Recently, methods to obtain truss or frame with high stiffness by the topology optimization problem of minimizing compliance has been studied actively. In general, cross-sectional areas or shapes of each member are defined as design variables in such a topology optimization problem. Practically, since the member cross-sections are to be selected from predetermined candidates, the design variables are treated as discrete variables. Therefore, heuristics or mixed-integer programming (MIP) is selected as the algorithm for solving truss or frame topology optimization problems. In particular, MIP has an advantage that the global optimality of the solution is guaranteed mathematically. However, it is difficult for MIP to treat large scale problems because of the high computational cost. In previous studies, the topology optimization using MIP is limited to small-scale problems, such as trusses or 2D-frames.
In recent years, second-order cone programming (SOCP) has attracted attention since it can be solved efficiently by using the interior-point method. It is reported in the latest study that mixed-integer second-order cone programming (MISOCP), i.e., the problem obtained by adding the integrity to some of the variables of SOCP, can deal with the large scale problem compared with MIP. In this paper, the topology optimization of 3D-frame, which is difficult to treat by using MIP, is solved by using MISOCP.
First, the 3D-frame topology optimization problem of minimizing compliance is formulated as MISCOP by using the dual problem of total potential energy minimization problem. From this formula, various 3D-frame topology optimization problems of minimizing compliance in which various sets of cross-sectional shapes are defined as design variables are considered. In this paper, 3D-cantilever with tip load, latticed dome with central concentrated load, and simple building with vertical and horizontal loads are optimized by SOCP in various design value cases. In almost all of the optimization cases, the solver has terminated with finding the global optimal solution. In only a few optimization cases which could not find the global optimal solutions, the MISOCP solver was run with time limit and the best feasible solutions found by the solver are reported. These solutions are not necesarily global optimal solutions, but these are good solutions in the cases of this paper. As numerical experiments for comparison, topology optimization problems treated in this paper were tried to be solved by using MIP as well, but the optimal solutions could not be found at all within realistic computational time.
The obtained main results are as follows:
•The 3D-frame topology optimization problem of minimizing compliance can be formulated as MISOCP by using the dual problem of total potential energy minimization problem.
•MISOCP can solve various large scale 3D-frame topology optimization problems which are difficult for MIP to treat because of the high computational cost.
•For a larger problem which is beyond the capability of MISOCP, MISOCP cannot find the global optimal solution but can find solution good enough.
The commercial solvers which treat MISOCP have been developed extensively, and calculation speed is improved year after year. Against this background, there exists a possibility that the MISOCP will become a powerful topology optimization tool for building structures.
