## Abstract

During the last decades, Finite Element (FEM) simulations of metal forming processes have become important tools for designing feasible production processes. In more recent years, several authors recognised the potential of coupling FEM simulations to mathematical optimization algorithms to design optimal metal forming processes instead of only feasible ones.

Within the current project, an optimisation strategy is being developed, which is capable of optimising metal forming processes in general using time consuming nonlinear FEM simulations. The expression “optimisation strategy” is used to emphasise that the focus is not solely on solving optimisation problems by an optimisation algorithm, but the way these optimisation problems in metal forming are modelled is also investigated. This modelling comprises the quantification of objective functions and constraints and the selection of design variables. This paper, however, is concerned with the choice for and the implementation of an optimisation algorithm for solving optimisation problems in metal forming. Several groups of optimisation algorithms can be encountered in metal forming literature: classical iterative, genetic and approximate optimisation algorithms are already applied in the field. We propose a metamodel based optimization algorithm belonging to the latter group, since approximate algorithms are relatively efficient in case of time consuming function evaluations such as the nonlinear FEM calculations we are considering. Additionally, approximate optimization algorithms strive for a global optimum and do not need sensitivities, which are quite difficult to obtain

for FEM simulations. A final advantage of approximate optimisation algorithms is the process knowledge, which can be gained by visualising metamodels.

In this paper, we propose a sequential approximate optimization algorithm, which incorporates both Response Surface Methodology (RSM) and Design and Analysis of Computer Experiments (DACE) metamodelling techniques. RSM is based on fitting lower order polynomials by least squares regression, whereas DACE uses Kriging interpolation functions as metamodels. Most authors in the field of metal forming use RSM, although this metamodeling technique was originally developed for physical experiments that are known to have a stochastic nature due to measurement noise present. This measurement noise is absent in case of deterministic computer experiments such as FEM simulations. Hence, an interpolation model fitted by DACE is thought to be more applicable in combination with metal forming simulations. Nevertheless, the proposed algorithm utilises both RSM and DACE metamodelling techniques. As a Design Of Experiments (DOE) strategy, a combination of a maximin spacefilling Latin Hypercubes Design and a full factorial design was implemented, which takes into account explicit constraints. Additionally, the algorithm incorporates cross validation as a metamodel validation technique and uses a Sequential Quadratic Programming algorithm for metamodel optimisation. To overcome the problem of ending up in a local optimum, the SQP algorithm is initialised from every DOE point, which is very time efficient since evaluating the metamodels can be done within a fraction of a second. The proposed algorithm

allows for sequential improvement of the metamodels to obtain a more accurate optimum.

As an example case, the optimisation algorithm was applied to obtain the optimised internal pressure and axial feeding load paths to minimise wall thickness variations in a simple hydroformed product. The results are satisfactory, which shows the good applicability of metamodeling techniques to optimise metal forming processes using time consuming FEM simulations.

Within the current project, an optimisation strategy is being developed, which is capable of optimising metal forming processes in general using time consuming nonlinear FEM simulations. The expression “optimisation strategy” is used to emphasise that the focus is not solely on solving optimisation problems by an optimisation algorithm, but the way these optimisation problems in metal forming are modelled is also investigated. This modelling comprises the quantification of objective functions and constraints and the selection of design variables. This paper, however, is concerned with the choice for and the implementation of an optimisation algorithm for solving optimisation problems in metal forming. Several groups of optimisation algorithms can be encountered in metal forming literature: classical iterative, genetic and approximate optimisation algorithms are already applied in the field. We propose a metamodel based optimization algorithm belonging to the latter group, since approximate algorithms are relatively efficient in case of time consuming function evaluations such as the nonlinear FEM calculations we are considering. Additionally, approximate optimization algorithms strive for a global optimum and do not need sensitivities, which are quite difficult to obtain

for FEM simulations. A final advantage of approximate optimisation algorithms is the process knowledge, which can be gained by visualising metamodels.

In this paper, we propose a sequential approximate optimization algorithm, which incorporates both Response Surface Methodology (RSM) and Design and Analysis of Computer Experiments (DACE) metamodelling techniques. RSM is based on fitting lower order polynomials by least squares regression, whereas DACE uses Kriging interpolation functions as metamodels. Most authors in the field of metal forming use RSM, although this metamodeling technique was originally developed for physical experiments that are known to have a stochastic nature due to measurement noise present. This measurement noise is absent in case of deterministic computer experiments such as FEM simulations. Hence, an interpolation model fitted by DACE is thought to be more applicable in combination with metal forming simulations. Nevertheless, the proposed algorithm utilises both RSM and DACE metamodelling techniques. As a Design Of Experiments (DOE) strategy, a combination of a maximin spacefilling Latin Hypercubes Design and a full factorial design was implemented, which takes into account explicit constraints. Additionally, the algorithm incorporates cross validation as a metamodel validation technique and uses a Sequential Quadratic Programming algorithm for metamodel optimisation. To overcome the problem of ending up in a local optimum, the SQP algorithm is initialised from every DOE point, which is very time efficient since evaluating the metamodels can be done within a fraction of a second. The proposed algorithm

allows for sequential improvement of the metamodels to obtain a more accurate optimum.

As an example case, the optimisation algorithm was applied to obtain the optimised internal pressure and axial feeding load paths to minimise wall thickness variations in a simple hydroformed product. The results are satisfactory, which shows the good applicability of metamodeling techniques to optimise metal forming processes using time consuming FEM simulations.

Original language | English |
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Title of host publication | Proceedings of the First Inveted COST 526 (APOMAT), Conference for Automatic Process Optimization in Materials Technology |

Editors | D. Buche, N. Hofmann |

Place of Publication | Morschach, Switzerland |

Pages | 242-251 |

Number of pages | 10 |

Publication status | Published - 2005 |

Event | 1st Conference for Automatic Process Optimization in Materials Technology, APOMAT 2005 - Morschach, Switzerland Duration: 1 Jan 2005 → 1 Jan 2005 |

### Conference

Conference | 1st Conference for Automatic Process Optimization in Materials Technology, APOMAT 2005 |
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Abbreviated title | APOMAT |

Country/Territory | Switzerland |

City | Morschach |

Period | 1/01/05 → 1/01/05 |

## Keywords

- METIS-229394
- IR-59539