A semi-implicit level set method for structural shape and topology optimization

Junzhao Luo, Zhen Luo, Liping Chen, Liyong Tong, Michael Yu Wang

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118 Citations (Scopus)


This paper proposes a new level set method for structural shape and topology optimization using a semi-implicit scheme. Structural boundary is represented implicitly as the zero level set of a higher-dimensional scalar function and an appropriate time-marching scheme is included to enable the discrete level set processing. In the present study, the Hamilton-Jacobi partial differential equation (PDE) is solved numerically using a semi-implicit additive operator splitting (AOS) scheme rather than explicit schemes in conventional level set methods. The main feature of the present method is it does not suffer from any time step size restriction, as all terms relevant to stability are discretized in an implicit manner. The semi-implicit scheme with additive operator splitting treats all coordinate axes equally in arbitrary dimensions with good rotational invariance. Hence, the present scheme for the level set equations is stable for any practical time steps and numerically easy to implement with high efficiency. Resultantly, it allows enhanced relaxation on the time step size originally limited by the Courant-Friedrichs-Lewy (CFL) condition of the explicit schemes. The stability and computational efficiency can therefore be greatly improved in advancing the level set evolvements. Furthermore, the present method avoids additional cost to globally reinitialize the level set function for regularization purpose. It is noted that the periodically applied reinitializations are time-consuming procedures. In particular, the proposed method is capable of creating new holes freely inside the design domain via boundary incorporating, splitting and merging processes, which makes the final design independent of initial guess, and helps reduce the probability of converging to a local minimum. The availability of the present method is demonstrated with two widely studied examples in the framework of the structural stiffness designs.

Original languageEnglish
Pages (from-to)5561-5581
Number of pages21
JournalJournal of Computational Physics
Issue number11
Publication statusPublished - 10 May 2008
Externally publishedYes


  • Level set methods
  • Semi-implicit schemes
  • Shape optimization
  • Topology optimization

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