Three-dimensional grayscale-free topology optimization using a level-set based r-refinement method

Shintaro Yamasaki*, Seiichiro Yamanaka, Kikuo Fujita

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


In this paper, we propose a three-dimensional (3D) grayscale-free topology optimization method using a conforming mesh to the structural boundary, which is represented by the level-set method. The conforming mesh is generated in an r-refinement manner; that is, it is generated by moving the nodes of the Eulerian mesh that maintains the level-set function. Although the r-refinement approach for the conforming mesh generation has many benefits from an implementation aspect, it has been considered as a difficult task to stably generate 3D conforming meshes in the r-refinement manner. To resolve this task, we propose a new level-set based r-refinement method. Its main novelty is a procedure for minimizing the number of the collapsed elements whose nodes are moved to the structural boundary in the conforming mesh; in addition, we propose a new procedure for improving the quality of the conforming mesh, which is inspired by Laplacian smoothing. Because of these novelties, the proposed r-refinement method can generate 3D conforming meshes at a satisfactory level, and 3D grayscale-free topology optimization is realized. The usefulness of the proposed 3D grayscale-free topology optimization method is confirmed through several numerical examples.

Original languageEnglish
Pages (from-to)1402-1438
Number of pages37
JournalInternational Journal for Numerical Methods in Engineering
Issue number10
Publication statusPublished - 2017 Dec 7
Externally publishedYes


  • finite element method
  • grayscale element
  • level-set method
  • r-refinement method
  • three-dimensional problem
  • topology optimization

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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