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First and second order unconditionally energy stable schemes for topology optimization based on phase field method

Author

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  • Yu, Qian
  • Wang, Kunyang
  • Xia, Binhu
  • Li, Yibao

Abstract

In this paper, we use the phase field method to deal with the compliance minimization problem in topology optimization. A modified Allen-Cahn type equation with two penalty terms is proposed. The equation couples the diffusive interface dynamics and the linear elasticity mechanics. We propose the first- and second-order unconditionally energy stable schemes for the evolution of phase field modeling. The linearly stabilized splitting scheme is applied to improve the stability. The Crank-Nicolson scheme is applied to achieve second-order accuracy in time. We prove the unconditional stabilities of our schemes in analysis. The finite element method and the projected conjugate gradient method combining with fast fourier transform are used to solve the compliance minimization problem. Several experimental results are presented to verify the efficiency and accuracy of the proposed schemes.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:405:y:2021:i:c:s0096300321003568
    DOI: 10.1016/j.amc.2021.126267
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    References listed on IDEAS

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    1. Delgado, G. & Hamdaoui, M., 2019. "Topology optimization of frequency dependent viscoelastic structures via a level-set method," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 522-541.
    2. Liu, Yisi & Wang, Xiaojun & Wang, Lei & Liu, Dongliang, 2019. "A modified leaky ReLU scheme (MLRS) for topology optimization with multiple materials," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 188-204.
    3. Hu, Xianliang & Li, Yixin & Ji, Hangjie, 2018. "A nodal finite element approximation of a phase field model for shape and topology optimization," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 675-684.
    4. 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.
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    Cited by:

    1. Zheng, Yongfeng & Gu, Yan & Gao, Liang & Wang, Yanzheng & Qu, Jinping & Zhang, Chuanzeng, 2022. "A new structural uncertainty analysis method based on polynomial expansions," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    2. Tan, Zhijun & Yang, Junxiang & Chen, Jianjun & Kim, Junseok, 2023. "An efficient time-dependent auxiliary variable approach for the three-phase conservative Allen–Cahn fluids," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    3. 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.

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