# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 5 Numerical Example: Synthesis of a Ten-Bar Truss Cantilever
### Discussion of Results  
The convergences of the performed truss syntheses illustrated in Tables 1-4 (without buckling) and Tables A1-A4 (with buckling) show that, in all four cases, the identical final/optimal result was obtained using both the MTP and STP MINLP strategies. Regarding the MTP strategy, the optimal result was always obtained after the first two optimization phases, i.e., after the topology optimization with continuous sizing variables followed by the discrete sizing optimization at the obtained fixed topology. Up to this point, the MTP strategy is identical to the original TP strategy [80], which is terminated after the first (and also final) solution with discrete sizing variables. The simple TP strategy would thus yield the same final results. Table 5 illustrates and compares the MINLP iterations required and the CPU times for different synthesis examples performed by the discussed MINLP strategies. The data captured in Table 5 show that the TP strategy required the lowest number of MINLP iterations as well as the lowest CPU times until the final/optimal result was attained in all four examples. With regard to the exposed outputs, the TP strategy proved to be the most effective from among all the compared MINLP strategies. However, as already mentioned, the TP strategy does not provide any information about the influence of discrete sizing variables on the possible change in topology. On the other hand, this  
Figure 8: Convergence to the optimal topology (initial MINLP phase of the **MTP** strategy), example **TC10b** (topology and shape optimization, without buckling constraints). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{**MINLP Strategy**} & \multicolumn{5}{c}{**Synthesis Example**} \\ \cline{3-6}  & & **TC10a** & **TC10b** & **TC10c** & **TC10d** \\ \hline \multirow{2}{*}{No. of MINLP iterations} & TP & 60 & 11 & 10 & 7 \\  & MTP & 95 & 16 & 21 & 13 \\  & STP & 130 & 23 & 21 & 35 \\ \hline \multirow{2}{*}{CPU time (s)} & TP & 9.657 & 13.951 & 1.485 & 4.324 \\  & MTP & 16.650 & 19.991 & 3.537 & 10.889 \\ \cline{1-1}  & STP & 23.148 & 21.478 & 3.344 & 43.806 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of optimization statistics for different MINLP strategies. influence is accounted for both in the MTP as well as in the STP MINLP strategy. From among these two strategies, the MTP strategy proved to be the more convenient one. It should be noted that the differences between the considered MINLP strategies become more expressed in cases when the NLP sub-problems become more extensive and difficult to solve. The CPU times of individual NLP sub-problems are considerably higher in examples TC10b and TC10d when compared to examples TC10a and TC10c. The former two examples include shape optimization. With the inclusion of shape variables (nodal coordinates), not only is the total number of variables higher but the problem is also more complex as the changing coordinates affect the calculation of internal forces (FE equations). In example TC10d, buckling constraints are also included. The inter-connection between buckling constraints and shape variables is additionally present, as the buckling resistance of a bar depends not only on its cross-sectional area but also on its buckling length. The CPU time of an individual NLP sub-problem in example TC10d is thus, on average, about 10 times higher than the CPU times of NLP sub-problems in example TC10a. Consequently, the differences between the applied MINLP strategies become more distinctive. Although the STP strategy proved to be the least favorable strategy regarding both the number of MINLP iterations required as well as the CPU times spent, its advantage lies in the fact that more intermediate solutions are obtained. The first solution with discrete sizing variables is obtained at the initial topology with all alternative structural elements active.