Topology optimization for worst load conditions based on the eigenvalue analysis of an aggregated linear system

Akihiro Takezawa*, Satoru Nii, Mitsuru Kitamura, Nozomu Kogiso

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

71 Citations (Scopus)


This paper proposes a topology optimization for a linear elasticity design problem subjected to an uncertain load. The design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set. Since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors, the idea of "aggregation" used in the field of control is introduced to assess the value of the robust compliance. The aggregation solution technique provides the direct relationship between the uncertain input load and output displacement, as a small linear system composed of these vectors and the reduced size of a symmetric matrix, in the context of a discretized linear elasticity problem, using the finite element method. Introducing the constraint that the Euclidean norm of the uncertain load set is fixed, the robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix according to the Rayleigh-Ritz theorem for symmetric matrices. Moreover, the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix. The proposed structural optimization method is implemented using topology optimization and the method of moving asymptotes (MMA). The numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures.

Original languageEnglish
Pages (from-to)2268-2281
Number of pages14
JournalComputer Methods in Applied Mechanics and Engineering
Issue number25-28
Publication statusPublished - 2011 Jun 15
Externally publishedYes


  • Eigenvalue analysis
  • Finite element method
  • Robust design
  • Sensitivity analysis
  • Topology optimization
  • Worst case design

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications


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