A posteriori error estimates for sequential laminates in shape optimization

TL;DR

This paper develops a posteriori error estimates for shape optimization problems involving microstructured materials, specifically sequential laminates, using dual weighted residual methods to improve accuracy in elastic structural design.

Contribution

It introduces a novel error estimation approach tailored for microstructure-based shape optimization, explicitly accounting for stress-dependent parameters.

Findings

Error estimates effectively identify regions with high approximation errors.

Numerical results demonstrate sharp interface resolution between different material densities.

The method enhances the robustness and practicality of shape optimization with microstructures.

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.