Modelling topology optimization problems as linear mixed 0–1 programs

Stolpe, Mathias; Svanberg, Krister

doi:10.1002/nme.700

Cited by 75 publications

(46 citation statements)

References 41 publications

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“…In the suggested method we make extensive use of the fact that these problems can be equivalently reformulated as convex mixed 0-1 programs, and thus solved by global optimization methods. Due to similar reformulation results presented in References [1,2] the suggested models and methods are, with minor modifications, applicable to a broad range of topology design problems with local as well as global constraints.…”

Section: Introductionmentioning

confidence: 85%

A hierarchical method for discrete structural topology design problems with local stress and displacement constraints

Stolpe

Stidsen

2006

Numerical Meth Engineering

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SUMMARYIn this paper, we present a hierarchical optimization method for finding feasible true 0-1 solutions to finite-element-based topology design problems. The topology design problems are initially modelled as non-convex mixed 0-1 programs.The hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design-dependent stress constraints. The method iteratively treats a sequence of problems of increasing size of the same type as the original problem. The problems are defined on a design mesh which is initially coarse and then successively refined as needed. At each level of design mesh refinement, a neighbourhood optimization method is used to treat the problem considered.The non-convex topology design problems are equivalently reformulated as convex all-quadratic mixed 0-1 programs. This reformulation enables the use of methods from global optimization, which have only recently become available, for solving the problems in the sequence. Numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented.

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Section: Introductionmentioning

confidence: 85%

A hierarchical method for discrete structural topology design problems with local stress and displacement constraints

Stolpe

Stidsen

2006

Numerical Meth Engineering

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“…Therefore, multiple researchers [Achtziger and Stolpe 2007;Kanno and Guo 2010;Rasmussen and Stolpe 2008;Stolpe and Svanberg 2003] restrict beam cross sections to a preassigned set and tackle the problem using mixed-integer programming, targeting a globally optimal solution. Alternatively, many meta-heuristic approaches have been proposed for the same problem, such as genetic algorithms [Kawamura et al 2002], ant colony optimization [Kaveh et al 2008], particle swarm optimization [Li et al 2009], and teachinglearning-based optimization [Camp and Farshchin 2014].…”

Section: Previous Workmentioning

confidence: 99%

Design and volume optimization of space structures

et al. 2017

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Fig. 1. Le : A statically sound space structure designed and optimized with our framework, motivated by the real architectural project shown in Figure 2. Right: The space structure is constructed with six types of customized beams to minimize the total volume of the material used for beams while maintaining moderate manufacturing complexity. Here, a ho er color indicates a larger beam cross-section area. Our framework automatically determines the optimal cross-section areas of the six types of beams as well as the assignment of beam types.We study the design and optimization of statically sound and materially e cient space structures constructed by connected beams. We propose a systematic computational framework for the design of space structures that incorporates static soundness, approximation of reference surfaces, boundary alignment, and geometric regularity. To tackle this challenging problem, we rst jointly optimize node positions and connectivity through a nonlinear continuous optimization algorithm. Next, with xed nodes and connectivity, we formulate the assignment of beam cross sections as a mixed-integer programming problem with a bilinear objective function and quadratic constraints. We solve this problem with a novel and practical alternating direction method based on linear programming relaxation. The capability and e ciency of the algorithms and the computational framework are validated by a variety of examples and comparisons.

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“…The artificial upper and lower boundsq ′ ij andq ′ ij ensure feasibility when profile j is not selected for member i and are calculated as follows [22]:…”

Section: Member Stiffness Relationsmentioning

confidence: 99%

“…u u u where u and u are the prescribed minimum and maximum allowed nodal displacements, respectively. Note that equations (12) and (13) are linear optimization problems with bound constraints, that can be solved without effort [22].…”

Section: Member Stiffness Relationsmentioning

confidence: 99%