On some fundamental properties of structural topology optimization problems

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    Abstract

    We study some fundamental mathematical properties of discretized structural topology optimization problems. Either compliance is minimized with an upper bound on the volume of the structure, or volume is minimized with an upper bound on the compliance. The design variables are either continuous or 0–1. We show, by examples which can be solved by hand calculations, that the optimal solutions to the problems in general are not unique and that the discrete problems possibly have inactive volume or compliance constraints. These observations have immediate consequences on the theoretical convergence properties of penalization approaches based on material interpolation models. Furthermore, we illustrate that the optimal solutions to the considered problems in general are not symmetric even if the design domain, the external loads, and the boundary conditions are symmetric around an axis. The presented examples can be used as teaching material in graduate and undergraduate courses on structural topology optimization.
    Original languageEnglish
    JournalStructural and Multidisciplinary Optimization
    Volume41
    Issue number5
    Pages (from-to)661-670
    ISSN1615-147X
    DOIs
    Publication statusPublished - 2010

    Keywords

    • Minimum compliance optimization
    • Topology optimization
    • Minimum volume optimization

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