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Truss topology design and sizing optimization with guaranteed kinematic stability

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.

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Notes

  1. https://github.com/shahabsafa/truss-data.git

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Acknowledgments

This research was supported by Air Force Office of Scientific Research Grant no. FA9550-15-1-0222. The authors would like to thank the editors and the referees to provide constructive suggestions which allowed us to improve the article significantly.

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Correspondence to Mohammad Shahabsafa.

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The authors declare that they have no conflict of interest.

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Responsible Editor: Fred van Keulen

Replication of results

The data of the Michell trusses that are solved in this study are made publicly available in the GitHub repository: https://github.com/shahabsafa/truss-data.git. The repository, for each test problem, includes both the raw data of the structure and the MPS file used as input for the Gurobi solver.

In the computational experiments, we used the commercial, state-of-the-art, mixed integer linear optimization software Gurobi (2019) to solve all test problems. The specification of the workstation used, and the specific parameter settings of Gurobi ver. 9.0.0 solver are presented on p. 10, at the beginning of Section 5.

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Appendices

Appendix 1

In the basic discrete model, the choice constraints for the TTDSO problem are defined as:

$$\begin{array}{lll} x_{i}=\sum\limits_{k\in \mathcal{K}} s_{k} z_{ik}, & i \in \mathcal{I},\\ \sum\limits_{k\in \mathcal{K}} z_{ik}=y_{i}, & i \in \mathcal{I},\ k \in \mathcal{K},\\ y_{i}\in \{0,1\}, &i\in \mathcal{I},\\ z_{ik} \in \{0,1\}, & i \in \mathcal{I}, k\in \mathcal{K}\cup \{0\}. \end{array}$$
(26)

To enforce equalities (8) and (9), the following set of constraints is needed:

$$ \begin{array}{rclr} (1-y_{i})\underline{\sigma}^{\text{d}}_{i} \ \le & \sigma^{\text{d}}_{i} &\le \ (1-y_{i})\overline{\sigma}^{\text{d}}_{i}, &i\in \mathcal{I},\\ \max \left( -\gamma_{i} s_{k}, \sigma_{i}^{\min}\right) z_{ik} \ \le& \sigma_{ik} & \leq \ \sigma_{i}^{\max} z_{ik}, & i \in \mathcal{I},\ k \in \mathcal{K}. \end{array}$$
(27)

The Euler buckling constraints are incorporated in the set of constraints (27) as well. The basic MILO model for TTDSO is defined as:

$$ \begin{array}{llll} \min \ & \sum\limits_{i\in \mathcal{I}}\rho l_{i} x_{i}, \\ \text{s.t.} & {R} {q} &= {f},\\ & R^{T} u &= {\Delta} l,\\ & x_{i}-\sum\limits_{k\in \mathcal{K}} s_{k} z_{ik}&=0, & i \in \mathcal{I}, \\ & \lambda_{i} {\Delta} l_{i}- \sum\limits_{k\in \mathcal{K}} \sigma_{ik} - \sigma^{\text{d}}_{i}& = 0,& i \in \mathcal{I}, \\ &q_{i} - \sum\limits_{k\in \mathcal{K}} s_{k}\sigma_{ik}& =0, & i \in \mathcal{I}, \\ &\sum\limits_{k\in \mathcal{K}}z_{ik}&=y_{i},& i\in \mathcal{I},\\ & \max \left( -\gamma_{i} s_{k}, \sigma_{i}^{\min}\right) z_{ik} & \le \sigma_{ik} \le \sigma_{i}^{\max} z_{ik} & i \in \mathcal{I},\ k \in \mathcal{K},\\ &u^{\min}_{\ell}&\leq u_{\ell} \leq u^{\max}_{\ell},&\ell\in \mathcal{L},\\ &(1-y_{i})\underline{\sigma}^{\text{d}}_{i} &\leq \sigma^{\text{d}}_{i}\leq (1-y_{i}) \overline{\sigma}^{\text{d}}_{i},&i\in \mathcal{I},\\ & y_{i_{1}}+ y_{i_{2}} & \le 1, & (i_{1},i_{2}) \in {\mathcal{A}}^{\text{p}}, \\ & \sum\limits_{\bar i\in {\mathcal{A}}_{i}} y_{\bar i} & \le |{\mathcal{A}}_{i}| (1-y_{i}), & i \in \mathcal{I},\\ & \sum\limits_{\bar i \in {\mathcal{C}}_{i}} y_{\bar i} & \le 1, & i\in \mathcal{I}^{B},\\ &y_{i}\in \{0,1\}, &i\in \mathcal{I},\\ & z_{ik}\in \{0,1\}, & i \in \mathcal{I},\ k \in \mathcal{K}. \end{array}$$
(28)

If yi = 1, for \(i\in \mathcal {I}\), then the problem reduces to a truss sizing optimization problem. In a truss sizing optimization problem, bar-crossing elimination and bar-overlapping elimination constraints are not needed, since the topology of the structure is pre-determined by the ground structure.

Appendix 2

The list of the definitions of the article is given in Table 6.

Table 6 List of definitions

The decision variables of the mathematical model and the parameters introduced in the article are presented in Tables 7 and 8, respectively.x

Table 7 List of the decision variables of the mathematical models
Table 8 List of the parameters of the article

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Shahabsafa, M., Fakhimi, R., Lei, W. et al. Truss topology design and sizing optimization with guaranteed kinematic stability. Struct Multidisc Optim 63, 21–38 (2021). https://doi.org/10.1007/s00158-020-02698-x

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Keywords

  • Truss topology optimization
  • Truss kinematic stability
  • Mixed integer linear optimization
  • Euler buckling constraints