Optimal Design Techniques in Non-Steady-State Metal Forming Processes - Manufacturing Systems Processes - page 173

FIGURE 5.40 Optimal billet after 7 BFGS iterations
and 17 simulations—Corresponding final part.
FIGURE 5.41 Intermediate shapes of the free surface of the billet and the resulting obtained part, for each simu-
lation during the optimization procedure.
authors using the same technique [5, 69]. The optimal preform shape is shown in Fig. 5.40 , as well as
the straight cylinder obtained at the end of the upsetting. The intermediate shapes which have been tested
by the optimization procedure are shown in Fig. 5.41 .
Shape Optimization of an Extrusion Die
Being a steady-state process, the design of optimal extrusion dies has been one of the first applications
of shape optimization in metal forming [3, 13, 49]. The incremental sensitivity analysis developed here
allows us to approach extrusion as a non steady-state process. The initial mesh and the extrusion dies
are shown in Fig. 5.42 . The materials behavior is viscoplastic. The simulation can be completed without
remeshing. When friction is neglected and when the material behavior does not depend on temperature
nor hardening, the process becomes steady as soon as the material front exits the die. This effect is shown
by the isovalues of the strain rates (see Fig. 5.42 ). The following standard optimization problem can be
proposed: find the shape of the die which minimizes the extrusion force. This shape is described by a
spline, using two active shape parameters (see Fig. 5.42 ).
We have used this example to study the convergence behavior of the optimization algorithm. Actually,
the graph of the objective function can be plotted, using a series of 80 simulations with different values
of the two parameters and interpolating it (see Fig. 5.43 ). The function is convex and its unique minimum
is easy to locate.
Figure 5.44 s hows a 2D representation of the same plot with isovalues. In order to evaluate the influence
of the initial guess on the convergence of the algorithm, we have chosen several starting points (circles
in Fig. 5.44 ) randomly distributed in the space of the two parameters. For the seven tests, the shapes of
the starting dies are shown in Fig. 5.45 , along with the optimal die.
Whatever the initial design, the algorithm converges to almost the same minimum (triangles in Fig. 5.44 )
but with a different number of iterations. Table 5.1 shows the number of simulations needed for each test.