During the last decades, Finite Element (FEM) simulations\ud
of metal forming processes have become important\ud
tools for designing feasible production processes. In more\ud
recent years, several authors recognised the potential of\ud
coupling FEM simulations to mathematical optimisation\ud
algorithms to design optimal metal forming processes instead\ud
of only feasible ones.\ud
Within the current project, an optimisation strategy is being\ud
developed, which is capable of optimising metal forming\ud
processes in general using time consuming nonlinear\ud
FEM simulations. The expression “optimisation strategy”\ud
is used to emphasise that the focus is not solely on solving\ud
optimisation problems by an optimisation algorithm, but\ud
the way these optimisation problems in metal forming are\ud
modelled is also investigated. This modelling comprises\ud
the quantification of objective functions and constraints\ud
and the selection of design variables.\ud
This paper, however, is concerned with the choice for\ud
and the implementation of an optimisation algorithm for\ud
solving optimisation problems in metal forming. Several\ud
groups of optimisation algorithms can be encountered in\ud
metal forming literature: classical iterative, genetic and\ud
approximate optimisation algorithms are already applied\ud
in the field. We propose a metamodel based optimisation\ud
algorithm belonging to the latter group, since approximate\ud
algorithms are relatively efficient in case of time consuming\ud
function evaluations such as the nonlinear FEM calculations\ud
we are considering. Additionally, approximate optimisation\ud
algorithms strive for a global optimum and do\ud
not need sensitivities, which are quite difficult to obtain\ud
for FEM simulations. A final advantage of approximate\ud
optimisation algorithms is the process knowledge, which\ud
can be gained by visualising metamodels.\ud
In this paper, we propose a sequential approximate optimisation\ud
algorithm, which incorporates both Response\ud
Surface Methodology (RSM) and Design and Analysis\ud
of Computer Experiments (DACE) metamodelling techniques.\ud
RSM is based on fitting lower order polynomials\ud
by least squares regression, whereas DACE uses Kriging\ud
interpolation functions as metamodels. Most authors in\ud
the field of metal forming use RSM, although this metamodelling\ud
technique was originally developed for physical\ud
experiments that are known to have a stochastic na-\ud
¤Faculty of Engineering Technology (Applied Mechanics group),\ud
University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands,\ud
email: [email protected]\ud
ture due to measurement noise present. This measurement\ud
noise is absent in case of deterministic computer experiments\ud
such as FEM simulations. Hence, an interpolation\ud
model fitted by DACE is thought to be more applicable in\ud
combination with metal forming simulations. Nevertheless,\ud
the proposed algorithm utilises both RSM and DACE\ud
metamodelling techniques.\ud
As a Design Of Experiments (DOE) strategy, a combination\ud
of a maximin spacefilling Latin Hypercubes Design\ud
and a full factorial design was implemented, which takes\ud
into account explicit constraints. Additionally, the algorithm\ud
incorporates cross validation as a metamodel validation\ud
technique and uses a Sequential Quadratic Programming\ud
algorithm for metamodel optimisation. To overcome\ud
the problem of ending up in a local optimum, the\ud
SQP algorithm is initialised from every DOE point, which\ud
is very time efficient since evaluating the metamodels can\ud
be done within a fraction of a second. The proposed algorithm\ud
allows for sequential improvement of the metamodels\ud
to obtain a more accurate optimum.\ud
As an example case, the optimisation algorithm was applied\ud
to obtain the optimised internal pressure and axial\ud
feeding load paths to minimise wall thickness variations\ud
in a simple hydroformed product. The results are satisfactory,\ud
which shows the good applicability of metamodelling\ud
techniques to optimise metal forming processes using\ud
time consuming FEM simulations
