# Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty

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## Abstract

Design of multiscale structures is a challenging task due to a vast design space of both materials and structures. Consideration of load uncertainty adds another level of complexity. In this paper, a robust concurrent TO (topology optimization) approach is developed for designing multiscale structures composed of multiple porous materials under random field loading uncertainty. To determine the optimal distribution of the porous materials at the macro/structural scale, our key idea is to employ the discrete material optimization method to interpolate the material properties for multiple porous materials. In addition, for the first time we interpret the interpolation schemes in the existing concurrent TO model of porous material with a clear physical meaning by putting forward a SIMP-like single interpolation scheme. This scheme integrates the SIMP (Solid Isotropic Material with Penalization) at the microscale and PAMP (Porous Anisotropic Material with Penalization) at the macroscale into a single equation. Efficient uncertainty characterization and propagation methods based on K-L expansion and linear superposition are introduced, and several important improvements in objective function evaluation and sensitivity analysis are presented. Improved sensitivity analysis equations are derived for volume preserving filtering, which is employed to deal with numerical instabilities at the macro and micro scales in the robust concurrent TO model. Measures to ensure manufacturability and to improve analysis accuracy and efficiency are devised. 2D and 3D examples demonstrate the effectiveness of the proposed approach in simultaneously obtaining robust optimal macro structural topology and material microstructural topologies.

## Keywords

Concurrent topology optimization Multiscale structure Porous materials Random field uncertainty## Notes

### Acknowledgements

Grant support from NIST 70NANB14H012 under Advanced Materials Center for Excellence: Center for Hierarchical Materials Design (CHiMaD) is greatly appreciated.

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