Truss geometry and topology optimization with global stability constraints
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RESEARCH PAPER
Truss geometry and topology optimization with global stability constraints Alemseged Gebrehiwot Weldeyesus1 Andrew Tyas3
· Jacek Gondzio1,2 · Linwei He3 · Matthew Gilbert3 · Paul Shepherd4 ·
Received: 12 February 2020 / Revised: 24 April 2020 / Accepted: 17 May 2020 © The Author(s) 2020
Abstract In this paper, we introduce geometry optimization into an existing topology optimization workflow for truss structures with global stability constraints, assuming a linear buckling analysis. The design variables are the cross-sectional areas of the bars and the coordinates of the joints. This makes the optimization problem formulations highly nonlinear and yields nonconvex semidefinite programming problems, for which there are limited available numerical solvers compared with other classes of optimization problems. We present problem instances of truss geometry and topology optimization with global stability constraints solved using a standard primal-dual interior point implementation. During the solution process, both the crosssectional areas of the bars and the coordinates of the joints are concurrently optimized. Additionally, we apply adaptive optimization techniques to allow the joints to navigate larger move limits and to improve the quality of the optimal designs. Keywords Geometry and topology optimization · Global stability · Nonlinear semidefinite programming · Interior point methods
1 Introduction Truss design problems are often formulated based on the so-called ground structure approach (Dorn et al. 1964), in which a set of joints are distributed in the design space and are connected by potential bars. We are here concerned with a truss design problem where the goal is to optimize both the topology and geometry of the structures, i.e., when the design variables are the cross-sectional areas of
Responsible Editor: Gengdong Cheng Alemseged Gebrehiwot Weldeyesus
[email protected] 1
School of Mathematics and Maxwell Institute for Mathematical Sciences, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK
2
NASK Research Institute, Kolska 12, 01-045 Warsaw, Poland
3
Department of Civil and Structural Engineering, University of Sheffield, Sir Frederick Mappin Building Mappin Street, Sheffield, S1 3JD, United Kingdom
4
Department of Architecture and Civil Engineering, University of Bath, Claverton Down, Bath BA2 7AY, UK
the bars and the coordinates of the joints. These problems have been studied in many articles, for example by Dobbs and Felton (1969), Kirsch (1990b), Ben-Tal et al. (1993), Bendsøe et al. (1994), Pedersen (1972), Sergeyev and Pedersen (1996), Achtziger (1998), Tejani et al. (2018), and Miguel and Miguel (2012), to mention just a few. The problems are highly nonlinear, mainly due to the variation of the joint coordinates. However, the models are known to obtain optimal designs that are more practically useful as they require less post-processing and contain fewer joints connecting bars in the design space. Alternatively, one can also obtain eff
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