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Triply periodic minimal surface using a modified Allen–Cahn equation

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  • Li, Yibao
  • Guo, Shimin

Abstract

In this paper, we propose a fast and efficient model for triply periodic minimal surface. The proposed model is based on the Allen–Cahn equation with a Lagrange multiplier term. The Allen–Cahn equation has the motion of mean curvature. And the Lagrange multiplier term corresponding to the constant volume constraint also relates to the average of mean curvature. By combining two terms, the mean curvature will be constant everywhere on the surface at the equilibrium condition. The proposed numerical method with the second-order accuracy of time and space exhibits excellent stability. In addition, the resulting discrete system is solved by a fast numerical method such as a multigrid method. Various numerical experiments are performed to demonstrate the accuracy and robustness of the proposed method.

Suggested Citation

  • Li, Yibao & Guo, Shimin, 2017. "Triply periodic minimal surface using a modified Allen–Cahn equation," Applied Mathematics and Computation, Elsevier, vol. 295(C), pages 84-94.
    • Handle: RePEc:eee:apmaco:v:295:y:2017:i:c:p:84-94
    DOI: 10.1016/j.amc.2016.10.005
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    References listed on IDEAS

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    1. Xu, Gang & Zhu, Yaguang & Wang, Guozhao & Galligo, André & Zhang, Li & Hui, Kin-chuen, 2015. "Explicit form of parametric polynomial minimal surfaces with arbitrary degree," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 124-131.
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    Cited by:

    1. Li, Yibao & Xia, Qing & Kang, Seungyoon & Kwak, Soobin & Kim, Junseok, 2024. "A practical algorithm for the design of multiple-sized porous scaffolds with triply periodic structures," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 481-495.
    2. Jin Wang & Zhengyuan Shi, 2021. "Multi-Reconstruction from Points Cloud by Using a Modified Vector-Valued Allen–Cahn Equation," Mathematics, MDPI, vol. 9(12), pages 1-15, June.
    3. Lee, Dongsun & Lee, Chaeyoung, 2022. "Numerical solutions of the Allen–Cahn equation with the p-Laplacian," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    4. Yu, Qian & Wang, Kunyang & Xia, Binhu & Li, Yibao, 2021. "First and second order unconditionally energy stable schemes for topology optimization based on phase field method," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    5. Binhu Xia & Yibao Li & Zhong Li, 2020. "Second-Order Unconditionally Stable Direct Methods for Allen–Cahn and Conservative Allen–Cahn Equations on Surfaces," Mathematics, MDPI, vol. 8(9), pages 1-12, September.

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