- Mathematical Optimization of Industrial Sheet Metal Forming Processes

# Mathematical Optimization of Industrial Sheet Metal Forming Processes

Publication date

2018

Document type

PhD thesis (dissertation)

Author

Rozgic, Marco

Advisor

Referee

Reese, Stefanie

Granting institution

Helmut-Schmidt-Universität / Universität der Bundeswehr Hamburg

Exam date

2018-04-24

Organisational unit

DDC Class

620 Ingenieurwissenschaften

Keyword

Sheet Metal Forming

Electromagnetic Forming

High-Speed Forming

Quasi-Static Forming

SAND

NAND

IPOPT

Abstract

In this work two methods for virtual process design in the context of coupled quasi-static electromagnetic impulse sheet metal forming applications are introduced. The first half of this thesis is concerned with solving the two main parameter identification problems arising when simulating and designing such a forming process: The identification of the employed material model and the identification of optimal process parameters. For both problems a similar black box framework based on LS-DYNA FEM/BEM simulations and IPOPT, an implementation of the inner point method is used. A thermodynamically consistent material model developed by Vladimirov et al. (2014), which is tailored for the efficient numerical treatment of anisotropic hyperelasticplastic materials is considered. This model is capable of representing rate dependent viscoplastic as well as rate independent elastoplastic materials. To account for material failure it is coupled to a scalar Lemaitre-type damage model. A framework for the reliable identification of the model’s parameters for the simulation of the behavior of the aluminum alloy EN AW-5083 at various strain rates based on uniaxial tensile tests and a corresponding LS-DYNA simulation coupled to IPOPT is introduced. For the comparison of data obtained by simulation and data coming from experiments a distance measure based on the L2????distance of functions is used. To this end an Akima spline developed by Akima (1970) is adapted to the experimental data. These special splines have the advantage of producing smooth interpolates, that are just as smooth as the according experimental curves, as long as no damage occurs. It is hence possible to interpolate the experimental data at every point needed and thus compare it efficiently to the data from simulation. It will be shown how a set of parameters suitable for the modeling of the high-speed situation can be derived from data obtained by quasi-static experiments via a linear extrapolation of the damage threshold parameter. The identified parameter set is suitable to represent a coupled forming process very precisely, as is shown in Kiliclar et al. (2016). The developed method for material parameter identification is adapted to perform a process parameter identification in a coupled sheet metal forming process. The task is to find a set of parameters describing a double exponential current pulse yielding a sharp cup radius. Double exponential pulses are used as a prototype for general monodirectional pulses. The material’s behavior is simulated by the previously identified material model, thus constraints for the damage can be employed efficiently. In contrast to the usual optimal shape determination, an objective function completely independent of the geometry parametrization is derived. It will turn out that a maximization of the major strain in the area of question can be traced back to a lower cup radius, thus yielding a sharper form. The subsequent application of electromagnetic forming reduces the drawing radius by roughly 5mm, compared to quasi-static deep drawing alone. The presented method can be seen as an extension to previous approaches by Taebi et al. (2012), since it uses a unified simulation framework and exploits a material model that incorporates damage. The second part of this work is concerned with treating the identification of process parameters as a PDE constrained optimization problems and deriving a fitting SAND formulation. The solution of the PDE, modeling the physical behavior, is seen as a state, and the parameters of the process are seen as controls. The first discretize, then optimize approach, is pursued in this work. Regularization methods keeping the controls in check can be avoided in this particular case, since the process parameters are already represented as a real valued finite dimensional vector. A class of problems mimicking a sheet metal forming problem in the linear elastic regime is introduced. This simplification is picked, because the required FEM can easily be implemented but the insights gained are useful for the development of a general optimization scheme for those types of technologically relevant problems. In the course of this work an efficient FEM solver for MATLAB, oFEM, was developed by Dudzinski et al. (2016) to provide the necessary discretizations of the PDEs. With oFEM and IPOPT the family of problems is thoroughly numerically analyzed. The derived SAND formulation is compared to a black box optimization problem based on earlier works by Taebi et al. (2012) and Rozgic et al. (2012). Compared to this so called NAND approach the SAND approach shows a potential gain in efficiency by two orders of magnitude.

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