Topology optimization of irregular multiscale structures with tunable responses using a virtual growth rule

Yingqi Jia, Ke Liu, Xiaojia Shelly Zhang

Research output: Contribution to journalArticlepeer-review


Many applications demand tunable structural responses through tailored organic microstructural distributions and spatially varied material properties. Notable progress has been made in discovering optimized designs using periodic material patterns and fixed material phases to achieve unusual structural responses. To enable the capability of exploring non-periodic material architectures with continuous material phase design space, we propose a topology optimization methodology that leverages a virtual growth rule for designing unique multiscale structures with tunable responses and irregular architectures, while naturally ensuring manufacturability. Our approach exploits the virtual growth algorithm to create a material database, delineating constitutive relations between the homogeneous frequency hints of building blocks responsible for generating microstructures and the resultant homogenized microstructural elasticity tensors. We then employ a neural network to yield a continuous and differentiable constitutive relation. Subsequently, a topology optimization framework is introduced to optimize both the macroscale material layout and the local frequency hints for building block distribution. Finally, we generalize the virtual growth algorithm to account for optimized heterogeneous frequency hints and grow irregular yet optimized structures at the microscale. We present four examples to showcase our proposed approach in programming several types of responses, including displacement cloaking, tunable strain energy density, and global structural stiffness, in both two and three dimensions. The optimized multiscale structures, characterized by their stochastic and irregular architectures, demonstrate programmed responses that closely match the desired targets. These structures also ensure microstructural connectivity and offer the flexibility to select building blocks with guaranteed minimal features. Consequently, we leverage such connectivity and minimal features to manifest the manufacturability of the optimized structures by 3D printing. Our proposed computational strategy, which precisely realizes programmed structural responses in multiscale structures with irregular architectures and facilitates manufacturing feasibility, can be beneficial for applications that prioritize structures exemplifying disorderedness, non-uniformity, and heterogeneity.

Original languageEnglish (US)
Article number116864
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - May 15 2024


  • Inverse design
  • Irregular architecture
  • Multiscale structures
  • Topology optimization
  • Tunable responses
  • Virtual growth

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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