Discrete topology and sizing optimization of frame structures with compliance constraints: A semidefinite programming-based approach

The discrete topology and sizing optimization of frame structures with compliance constraints is studied using a novel approach, which is capable of finding the theoretical lower bounds and high-quality discrete solutions in an efficient manner. The proposed approach works by reformulating the discrete problem as a relaxed semidefinite programming (SDP) problem. This reformulation is made possible by a linear relaxation of the original discrete space and the elimination of the nonconvex equilibrium equation using a semidefinite constraint. A continuous global optimum is first derived using existing solvers and then the discrete solution is discovered by the neighborhood search. Numerical examples are presented, including the sizing optimization of 2-Bay 6-Story frame and 3-Bay 10-Story frame, the topology and sizing optimization of 2-Bay 6-Story braced frame. A topology and sizing example with multiple load cases is also provided. The proposed approach and three other metaheuristic algorithms are used to solve these examples. Theoretical lower bounds for these examples can be efficiently discovered by the proposed approach. For the sizing problems, the discrete solutions by the proposed approach are all better than the other algorithms. For the topology and sizing problems, the proposed approach achieves discrete solutions better than genetic algorithm, but worse than the other metaheuristics. The computational superiority of the proposed approach is validated in all the examples.