Unconditionally energy stable schemes for fluid-based topology optimization

Yibao Li, Kunyang Wang, Qian Yu, Qing Xia, Junseok Kim

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


We present first- and second-order unconditionally energy stable schemes for fluid-based topology optimization problems. Our objective functional composes of five terms including mechanical property, Ginzburg–Landau energy, two penalized terms for solid, and the volume constraint. We consider the steady-state Stokes equation in the fluid domain and Darcy flow through porous medium. By coupling a Stokes type equation and the Allen–Cahn equation, we obtain the evolutionary equation for the fluid-based topology optimization. We use the backward Euler method and the Crank–Nicolson method to discretize the coupling system. The first- and second-order accurate schemes are presented correspondingly. We prove that our proposed schemes are unconditionally energy stable. The preconditioned conjugate gradient method is applied to solve the system. Several numerical tests are performed to verify the efficiency and accuracy of our schemes.

Original languageEnglish
Article number106433
JournalCommunications in Nonlinear Science and Numerical Simulation
Publication statusPublished - 2022 Aug

Bibliographical note

Funding Information:
Y.B. Li is supported by the Fundamental Research Funds for the Central Universities, China (No. XTR042019005 ). The corresponding author (J.S. Kim) was supported by Korea University Grant, Republic of Korea. The authors are grateful to the reviewers whose valuable suggestions and comments significantly improved the quality of this paper.

Publisher Copyright:
© 2022 Elsevier B.V.


  • Phase-field methods
  • Stokes equation
  • Topology optimization
  • Unconditionally energy stable

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Applied Mathematics


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