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Relaxation approach to topology optimization of frame structure under frequency constraint

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Abstract

This paper deals with the frame topology optimization under the frequency constraint and proposes an algorithm that solves a sequence of relaxation problems to obtain a local optimal solution with high quality. It is known that an optimal solution of this problem often has multiple eigenvalues and the feasible set is disconnected. Due to these two difficulties, conventional nonlinear programming approaches often converge to a local optimal solution that is unacceptable from a practical point of view. In this paper, we formulate the frequency constraint as a positive semidefinite constraint of a certain symmetric matrix, and then relax this constraint to make the feasible set connected. The proposed algorithm solves a sequence of the relaxation problems with gradually decreasing the relaxation parameter. The positive semidefinite constraint is treated with the logarithmic barrier function and, hence, the algorithm finds no difficulty in multiple eigenvalues of a solution. Numerical experiments show that global optimal solutions, or at least local optimal solutions with high qualities, can be obtained with the proposed algorithm.

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Notes

  1. This formula was suggested by Broyden (1970a, b), Fletcher (1970), Goldfarb (1970), and Shanno (1970) independently; see, e.g., Fletcher (1987) and Nocedal and Wright (2006) for more accounts.

  2. See, e.g., Calafiore and El Ghaoui (2014) and Nocedal and Wright (2006) for accounts of the Armijo condition.

  3. See, e.g., (Yamashita and Yabe 2015) for survey on numerical solutions of nonlinear SDP.

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Acknowledgments

The work of the second author is partially supported by JSPS KAKENHI (C) 26420545.

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Authors and Affiliations

  1. Department of Mathematical Informatics, University of Tokyo, Tokyo, 113-8656, Japan

    Shinji Yamada & Yoshihiro Kanno

  2. Materials and Structures Laboratory, Tokyo Institute of Technology, Yokohama, 226-8503, Japan

    Yoshihiro Kanno

Authors
  1. Shinji Yamada
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  2. Yoshihiro Kanno
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Correspondence to Yoshihiro Kanno.

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Yamada, S., Kanno, Y. Relaxation approach to topology optimization of frame structure under frequency constraint. Struct Multidisc Optim 53, 731–744 (2016). https://doi.org/10.1007/s00158-015-1353-6

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  • DOI: https://doi.org/10.1007/s00158-015-1353-6

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