Optimal Design Techniques in Non-Steady-State Metal Forming Processes - Manufacturing Systems Processes - page 168

FIGURE 5.30 Time evolution of d ene / dp using a time step adapted to the contact events—comparison of the

analytical and numerical derivatives.

reasonable volume loss of 0.4% at the end of forging. However, when it comes to the sensitivity analysis,
this level of accuracy is not acceptable. In fact, Fig. 5.31 shows that in this case the time evolution of
d

ene / dp is almost negligible. This is explained by the numerical volume variations which are of the same
magnitude as the variations due to the shape changes, and so hide the shape sensitivities. On the other
hand, with the incremental formulation, the time evolution of the derivatives is quite satisfactory, as
shown in Fig. 5.31 .

Mesh Refinement
The same kind of phenomenon arises with respect to mesh refinement. It is well known that the finer
the mesh, the better the finite element solution. However, numerical experiments show that the derivative
calculations require more accurate solutions using finer meshes than the direct forming problem. For
instance, in the extrusion example described (see Fig. 5.42 ) , three different meshes have been tested (see
Fig. 5.32 ). In Fig. 5.33 , the extrusion force evolution is plotted for the three meshes. It shows that after
120 time increments a quasi-steady-state is reached. Due to discrete contact events, small oscillations can
be noticed. Their frequency and amplitude depend on the mesh size, but, even for the coarser mesh, the
forming force is estimated with a satisfactory accuracy. However, for the sensitivity analysis, Fig. 5.34
shows that the amplitude of the corresponding oscillations can be quite large. In fact, for the coarser
mesh, it is not possible to conclude whether the derivative is positive or negative. With the intermediate
mesh, the oscillations are reduced. Only the finer mesh makes it possible to estimate a steady-state value
of the derivative.
This example shows the influence of the mesh size on the accuracy of the derivatives. For an actual
non-steady-state forming process with remeshings, it is then also important to control the mesh size during
the entire simulation. With the academic forging example presented in Fig. 5.13 , three different values of
the remeshing criteria have been tested. The corresponding meshes are shown in Fig 5.35 at the end of
the simulations. Figure 5.36 shows the time evolution of d

ene / dp for the different meshes. For the coarsest
mesh, it seems that the derivatives are diverging after the second remeshing, while the finer meshes provide
close solutions, so showing that the method is converging toward a given value of the derivative.