# Topology and Parametric Optimization-Based Design Processes for Lightweight Structures
## 4 Results
### Hollow Plate
#### 4.1.2 Parametric Optimization of a Hollow Plate  
At this design workflow, a PO of the Hollow Plate was conducted at the first level before the TO, which now was implemented at the second level. In other words, a size/shape PO was carried out before the procedure presented in Section 4.1.1. The intention was to decrease the design space for the TO. The processes presented here are (4) parametric size/shape optimization (PO), (5) parametric size/shape, and topology optimization with no redesign (PO_TO_NR), (6) parametric size/shape, and topology optimization with redesign (PO_TO_R), and (7) parametric size/shape, topology optimization with redesign, and parametric shape optimization (PO_TO_R_PO). Concerning the size/shape parametric optimization in process (4), the same procedure was followed as process (3), but in this case, 100 samples were used in order to increase the prediction accuracy of the statistical model. The following rules had to be followed during the selection of the factors and their range in this process:  
\[\mathrm{L}>2\times\mathrm{r}, \tag{13}\]  
\[\mathrm{L}>\mathrm{FA}, \tag{14}\]  
\[\mathrm{r}<\mathrm{l}1<\mathrm{L}-\mathrm{r}, \tag{15}\]  
\[\mathrm{H}>2\times\mathrm{r}, \tag{16}\]  
\[\mathrm{and}\mathrm{r}<\mathrm{h}1<\mathrm{H}-\mathrm{r} \tag{17}\]  
The equality sign in these inequality constraints was overlooked. In this way, we could avoid infeasible design solutions that change the model's topology, as shown in Figure 10.  
For this reason, three new percentage parameters were defined: The allowable range for the hole at the horizontal direction, the allowable range for the hole at the vertical direction, and the allowable range for the force placement denoted with hole horizontal position (hhp), hole vertical position (hvp), and force position (fcp), respectively. The values of these parameters were ranged from 10% to 90% with a value increment (step) equal to 10. Hence, the l1, h1, and FA are now dependent parameters and were calculated by the following formulas:  
\[\mathrm{l1}=\mathrm{r}+\mathrm{hhp}\times(\mathrm{L}-2\times\mathrm{r}), \tag{18}\]  
\[\mathrm{h1}=\mathrm{r}+\mathrm{hvp}\times(\mathrm{H}-2\times\mathrm{r}), \tag{19}\]  
\[\mathrm{and}\mathrm{d1}=\mathrm{fcp}\times(\mathrm{L}-\mathrm{FA}) \tag{20}\]  
An overview of all the used design parameters in the case of the Hollow Plate is presented in Table 2.  
In addition, a sensitivity analysis was carried out in order to identify the factors that had the most significant influence on the output parameters (see Figure 11). The length and thickness are the factors with the most substantial impact at Hollow Plate's mass reduction. On the other hand, the parameters that affected the stress most were the thickness and the hole radius. Furthermore, many iterations were carried out at processes (5) and (6), creating a manual loop between the TO and the validation study. The maximum identified mass reduction was 34.7%, and it could be achieved at the second optimization level with the TO. Finally, a new round of parametric size optimization was held at process (6). The final designs of the processes in this workflow are illustrated in \[\mathrm{L}>\mathrm{FA}, \tag{14}\]0