Reaction–diffusion equation driven topology optimization of high-resolution and feature-rich structures using unstructured meshes

Hao Li*, Minghao Yu, Pierre Jolivet, Joe Alexandersen, Tsuguo Kondoh, Tiannan Hu, Kozo Furuta, Kazuhiro Izui, Shinji Nishiwaki

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


Motivated by the need for porous structures in the design of biodegradable implants, as well as for the diverse and competitive designs in architecture, this paper builds upon the recent advances in the single-scale (macroscopic) topology optimization (TO) approach and proposes a level-set method (LSM) for the design of lattice structures. The key idea is to introduce the maximum length-scale constraint, realized by a PDE filter, into the reaction–diffusion equation (RDE)-based LSM which can end up with feature-rich shape, starting from scratch. Then, the proposed TO technique is coupled with distributed computing algorithms based on PETSc, interfaced in the open-source software FreeFEM using unstructured meshes, to deliver both two- and three-dimensional designs with complex-shape geometry. The parallel efficiency is validated by solving large-scale 3D benchmarks with 20 million tetrahedral elements on a cloud-based cluster. Furthermore, we integrate local mesh refinement (h-adaptation scheme) into this workflow, allowing one to reach high-resolution and feature-rich 3D designs on a desktop, paving the way to future investigations, aimed at including the maximum length-scale constraint in large-scale multiphysics topology optimization at a reasonable cost.

Original languageEnglish
Article number103457
JournalAdvances in Engineering Software
Number of pages21
Publication statusPublished - Jun 2023

Bibliographical note

Publisher Copyright:
© 2023 Elsevier Ltd


  • FreeFEM-PETSc-ParMmg
  • Lattice designing
  • Maximum length-scale constraint
  • PDE filter
  • Reaction–diffusion equation
  • Topology optimization
  • Unstructured mesh


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