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.

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 y_i = 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