Civil Engineering Reference
In-Depth Information
Gradient-Based Optimization Algorithm
Several classical algorithms can be used for the minimization of the objective function
with respect
to the shape parameters
p
. Quasi-Newton type algorithms are the most effective, and among them, the
most widely used is the BFGS algorithm [20]. It consists of the following steps:
1. initialize
p
[choice of the initial shape and/or process parameters]
2. compute
/
dp
[simulation of the successive operations of the forging sequence]
3. compute
H
BFGS
, a numerical approximation of the hessian matrix:
,
d
d
2
dp
2
----------
H
BFGS
(5.111)
1
4. compute
P
(
)
(5.112)
BFGS
p
5. update
p
into
p
, eventually using a line search procedure:
p
p
p
(5.113)
where
is the line search coefficient
6. check convergence; if verified stop; otherwise go to step 2.
H
BFGS
is incrementally calculated as follows:
t
P
t
P
t
P
i
grad
()
()
P
i
grad
(5.114)
------------------------------------
-----------------------------------
-----------------------------------
H
i
1
I
H
i
I
P
t
P
t
P
t
grad
()
grad
()
grad
()
where
grad
()
grad
(
i
1
)
grad
()
(5.115)
P
i
P
i
1
P
i
(5.116)
and I is the identity matrix
From a technical standpoint, the optimization software consists of three main modules. First, for any set
of
p
values and the corresponding shapes, the forging simulation module, which is enriched with the
objective function and gradient calculations, computes
/
dp
(
p
). These values are inputs for
the optimization module which computes the new values of the shape parameters,
p
(
p
) and
d
. Finally, a conversion
module generates the shapes of the forging dies, which correspond to the actual values of the parameters,
in the format required by the simulation software.
5.5
Examples
Shape Optimization of the Initial Billet
After forging a straight cylinder, the free surface bulges (see
Fig. 5.37
).
We look for the initial design of
the billet which produces a straight side at the end of forging (see
Fig. 5.38
).
First, the free surface of the
billet is discretized by a spline curve. Then, the optimization algorithm is used to minimize the gap
between the forged part and the desired cylinder. The
fil
function is used, without the 1
desired
characteristic
