# Sketch-Guided Topology Optimization with Enhanced Diversity for Innovative Structural Design
## 4 Experiments
### Experiment for Generating Innovative Structural Designs  
The TO framework is tested using the hyper-parameters summarized in Table 1 to generate innovative structural designs of automobile wheels. \(\alpha\), \(\beta\), and \(\gamma\) are initialized as 0. \(\alpha_{Inc}\), \(\beta_{Inc}\), and \(\gamma_{Inc}\) are the increment values in each TO iteration. \(n_{FEA}\) is the grid sampling resolution for FEA computation. An adaptive parallel scale is not implemented by setting \(N_{0}\) equal to \(N\). Sketch images are preprocessed by applying a morphological dilation operator of kernel size (2,2). Figure 4 illustrates the loss curves and intermediate solutions during the iterative TO process. The two sketch-based constraint penalty losses take effect on the density distributions from the beginning of the iterations. As it iterates, the volume constraint has a stronger effect, with a decreased penalty loss value, the compliance gradually converges, while the sketch-based constraint penalty losses exhibit some degree of fluctuation. To ensure the comparability of results across multiple TO processes, a fixed number of iterations is set in the experiments. The convergence of each solving process is verified according to the relative changes in the loss terms for compliance and volume constraint. Over the last 30 iterations, if the relative changes in both losses remain below a threshold, the conversion of the process is validated. Figures 5 and 6 show the generated innovative structural designs constrained by two examples of free-hand sketches. Using a parallel scale of \(M=20\) without an adaptive scale, a total of 20 designs are generated for each sketch image; from them, 8 randomly selected designs are displayed. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline Desired volume & 0.35 & \(N_{0}\) & 30 & \(\alpha_{\max}\) & 100 \\ NN layers & \(1\times(100,)\) & \(n\) & 5 & \(\beta_{\max}\) & 1 \\ \(n_{EA}\) & 180 & \(M\) & 20 & \(\gamma_{\max}\) & 0.1 \\ \(l_{\min}\) & 9 & \(\alpha_{Inc}\) & 0.3 & \(p_{\max}\) & 6 \\ \(l_{\max}\) & 80 & \(\beta_{Inc}\) & \(5\times 10^{-4}\) & E & 1 \\ \(n_{f}\) & 1000 & \(\gamma_{Inc}\) & \(5\times 10^{-5}\) & Nu & 0.3 \\ \(n_{f}\) & 30 & \(p_{Inc}\) & 0.04 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hyper-parameters of multi-solution TO framework. Figure 4: Loss curves and intermediate solutions during the iterative TO process. The sketch in Figure 5 contains sharp, angular strokes, reflecting a bold and austere style, while the sketch in Figure 6 displays droplet-shaped forms, reflecting a soft and graceful style. Qualitatively, it is evident that the generated solutions embody the visual styling features of those sketches. Also, they exhibit large differences from each other in terms of structural topology and geometric features, verifying the effectiveness of the proposed TO framework in generating innovative structural designs that conform to free-hand sketches and are diverse. In most of the generated structures, a cavity that closely matches the closed-shape contour is generated, though subtle differences between them appear in some solutions. This is due to the closed-shape-based constraint being assigned with a large weight coefficient (\(\gamma_{Inc}=0.1\beta_{Inc}\)). In a special case of solution 4 in Figure 6, the contour is matched by a solid structure rather than a cavity. This is caused by the undirected boundary constraint suppressing the closed-shape-based constraint. These experimental results show that the former constraint is beneficial for promoting solution diversity while achieving an effective sketch-based constraint. Also, the competition between two different constraint terms further facilitates the diverse design exploration. Figure 7 presents more examples of generated solutions. Among them, two sketch images that only contain stroke elements also achieve effective human control.