t)
+ (Δt) ˙
q
(t) +
(Δt)
2
2
¨
q
(t) ,
(5.104)
where q
(t) and ˙
q
(t) are prescribed, and ¨
q
(t) can be determined by
¨
q
(t) = (M
(t)
uu )
−1
F
(t)
ue − F
(t)
ui − C
(t)
uu ˙
q
(t) − K
(t)
uu q
(t)
.
(5.105)
5.8.3 Newton-Raphson Method
For nonlinear static equilibrium equation, the calculation iteration is investigated.
The kth iteration of static equilibrium equation is defined as
K
(k)
uu q
(k)
= F
(k)
ue − F
(k)
ui .
(5.106)
Thus, the incremental displacement vector in the kth iteration can be solved by
q
(k)
= (K
(k)
uu )
−1
F
(k)
ue − F
(k)
ui
.
(5.107)
Consequently, the displacement vector in iteration k + 1 can be obtained as
q
(k+1)
= q
(k)
+ q
(k)
.
(5.108)
Using the current displacement matrix, the system matrices and vectors will be
updated to K
(k+1)
uu
, F
(k+1)
ue
and F
(k+1)
ui
. In such a way, the equilibrium equation changes
to the (k + 1)th iteration. Again, q can be calculated, and new equilibrium equation
is formed. Following the above loops, until it converges within an accepted error
as