An approach for robust PDE-constrained optimization with application to shape optimization of electrical engines and of dynamic elastic structures under uncertainty

Abstract

We present a robust optimization framework that is applicable to general nonlinear programs (NLP) with uncertain parameters. We focus on design problems with partial differential equations (PDE), which involve high computational cost. Our framework addresses the uncertainty with a deterministic worst-case approach. Since the resulting min–max problem is computationally intractable, we propose an approximate robust formulation that employs quadratic models of the involved functions that can be handled efficiently with standard NLP solvers. We outline numerical methods to build the quadratic models, compute their derivatives, and deal with high-dimensional uncertainties. We apply the presented approach to the parametrized shape optimization of systems that are governed by different kinds of PDE and present numerical results.

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Acknowledgements

This work was supported by Deutsche Forschungsgemeinschaft within SFB 805 and by Bundesministerium für Bildung und Forschung within SIMUROM.

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Correspondence to Stefan Ulbrich.

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Kolvenbach, P., Lass, O. & Ulbrich, S. An approach for robust PDE-constrained optimization with application to shape optimization of electrical engines and of dynamic elastic structures under uncertainty. Optim Eng 19, 697–731 (2018). https://doi.org/10.1007/s11081-018-9388-3

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Keywords

  • PDE-constrained optimization
  • Robust optimization

Mathematics Subject Classification

  • 35Q60
  • 35Q72
  • 49K20
  • 49K35