Minimum length-scale constraints for parameterized implicit function based topology optimization

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This paper introduces explicit minimum lengthscale constraint functions suitable for parameterized implicit function based topology optimization methods. Length-scale control in topology optimization has many potential benefits, such as removing numerical artefacts, mesh independent solutions, avoiding thin, or single node hinges in compliant mechanism design and meeting manufacturing constraints. Several methods have been developed to control length-scale when using densitybased or signed-distance-based level-set methods. In this paper a method is introduced to control length-scale for parameterized implicit function based topology optimization. Explicit constraint functions to control the minimum length in the structure and void regions are proposed and implementation issues explored in detail. Several examples are presented to show the efficacy of the proposed method. The examples demonstrate that the method can simultaneously control minimum structure and void length-scale, design hinge free compliant mechanisms and control minimum length-scale for three dimensional structures.
Original languageEnglish
Pages (from-to)155–169
Number of pages15
JournalStructural and multidisciplinary optimization
Issue number1
Early online date12 Jan 2018
Publication statusPublished - Jul 2018

Bibliographical note

Open access via Springer Compact Agreement The author would like to thank the Numerical Analysis Group at the Rutherford Appleton Laboratory for their FORTRAN HSL packages (HSL, a collection of Fortran codes for large-scale scientific computation. See The author also would like to acknowledge the support of the Maxwell compute cluster funded by the University of Aberdeen. Finally, the author thanks the anonymous reviewers for their helpful comments and suggestions that improved this paper.


  • length-scale constraints
  • topology optimization
  • implicit function
  • parameterization


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