Civil Engineering Reference
In-Depth Information
Derivatives Calculation
In order to simplify the derivative calculations and focus on their basic ideas, this presentation is carried
out in the same frame of hypothesis as in Section 5.3.
According to the velocity formulation which has been used to solve the forging problem equations,
the different variables of the objective and constraint functions, such as
˙
,
,
,
…
v
t
,
can be expressed
as functions of the velocities and coordinates. Consequently,
and
C
can be considered as functions
explicit of the optimization parameters,
p
,
x
, and
v
. In the following, only the objective function will be
considered while the same method applies to the constraint function.
()
p
(
vp
()
,
xp
()
,
p
)
(5.80)
For the discrete method, the discretized form of
is considered:
N
inc
h
p
()
i
(
V
i
p
()
,
X
i
p
()
,
p
)
(5.81)
i
0
where
i
is the contribution to the
value which is computed at increment
i
, for instance:
for
fil
, if
i
N
inc
, then
i
0 while
Ninc
1
desired
d
(5.82)
end
des
\
end
des
m
1
˙
q
1
for
ene
,
i
,
i
K
(
3
)
d
K
v
t
d
t
i
(5.83)
t
c
i
The chain rule differentiation gives:
N
inc
d
h
V
i
i
d
d
V
i
X
i
i
d
d
X
i
p
i
--------
--------
--------
--------
--------
(5.84)
d
p
i
0
The partial derivatives and are straightforwardly calculated from the differ-
entiation of the discrete equations which define the objective function. On the other hand,
dV
i
/
dp
is
calculated from the differentiation of the discrete equations of the forging problem, while
dX
i
/
dp
is obtained
by the time integration of
dV
i
/
dp
. As the direct forging problem is solved incrementally, the sensitivity
analysis is carried out in the same way.
i
V
i
,
i
X
i
i
P
,
Incremental Calculation of the Basic Derivatives
In fact, at time
t
0,
dX
0
/
dp
is known. It is not equal to zero only if the initial shape of the workpiece
is also to be optimized. In this specific case,
dX
0
/
dp
is obtained by the differentiation of the functions
which parameterize the shape and provide
X
0
(
p
). Possibly the meshing itself could also have to be
differentiated, but it is a more complex issue which has been addressed in structural mechanics for
instance, and which is not relevant for most of the present studied problems.
For any time increment
i
,
dX
i
/
dp
is assumed to be known, so
dV
i
/
dp
can be calculated as it will be
described. Then,
dX
i
1
/
dp
is provided by differentiation of the time integration scheme. For the one step
Euler scheme, we have:
X
i
1
X
i
V
i
t
i
(5.85)
d
X
i
1
d
d
i
i
--------
--------
so:
-------------
t
i
(5.86)
d
d
d
