On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    We consider equivalent reformulations of nonlinear mixed 0–1 optimization problems arising from a broad range of recent applications of topology optimization for the design of continuum structures and composite materials. We show that the considered problems can equivalently be cast as either linear or convex quadratic mixed 0–1 programs. The reformulations provide new insight into the structure of the problems and may provide a foundation for the development of new methods and heuristics for solving topology optimization problems. The applications considered are maximum stiffness design of structures subjected to static or periodic loads, design of composite materials with prescribed homogenized properties using the inverse homogenization approach, optimization of fluids in Stokes flow, design of band gap structures, and multi-physics problems involving coupled steady-state heat conduction and linear elasticity. Several numerical examples of maximum stiffness design of truss structures are presented.
    Original languageEnglish
    JournalOptimization and Engineering
    Volume8
    Issue number2
    Pages (from-to)163-192
    ISSN1389-4420
    DOIs
    Publication statusPublished - 2007

    Keywords

    • Convex mixed integer programming
    • Topology optimization

    Cite this

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    title = "On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs",
    abstract = "We consider equivalent reformulations of nonlinear mixed 0–1 optimization problems arising from a broad range of recent applications of topology optimization for the design of continuum structures and composite materials. We show that the considered problems can equivalently be cast as either linear or convex quadratic mixed 0–1 programs. The reformulations provide new insight into the structure of the problems and may provide a foundation for the development of new methods and heuristics for solving topology optimization problems. The applications considered are maximum stiffness design of structures subjected to static or periodic loads, design of composite materials with prescribed homogenized properties using the inverse homogenization approach, optimization of fluids in Stokes flow, design of band gap structures, and multi-physics problems involving coupled steady-state heat conduction and linear elasticity. Several numerical examples of maximum stiffness design of truss structures are presented.",
    keywords = "Convex mixed integer programming, Topology optimization",
    author = "Mathias Stolpe",
    year = "2007",
    doi = "10.1007/s11081-007-9005-3",
    language = "English",
    volume = "8",
    pages = "163--192",
    journal = "Optimization and Engineering",
    issn = "1389-4420",
    publisher = "Springer New York",
    number = "2",

    }

    On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs. / Stolpe, Mathias.

    In: Optimization and Engineering, Vol. 8, No. 2, 2007, p. 163-192.

    Research output: Contribution to journalJournal articleResearchpeer-review

    TY - JOUR

    T1 - On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs

    AU - Stolpe, Mathias

    PY - 2007

    Y1 - 2007

    N2 - We consider equivalent reformulations of nonlinear mixed 0–1 optimization problems arising from a broad range of recent applications of topology optimization for the design of continuum structures and composite materials. We show that the considered problems can equivalently be cast as either linear or convex quadratic mixed 0–1 programs. The reformulations provide new insight into the structure of the problems and may provide a foundation for the development of new methods and heuristics for solving topology optimization problems. The applications considered are maximum stiffness design of structures subjected to static or periodic loads, design of composite materials with prescribed homogenized properties using the inverse homogenization approach, optimization of fluids in Stokes flow, design of band gap structures, and multi-physics problems involving coupled steady-state heat conduction and linear elasticity. Several numerical examples of maximum stiffness design of truss structures are presented.

    AB - We consider equivalent reformulations of nonlinear mixed 0–1 optimization problems arising from a broad range of recent applications of topology optimization for the design of continuum structures and composite materials. We show that the considered problems can equivalently be cast as either linear or convex quadratic mixed 0–1 programs. The reformulations provide new insight into the structure of the problems and may provide a foundation for the development of new methods and heuristics for solving topology optimization problems. The applications considered are maximum stiffness design of structures subjected to static or periodic loads, design of composite materials with prescribed homogenized properties using the inverse homogenization approach, optimization of fluids in Stokes flow, design of band gap structures, and multi-physics problems involving coupled steady-state heat conduction and linear elasticity. Several numerical examples of maximum stiffness design of truss structures are presented.

    KW - Convex mixed integer programming

    KW - Topology optimization

    U2 - 10.1007/s11081-007-9005-3

    DO - 10.1007/s11081-007-9005-3

    M3 - Journal article

    VL - 8

    SP - 163

    EP - 192

    JO - Optimization and Engineering

    JF - Optimization and Engineering

    SN - 1389-4420

    IS - 2

    ER -