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# Multi-Material Topology Optimization for Spatial-Varying Porous Structures

Chengwan Zhang1, Kai Long1,*, Zhuo Chen1,2, Xiaoyu Yang1, Feiyu Lu1, Jinhua Zhang3, Zunyi Duan4

1 State Key Laboratory for Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing, 102206, China
2 School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, 510640, China
3 Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing, 100124, China
4 Institute of Structural Health Monitoring and Control, School of Mechanics, Civil Engineering & Architecture, Northwestern Polytechnical University, Xi’an, 710072, China

* Corresponding Author: Kai Long. Email:

Computer Modeling in Engineering & Sciences 2024, 138(1), 369-390. https://doi.org/10.32604/cmes.2023.029876

## Abstract

This paper aims to propose a topology optimization method on generating porous structures comprising multiple materials. The mathematical optimization formulation is established under the constraints of individual volume fraction of constituent phase or total mass, as well as the local volume fraction of all phases. The original optimization problem with numerous constraints is converted into a box-constrained optimization problem by incorporating all constraints to the augmented Lagrangian function, avoiding the parameter dependence in the conventional aggregation process. Furthermore, the local volume percentage can be precisely satisfied. The effects including the global mass bound, the influence radius and local volume percentage on final designs are exploited through numerical examples. The numerical results also reveal that porous structures keep a balance between the bulk design and periodic design in terms of the resulting compliance. All results, including those for irregular structures and multiple volume fraction constraints, demonstrate that the proposed method can provide an efficient solution for multiple material infill structures.

## Keywords

Zhang, C., Long, K., Chen, Z., Yang, X., Lu, F. et al. (2024). Multi-Material Topology Optimization for Spatial-Varying Porous Structures. CMES-Computer Modeling in Engineering & Sciences, 138(1), 369–390.

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