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Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

A posteriori error estimates for sequential laminates in shape optimization

Pages: 1377 - 1392, Volume 9, Issue 5, October 2016      doi:10.3934/dcdss.2016055

 
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Benedict Geihe - Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universitat Bonn, Endenicher Allee 60, 53111 Bonn, Germany (email)
Martin Rumpf - Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universitat Bonn, Endenicher Allee 60, 53111 Bonn, Germany (email)

Abstract: A posteriori error estimates are derived in the context of two-dimensional structural elastic shape optimization under the compliance objective. It is known that the optimal shape features are microstructures that can be constructed using sequential lamination. The descriptive parameters explicitly depend on the stress. To derive error estimates the dual weighted residual approach for control problems in PDE constrained optimization is employed, involving the elastic solution and the microstructure parameters. Rigorous estimation of interpolation errors ensures robustness of the estimates while local approximations are used to obtain fully practical error indicators. Numerical results show sharply resolved interfaces between regions of full and intermediate material density.

Keywords:  Elastic shape optimization, two scale optimization, homogenization, a posteriori error estimates, nested laminates, adaptive meshes.
Mathematics Subject Classification:  35B27, 65N15, 74B05, 74P05, 74Q05, 74S05, 49M29, 49Q10.

Received: December 2014;      Revised: July 2015;      Available Online: October 2016.

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