Truss topology design and sizing optimization with guaranteed kinematic stability
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RESEARCH PAPER
Truss topology design and sizing optimization with guaranteed kinematic stability Mohammad Shahabsafa1 · Ramin Fakhimi1 · Weiming Lei1 · Sicheng He2 · Joaquim R. R. A. Martins2 · ´ Terlaky1 · Luis F. Zuluaga1 Tamas Received: 19 February 2020 / Revised: 10 June 2020 / Accepted: 20 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Kinematic stability is an often overlooked, but crucial, aspect when mathematical optimization models are developed for truss topology design and sizing optimization (TTDSO) problems. In this paper, we propose a novel mixed integer linear optimization (MILO) model for the TTDSO problem with discrete cross-sectional areas and Euler buckling constraints. Random perturbations of external forces are used to obtain kinematically stable structures. We prove that, by considering appropriate perturbed external forces, the resulting structure is kinematically stable with probability one. Furthermore, we show that necessary conditions for kinematic stability can be used to speed up the solution of discrete TTDSO problems. Using the proposed TTDSO model, the MILO solver provides optimal or near optimal solutions for trusses with up to 990 bars. Keywords Truss topology optimization · Truss kinematic stability · Mixed integer linear optimization · Euler buckling constraints
1 Introduction The truss design problem is an important problem in the field of structural design optimization (Haftka and G¨urdal 2012). In the past, various formulations and solution methodologies for truss design problems have been developed (see, e.g., Arora and Wang, 2005; Rozvany; 2009; Stolpe; 2016). Dorn et al. (1964) considered a ground structure framework for the truss design problem and used numerical optimization to solve the problem. In a ground structure framework, the set of potential bars of a truss structure is given and the optimal cross-sectional areas of the bars are to be determined. Two objectives are commonly used in truss design problems. A wide range of mathematical models for truss design problems consider the structure’s compliance Responsible Editor: Fred van Keulen Mohammad Shahabsafa
[email protected] 1
Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA
2
Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USA
minimization as the objective (Ben-Tal and Bendsøe 1993; Bendsøe et al. 1994; Stolpe 2007). In turn, several second-order cone optimization and semi-definite optimization models have been suggested to address compliance minimization for truss topology design and sizing optimization (TTDSO) problems (Kanno 2016; Kanno and Fujita 2018; Ben-Tal and Bendsøe 1993). Kanno (2018), for example, proposed a robust mixed integer semi-definite optimization (MISDO) model for the minimum-compliance truss topology design and sizing optimization problem with uncertainty of the external loads and developed a heuristic method to solve the proposed MISDO model. Another frequently used objective in t
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