Convolution Hierarchical Deep-Learning Neural Network Tensor Decomposition (C-HiDeNN-TD) for high-resolution topology optimization

Hengyang Li, Stefan Knapik, Yangfan Li, Chanwook Park, Jiachen Guo, Satyajit Mojumder, Ye Lu*, Wei Chen, Daniel W. Apley, Wing Kam Liu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


High-resolution structural topology optimization is extremely challenging due to a large number of degrees of freedom (DoFs). In this work, a Convolution-Hierarchical Deep Learning Neural Network-Tensor Decomposition (C-HiDeNN-TD) framework is introduced and applied to solve the computationally challenging giga-scale topology optimization problem using only a single personal computer (PC). Utilizing the idea of convolution, the C-HiDeNN opens a new avenue for the development of topology optimization theory with arbitrarily high-order smoothness under given DoFs. The convolution involves a set of controllable parameters, including patch size, dilation parameter, and polynomial order. These parameters allow built-in control of the design length-scale, accuracy, and smoothness of solutions. From the point of view of neural networks, increasing the “patch size” is analogous to adding an extra hidden layer with extra neurons, leading to an enhanced approximation capability. Under the separation of variables, the C-HiDeNN-TD can greatly reduce the computational cost of finding a 3D high-resolution topology design by decomposing the ultra-large-scale 3D mechanical problem into several tractable small 1D problems. Orders of magnitude speedups compared to traditional finite element-based topology optimization have been demonstrated through numerical examples. Furthermore, the C-HiDeNN-TD method enables much more efficient concurrent multi-scale topology design than traditional approaches. The proposed framework opens numerous opportunities for high-resolution design, n-scale concurrent design, and structure-lattice-materials design.

Original languageEnglish (US)
Pages (from-to)363-382
Number of pages20
JournalComputational Mechanics
Issue number2
StatePublished - Aug 2023


  • 3D printing and metamaterial design
  • Hierarchical deep-learning neural networks with GPU
  • High resolution topology optimization
  • Multi-scale concurrent design
  • Reduced-order modeling

ASJC Scopus subject areas

  • Computational Mechanics
  • Ocean Engineering
  • Mechanical Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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