# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 5 Numerical Example: Synthesis of a Ten-Bar Truss Cantilever
### The MINLP Syntheses
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Optimization** & \multirow{2}{*}{**MINLP Iteration**} & \multicolumn{2}{c}{**Value of Objective Function**} & \multicolumn{2}{c}{**No. of Active Bin. Var.**} & \multicolumn{1}{c}{**CPU Time (s)**} \\ \hline
**Cycle** & **Phase** & & & **(N)** & \\ \hline \multirow{6}{*}{\(n=2\)} & & 1 & 1. NLP & 17,025.3822 & - & 0.441 \\  & & 2 & 1. MILP & 13,466.9650 & 8 & 0.020 \\  & & & 2. NLP & 17,919.2246 & & 1.020 \\  & T & 3 & 2. MILP & 16,313.8410 & & 0.030 \\  & & & 3. NLP & 17,346.7775 & & 2.000 \\  & & 4 & 3. MILP & 17,148.3880 & & 0.040 \\  & & & 4. NLP & 16,767.5972 & (\(z^{l}\)) & 1.020 \\  & & &... &... &... \\  & D & 11 & 10. MILP & 15,609.8720 & & 16 \\  & & & 11. NLP & **16,936.2314** & (\(z_{\text{\bf OPT}}\)) & (\({}^{\circ}\)) 9.380 \\ \hline \multirow{6}{*}{\(n=2\)} & & 12 & 11. MILP & 16,054.3400 & 8 & 0.040 \\  & & & 12. NLP & 17,919.2246 & (\(z_{1}{}^{l}>z_{\text{\rm OPT}}\)) & 2.469 \\  & & 13 & 12. MILP & 17,259.0410 & & 0.050 \\  & & & 13. NLP & 17,346.7775 & (\(z_{2}{}^{l}>z_{\text{\rm OPT}}\)) & 0.492 \\  & & & 13. MILP & 18,156.3480 & & 0.060 \\  & & & 14. NLP & 16,978.6881 & (\(z_{3}{}^{l}>z_{\text{\rm OPT}}\)) & 2.137 \\  & & & 15 & 14. MILP & 22,324.8906 & & 0.060 \\  & & & 15. NLP & 19,004.3514 & (\(z_{4}{}^{l}>z_{\text{\rm OPT}}\)) & 0.449 \\  & & & 15. MILP & 26,279.0820 & & 0.080 \\  & & & 16. NLP & 18,044.2047 & (\(z_{5}{}^{l}>z_{\text{\rm OPT}}\)) & 0.203 \\  & & & & \(\sum\) = **19.991** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Convergence of the Modified OA/ER algorithm and the **MTP** strategy, example **TC10b** (topology and shape optimization, without buckling constraints). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Optimization** & \multicolumn{3}{c}{**MINLP Iteration**} & \multicolumn{3}{c}{**Value of Objective Function**} & \multicolumn{1}{c}{**No. of Active Bin. Var.**} & \multicolumn{1}{c}{**CPU Time (s)**} \\ \hline
**Cycle** & **Phase** & & & & & \\ \hline \multirow{6}{*}{\(n=1\)} & T & 1 & 1. NLP & 17,025.3822 & \((z^{l})\) & 0.430 \\  & - & 1 &... &... &... & 20 \\  & D & 8 & 7. MILP & 15,967.6240 & & \\  & & & 8. NLP & 17,168.1830 & \((z_{\text{OPT}})\) & (\({}^{\circ}\)) 2.550 \\ \hline \multirow{6}{*}{\(n=2\)} & & 9 & 8. MILP & 13,466.9700 & & 8 & 0.030 \\  & & & 9. NLP & 17,919.2246 & & & 1.020 \\  & T & 10 & 9. MILP & 16,313.8410 & & & 0.030 \\  & T & & 10. NLP & 17,346.7775 & & & 2.000 \\  & & 11 & 10. MILP & 17,148.3880 & & & 0.040 \\  & & & 11. NLP & 16,767.5972 & \((z^{l}<z_{\text{OPT}})\) & & 1.031 \\  & & &... &... &... &... &... \\  & D & 18 & 17. MILP & 15,609.8720 & & & \\  & & & 18. NLP & **16,936.2314** & \((z_{\text{OPT}})\) & & (\({}^{\circ}\)) 9.334 \\ \hline \multirow{6}{*}{\(n=3\)} & & 19 & 18. MILP & 19,935.8230 & & 8 & 0.050 \\  & & & 19. NLP & 16,978.6881 & \((z_{1}{}^{l}>z_{\text{OPT}})\) & & 2.160 \\  & & 20 & 19. MILP & 19,949.4000 & & & 0.051 \\  & & & 20. NLP & 19,004.3514 & \((z_{2}{}^{l}>z_{\text{OPT}})\) & & 1.309 \\  & & 21 & 20. MILP & 21,562.9020 & & & 0.060 \\  & T & & 21. NLP & 18,044.2047 & \((z_{3}{}^{l}>z_{\text{OPT}})\) & & 0.199 \\  & & 22 & 21. MILP & 25,832.8800 & & & 0.070 \\  & & & 22. NLP & 18,831.2099 & \((z_{4}{}^{l}>z_{\text{OPT}})\) & & 0.563 \\  & & 23 & 22. MILP & 28,257.8040 & & & 0.090 \\  & & & 23. NLP & 17,898.1026 & \((z_{5}{}^{l}>z_{\text{OPT}})\) & & 0.461 \\  & & & & & & \(\sum\) = **21.478** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Convergence of the Modified OA/ER algorithm and the **STP** strategy, example **TC10b** (topology and shape optimization, without buckling constraints). Figure 7: Convergence to the optimal topology, example TC10a (initial phase of the **MTP** strategy), example **TC10a** (topology optimization, without buckling constraints).