Large-scale topology optimization using preconditioned Krylov subspace methods with recycling

Shun Wang, Eric de Sturler, Glaucio H. Paulino

Research output: Contribution to journalArticlepeer-review

159 Scopus citations


The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large three-dimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (minimum residual method) version with recycling (and a short-term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC.

Original languageEnglish (US)
Pages (from-to)2441-2468
Number of pages28
JournalInternational Journal for Numerical Methods in Engineering
Issue number12
StatePublished - Mar 19 2007
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • General Engineering
  • Applied Mathematics


  • Iterative methods
  • Krylov subspace recycling
  • Large-scale computation
  • Preconditioning
  • Three-dimensional analysis
  • Topology optimization


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