Optimal Design Techniques in Non-Steady-State Metal Forming Processes - Manufacturing Systems Processes - page 167

FIGURE 5.29 Time evolution of d ene / dp using either a constant time step or a time step adapted to the contact

events.

the present forging model (time integration, contact evolution, mass conservation, remeshings) influence
the derivatives and affect their accuracy.

Time Integration
We study the simple upsetting process described in Fig. 5.37 . The contact is assumed to be perfectly
sticking and the shape of the initial billet is described by a spline curve to be optimized. This academic
problem requires a single forging step without contact evolution and without remeshing. It is well suited
for testing the accuracy of the sensitivity analysis by comparing the analytical derivatives d

EVE / dp of the
total forging energy with the numerical values obtained by a finite difference scheme, as defined by Eqs.
(5.4) and (5.5).
The estimated error on d

EVE / dp is about 0.8 10 4 % at the first time increment. As the process
simulation goes on, the accuracy decreases due to the cumulating of the discretization errors. Neverthe-
less, after 70 time increments, the error is still smaller than 0.1%, which is quite satisfactory, although
this accuracy can be increased by reducing the time step size. In fact, it is noticed that, in logarithmic
scale, the estimated error on dX / dp at the end of the upsetting process is a quasi-linear function of the
time step size, as suggested by the discretization order of the time integration scheme (see Eq. (5.31)).

Contact Algorithm—Time Step Adaptation
We study again the academic forging example presented in Fig. 5.13 . This is the forging of a simple
cylinder by a die which shape is discretized by a three points spline. The shape sensitivity of the forming
energy, d ene / d p evolves in time as shown in Fig. 5.29 . It is initially equal to zero and continuously

changes along with the material deformation and the nodal contact events. These events produce jumps
in the sensitivity evolution, because when a node slightly penetrates the die, it is projected back onto the
die surface (see Fig. 5.18 ). A more accurate strategy has been proposed by reducing the time step in order
to avoid the penetrations (see Fig. 5.19 ) . Figure 5.29 shows that d

EVE / dp is strongly affected by the
choice of the time step strategy, so the more accurate has to be used. Figure 5.30 shows that it provides
a good agreement with the numerical derivatives: after a height reduction of nearly 40%, the difference
between the two values of the objective function is about 0.5 10 5 %. Such a good result could not be
obtained with a constant time step as

ene was a discontinuous function of the die shape parameter p .

Incremental Mass Conservation
We previously described an incremental mass conservation formulation which makes it possible to avoid
the numerical volume losses. In the present example, the classical formulation [Eq.(5.12)] provides a