Topology optimization of truss structure considering kinematic stability based on mixed-integer programming approach.

Kinematic instability due to unstable nodes is an often neglected but critical aspect of mathematical optimization models in truss topology optimization problems. On the one hand, kinematically unstable structures cannot be used in the actual structural design. On the other hand, unstable nodes within continuous parallel bars can make the calculation of bar length wrong and affect the optimization effect. To avoid kinematic instability, a computationally efficient nominal disturbing force (NDF) approach for truss topology optimization is presented in this paper. Using the NDF approach, the most favorable structure for the optimization goal can be selected in three schemes: (1) adding bracings at unstable nodes, (2) removing unstable nodes and replacing short bars with long ones, or (3) selecting a new topology form to avoid containing unstable nodes. Compared with the widely used nominal lateral force (NLF) approach in the literature, the NDF approach can not only improve the optimization efficiency but also obtain lighter optimization results. Moreover, using the NDF approach, a mixed-integer linear optimization model for minimizing the weight of truss with discrete cross-sectional areas subject to constraints on kinematic stability, bar buckling, allowable stress, nodal displacement, bar crossing, and overlapping is proposed in this study. Because the objective and constraint functions are linear expressions in terms of variables, the globally optimal structures can be obtained by using the proposed model. In addition, two necessary conditions for kinematic stability are proposed to speed up the computational efficiency and delete unnecessary nodes within consecutive tension bars. Finally, the effectiveness of the proposed NDF method and the necessary conditions for kinematic stability are studied on four truss topology optimization problems in two and three dimensions. [ABSTRACT FROM AUTHOR]