A simple and effective inverse projection scheme for void distribution control in topology optimization

S. R.M. Almeida, G. H. Paulino, E. C.N. Silva

Research output: Contribution to journalArticlepeer-review

42 Scopus citations


The ability to control both the minimum size of holes and the minimum size of structural members are essential requirements in the topology optimization design process for manufacturing. This paper addresses both requirements by means of a unified approach involving mesh-independent projection techniques. An inverse projection is developed to control the minimum hole size while a standard direct projection scheme is used to control the minimum length of structural members. In addition, a heuristic scheme combining both contrasting requirements simultaneously is discussed. Two topology optimization implementations are contributed: one in which the projection (either inverse or direct) is used at each iteration; and the other in which a two-phase scheme is explored. In the first phase, the compliance minimization is carried out without any projection until convergence. In the second phase, the chosen projection scheme is applied iteratively until a solution is obtained while satisfying either the minimum member size or minimum hole size. Examples demonstrate the various features of the projection-based techniques presented.

Original languageEnglish (US)
Pages (from-to)359-371
Number of pages13
JournalStructural and Multidisciplinary Optimization
Issue number4
StatePublished - Oct 2009
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Systems Engineering
  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design
  • Control and Optimization


  • Combined projection
  • Direct projection
  • Inverse projection
  • Projection functions
  • Topology optimization
  • Two-phase optimization


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