# Optimisation of Cable Dome Structure Design for Progressive Collapse Resistance
## 4 Optimised Design
### Locally Optimised Design  
Building upon the results of global optimisation, this subsection presents a local optimisation process for Dome 4, that is, the Levy cable dome structure that exhibited the highest global resistance to progressive collapse among all six designs. According to the member importance analysis results, the removal of a member that has a large importance coefficient can result in a high dynamic response in the structure and may cause progressive collapse. Therefore, in this study, an optimisation model minimising the maximum member importance coefficient (\(\beta_{\text{max}}\)) was developed. In the current structure, the optimisation target was the importance coefficient of the OHC (Member 10), namely min\(\beta_{10}\). Parameter analysis revealed that, for structure systems with fixed topological relationships, the shape parameters (e.g., hoop cable radius and strut length) more sensitively influenced the importance coefficients of the members compared with the design parameters (e.g., cross-sectional area and prestress level). Therefore, the following mathematical model for local optimisation was proposed:  
\[\left\{ \begin{aligned} \min\beta_{10}=\rho_{\text{j}}/\sum\limits_{ \text{j=1}}^{13}\rho_{\text{j}}\\ \text{X}_{\text{i,min}}\leq\text{X}_{\text{i}}\leq\text{X}_{\text{i, max}}\end{aligned} \right. \tag{6}\]  
where Xi is the optimisation variable. As depicted in Figure 9, three combinations of shape parameters were used as the optimisation variables: (1) height differences of the support node (Node 1), with Node 3 on the OS (S1) and Node 5 on the MS (S2); (2) lengths of the OS (H1) and MS (H2); and (3) radii of the OHC (R1) and IHC (R2). In (6), Xi,max and Xi,min refer to the upper and lower limits, respectively, of the optimisation variable in question and are set as 120% and 80% of the variable's initial value, respectively (Table 3). An interactive genetic algorithm was used in MATLAB and Ansys LS-DYNA for design optimisation. The optimisation process was as follows: First, individuals with appropriate lengths were generated to form the initial population in MATLAB according to the solution preciseness and number of variables, and the population size was set to 20. Second, the basic information of each individual in the initial population in MATLAB was input into Ansys LS-DYNA to construct a basic structural model and calculate the prestress of each member through an iterative approach. Third, the fitness value was calculated according to the optimisation index in Ansys LS-DYNA, and the result was saved and then sent back to MATLAB. Fourth, the calculation results of each individual were summarized and analysed, the best individual was selected, and both its individual code and individual  
Figure 9: Optimisation variables. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Optimisation** & **Initial Values** & **Lower Limits** & **Upper Limits** & **Optimisation** \\
**Variables** & & & & **Results** \\ \hline S1 & 2.708 & S1 min = 2.166 & S1 max = 3.249 & 2.712 \\ S2 & 4.418 & S2 min = 3.534 & S2 max = 5.302 & 4.745 \\ H1 & 6.800 & H1 min = 5.440 & H1 max = 8.160 & 8.096 \\ H2 & 5.800 & H2 min = 4.640 & H2 max = 6.960 & 6.878 \\ R1 & 24.233 & R1 min = 19.386 & R1 max = 29.080 & 29.080 \\ R2 & 12.866 & R2 min = 10.293 & R2 max = 15.439 & 13.575 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Control range of optimisation variables. solution were recorded. Fifth, a new code was generated through crossover and mutation on the basis of the code of the fittest individual. Here, the probabilities of crossover and mutation were set to 0.8 and 0.2, respectively. Sixth, the number of iterations (200 in this study) determined whether the optimisation process continued or stopped. In case the process continued, the optimisation process started over from the first step. However, in case the process stopped, the optimisation process proceeded to produce the final optimal solution. Figure 10 illustrates the iterative optimisation of the importance coefficient of the OHC.