Sergio Conti, Benedict Geihe, Martin Lenz and Martin Rumpf A posteriori modeling error estimates in the optimization of two-scale elastic composite materials ESAIM: Mathematical Modelling and Numerical Analysis, 52: 1457-1476 2018 10.1051/m2an/2017004
2015
Benedict Geihe and Martin Rumpf A posteriori error estimates for sequential laminates in shape optimization In DCDS-S Special issue on Homogenization-Based Numerical Methods 2015 http://arxiv.org/abs/1501.07461
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.