Civil Engineering Reference
In-Depth Information
In practice, in order to reduce the number of unknowns and obtain a pure velocity formulation, a penalty
formulation is often used. So, the following penalty term is substituted to the pressure term of Eq. (5.24):
1
m
1
˙
h
˙
h
:
B
k
d
w
h
tr
B
k
3
k
,
2
K
(
)
(
d
w
h
)
pen
div
v
()
d
w
h
()
d
w
h
1
1
1
1
c
N
k
ds
h
K
v
ht
0
(5.26)
where
pen
is a large numerical coefficient.
Equation (5.27) is then substituted to Eq. (5.24):
1
1
pen
1,
d
w
h
div
v
()
d
w
h
(5.27)
1
1
Finally, the discrete equations of the problems can be summarized by:
k
,
R
k
(
X
,
V
)
0
V
k
.
n
k
(5.28)
(
V
die
)
0
c
k
,
R
k
k
(
X
,
V
)
t
0
where
X
and
V
are the arrays of the nodal coordinates and velocities of the finite element mesh,
At each time step of the process, these non linear equations are solved by a Newton-Raphson algorithm:
()
0 or
V
0
()
0
()
0
V
is estimated from the previous time step
()
is known
i
()
i
⇒
(
i
1
)
V
(
i
1
)
()
i
()
i
V
V
V
which is the solution of:
RXV
i
(
1
)
RXV
()
XV
()
(
,
)
(
,
)
R
/
V
(
,
)
V
0
(5.29)
()
i
1
V
R
()
XV
()
RXV
()
------
so:
V
(
,
)
(
,
)
(5.30)
Contact Algorithm and Time Integration
In order to satisfy the unilateral contact condition, the following algorithm is used. Initially, all the nodes
which are geometrically in contact with the dies are treated as contact nodes of
∂
c
[Eq. (5.19)] and the
velocity field is so computed. If the calculated normal stress value of any node of
∂
c
is not negative,
then the node is released from the contact surface
∂
c
, and a new velocity field is computed. This is
repeated until all the nodes of
∂
c
exhibit a negative normal stress value.
At time
t
t
, the domain
t
is updated according to:
k
X
t
k
V
t
k
k
,
X
t
t
t
(5.31)
A node which penetrates the tool during the time interval
t
has to be projected onto the tool surface
at time
t
at time
t
and which is then supposed to remain on the tool surface during
t
.
