Structural and Multidisciplinary Optimization

, Volume 60, Issue 4, pp 1687–1707 | Cite as

Free sizing optimization of a front hood using the ESL method: overcoming challenges and traps

  • Artem KarevEmail author
  • Lothar Harzheim
  • Rainer Immel
  • Matthias Erzgräber
Industrial Application


Topology and free sizing optimization are very important tools in the early design phase. Yet, they are only well established for optimization problems based on linear analysis. In the nonlinear analysis area—in particular crash—such kind of optimization cannot be applied due to the unavailability of gradients. A workaround is to create linear auxiliary load cases approximating the nonlinear load case at different time steps, which can be used in optimization based on linear statics analysis. The equivalent static load (ESL) method provides a procedure to create such auxiliary load cases in a well-defined way. However, it is a great challenge to translate nonlinear requirements and responses to linear statics system. For this reason, strain energy is often used as objective function even if it does not reflect the real objective in the optimization task. In this study, different formulations for a front hood free sizing optimization problem are assessed for their ability to translate the actual objective of the nonlinear system to the linear statics system. It turns out that even though such formulations can be found, they show no advantage in comparison to simpler formulations. The best performance was obtained using mass as objective function, whereas the strain energy formulation fails due to the tendency to favor huge mesh deformations in void areas. The reason for such behavior and the related issues appearing especially in topology and free sizing optimization are discussed in detail. Finally, recommendations are given on how to improve the performance of ESL-based optimization.


Equivalent static loads Topometry Crash Pedestrian protection 


Compliance with Ethical Standards

Conflict of interests

Artem Karev, Lothar Harzheim, Rainer Immel, and Matthias Erzgräber declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Numerical Methods in Mechanical EngineeringTechnische Universität Darmstadt|DarmstadtGermany
  2. 2.Opel Automobile GmbH, BahnhofsplatzRüsselsheim am MainGermany

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