Topology Optimization of Differentiable Microstructures

Xiaoya Zhai, Weiming Wang, Falai Chen, Jun Wu

Research output: Contribution to journalArticlepeer-review


Recent years have seen a growing interest in topology optimization of functionally graded microstructures, characterized by an array of microstructures with varying volume fractions. However, microstructures optimized at slightly different volume fractions do not necessarily connect well when placed adjacently. Furthermore, optimization is commonly performed on a finite set of volume fractions, limiting the number of microstructure configurations. In this paper, we introduce the concept of differentiable microstructures, which are parameterized microstructures that exhibit continuous variations in both geometry and mechanical properties. To construct such microstructures, we propose a novel formulation for topology optimization. In this approach, a series of 2-dimensional microstructures is represented using a height field, and the objective is to maximize the bulk modulus of the entire series. Through this optimization process, an initial microstructure with a small volume fraction undergoes non-uniform transformations, generating a series of microstructures with progressively increasing volume fractions. Notably, when compared to traditional uniform morphing methods, our proposed optimization approach yields a series of microstructures with bulk moduli that closely approach the theoretical limit.

Original languageEnglish
Article number116530
JournalComputer Methods in Applied Mechanics and Engineering
Issue numberA
Early online date13 Oct 2023
Publication statusPublished - 1 Jan 2024


  • Functionally graded microstructures
  • Inverse homogenization
  • Topology optimization


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