Optimal Design Techniques in Non-Steady-State Metal Forming Processes - Manufacturing Systems Processes - page 145

FIGURE 5.13 Test problem—forging of a cylinder
using a die defined by a four points cubic spline—the
arrow shows the perturbed characteristic point.

FIGURE 5.14 Numerical and analytical derivatives of the energy objective function versus the increment number.

This shows that, for standard non steady state forming problems, numerical derivatives cannot be
used and that an effort has to be made to compute them analytically.
Introduction to the Gradient Calculation
In order to carry out the derivatives calculations, which is also called the sensitivity analysis , several
methods can be considered. They can be split in two main classes: the methods based on the direct
differentiation and the methods based on the adjoint state. These two approaches are developed and
compared later. However it is still possible to split each of these methods into two other classes, whether
the differentiation is first carried out at the discrete or at the continuous level.
Variational Differentiation Methods
Following the variational approaches, the problem equations are differentiated before being discretized.
From a mathematical standpoint, it could probably be demonstrated that these two operations are sym-
metric, or almost symmetric, as numerical experiments show. However, from a strict numerical standpoint,
the symmetry is not obtained due to the round off errors. According to this, the variational approaches
are sometimes considered to be less consistent, as the resolutions of the two problems, the direct forging