Optimal Design Techniques in Non-Steady-State Metal Forming Processes - Manufacturing Systems Processes - page 144

FIGURE 5.12 Error on the finite difference derivatives resulting from the discretization error on the objective

function.

i th

where p i is a vector of parameter values which is almost equal to zero, but the coordinate,

0
0
0

which is equal to p : p i

pi th coordinate ( )

0
0

This is an easy way to compute the function derivatives, but it has two major disadvantages: it is quite
time consuming as it requires N par or 2 N par additional simulations of the process, and it is difficult to
control the accuracy of the derivatives. In fact, it is all the better that p is smaller. However, the round

off and discretization errors have to be taken into account, so p cannot get a smaller value than the

accuracy of the simulation. If p is too small, as shown in Fig. 5.12 , the difference between (p pi)

and p

must be sufficiently large, as shown in Fig. 5.12 , but this also reduces the accuracy of the derivative.
Although this method has been used in the beginnings, for instance for the preform shape optimization
in forging [Bec89], the number of required simulations for reaching an optimal solution was large (more
than 250), probably due to the poor accuracy of the numerical derivatives.
On the other hand, for more complex forging processes which require several remeshings, the discretization
error is even increased during the transfer of domain and variables from one mesh onto the other. It can get
so large that the numerical derivatives do not make sense any longer. The following simple example shows
it. This is the forging of a simple cylinder by a die, the shape of which is discretized by a three points spline
(see Fig. 5.13 ) . Three remeshings are required to compute the prescribed deformation.
The time evolution of the derivative of the energy objective function is shown in Fig. 5.14 . This
derivative has been computed both analytically, as it will be described in the next sections, and using the
finite difference method. Two

( p ) belongs to the discretization error so the numerical derivative can get any value. Indeed,

10%, after the first
remeshing the numerical derivative is just slightly different from the analytical value, while after the third
remeshing, the numerical derivatives oscillate so much that they do not make sense. With

p values, 10% and 100%, have been tested. With p

100%, the
oscillations disappear, but the accuracy of the derivatives is poor with respect to the analytical values, as
shown in Fig. 5.14 .

p