Simple and efficient volume merging method for triply periodic minimal structures

Yibao Li, Qing Xia, Sungha Yoon, Chaeyoung Lee, Bingheng Lu, Junseok Kim

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31 Citations (Scopus)


Triply periodic minimal surfaces (TPMSs), which are periodic in all three directions and are surfaces of zero mean curvature, have been proven experimentally to be highly suitable for tissue scaffolds. However, simply gluing different TPMS units with different porosities and pore sizes could induce discontinuous structures and destroy the physical properties. In this study, we propose a simple and efficient volume merging method for triply periodic minimal structures. The proposed method can be divided into two steps. The first step is a novel merging algorithm for unit triply periodic minimal structures in the implicit function framework. The composite scaffold can be designed by merging different unit structures to satisfy the properties of internal connectivity. The second step is to optimize the designed composite scaffolds to satisfy the properties of TPMSs. A modified Allen–Cahn-type equation with a correction term is proposed. The mean curvature on the surface is constant at all points in the equilibrium state. Typically, the obtained structure is smooth owing to the motion by mean curvature flow. Therefore, the quality of the structure is significantly improved. Based on the operator splitting method, the proposed algorithm consists of two analytical evaluations for the ordinary differential equations and one numerical solution for the implicit Poisson-type equation. The proposed numerical scheme can be applied in a straightforward manner to a GPU-accelerated discrete cosine transform (DCT) implementation, which can be executed multiple times faster than CPU-only alternatives. Computational experiments are presented to demonstrate the efficiency and robustness of the proposed method.

Original languageEnglish
Article number107956
JournalComputer Physics Communications
Publication statusPublished - 2021 Jul

Bibliographical note

Funding Information:
Y.B. Li is supported by National Natural Science Foundation of China (No. 11871056 , No. 11771348 ). B.H. Lu are supported by Shaanxi Provincial Science and Technology Planning Project ( 2017KTZD6-01 ) and Dongguan University of Technology High-level Talents (Innovation Team) Research Project ( KCYCXPT2016003 ). C. Lee expresses thanks for the support from the BK21 FOUR program. The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( NRF-2016R1D1A1B03933243 ). The authors would like to thank the reviewers for their constructive and helpful comments regarding the revision of this article.

Publisher Copyright:
© 2021 Elsevier B.V.


  • Allen–Cahn equation
  • Implicit method
  • Tissue engineering scaffolds
  • Triply periodic minimal surfaces
  • Volume merging method

ASJC Scopus subject areas

  • Hardware and Architecture
  • General Physics and Astronomy


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