## 9 Topology Optimization Results  
Figure 5 summarizes the results obtained for the compliance minimization of the Cantilever beam. Figure 6 summarizes the results obtained for the topology optimization of the L-bracket considering the different problem statements, plotting the objective function at each iteration and illustrating one of the final geometries obtained. Finally, Figure 7 shows the topology-optimized geometry of the bonded support. The color code adopted here displays elements with \(\rho=0\) in white, intermediate densities in grey, and elements with \(\rho=1.0\) in black. These results were not adimensionalized. The results obtained for each problem statement and any particular choice of parameters are discussed in the following subsections. In common, these examples share the volume constraint \(V^{*}=0.5\). In the particular case of the solution \"OC--Discrete and decreasing\", the initial density was set to 1.0 and gradually reduced by a factor of \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Mesh Size** & \(\rho\) & \(\frac{\partial\sigma^{PN}(\rho)}{\partial\rho}\) & \(\partial\sigma^{PN}_{spf}(\rho)\) & \(\partial\sigma^{PN}_{u}(\rho)\) & \(\frac{|\partial\sigma^{PN}_{spf}(\rho)|}{|\partial\sigma^{PN}_{spf}(\rho)|}\) \\ \hline \multirow{4}{*}{0.5} & 1 & \(-0.0088\) & \(0.0004\) & \(-0.0092\) & \(24.037\) \\  & 0.5 & \(-2304.3703\) & \(0.5426\) & \(-2304.9129\) & \(4248.295\) \\  & 0.1 & \(-90.9833\) & \(1.2132\) & \(-92.1965\) & \(75.996\) \\ \hline \multirow{4}{*}{1} & 1 & \(-0.0305\) & \(0.0013\) & \(-0.0318\) & \(23.864\) \\  & 0.5 & \(-7949.4335\) & \(1.8849\) & \(-7951.3184\) & \(4218.445\) \\  & 0.1 & \(-313.8380\) & \(4.2147\) & \(-318.0527\) & \(75.462\) \\ \hline \multirow{4}{*}{3} & 1 & \(-0.2140\) & \(0.0096\) & \(-0.2236\) & \(23.289\) \\  & 0.5 & \(-3.4697\) & \(0.1086\) & \(-3.5784\) & \(32.936\) \\ \cline{1-1}  & 0.1 & \(-2206.1706\) & \(30.3676\) & \(-2236.5382\) & \(73.649\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average value of the stress derivative, and its components, observed in the L-bracket model at different design density values, considering three different mesh sizes. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Mesh Size** & **Number of Elements** & **p-Norm (MPa)** & **Element Location** & **Element Label** & \(\Delta\rho\) & \(\rho=1-\Delta\rho\) & **Finite Differences** & **Continuous Derivative** \\ \hline \multirow{4}{*}{0.5} & \multirow{4}{*}{102,400} & \multirow{4}{*}{314.27741} & \multirow{4}{*}{B} & \multirow{4}{*}{102,332} & \(0.1\) & \(314.27740\) & \(0.0000\) & \multirow{4}{*}{0.000} \\  & & & & & \(0.01\) & \(314.27741\) & \(0.0000\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(314.27740\) & \(0.0000\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(314.28104\) & \(-0.0363\) & & \\ \hline \multirow{4}{*}{1} & \multirow{4}{*}{25,600} & \multirow{4}{*}{272.96255} & \multirow{4}{*}{B} & \multirow{4}{*}{25,560} & \(0.1\) & \(272.87736\) & \(0.8519\) & \multirow{4}{*}{0.4426} \\  & & & & \(0.01\) & \(272.95502\) & \(0.7534\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(272.96249\) & \(0.0006\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(272.96255\) & \(0.0000\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(272.97007\) & \(-0.0752\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(272.96326\) & \(-0.0713\) & & \\ \hline \multirow{4}{*}{3} & \multirow{4}{*}{2889} & \multirow{4}{*}{221.08948} & \multirow{4}{*}{A} & \multirow{4}{*}{1953} & \(0.1\) & \(22.36352\) & \(-112.7404\) & \multirow{4}{*}{\(-65.2413\)} \\  & & & & \(0.01\) & \(221.48312\) & \(-39.3637\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(221.08923\) & \(0.0025\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(221.08941\) & \(0.0066\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(221.13180\) & \(-0.4232\) & & \\ \cline{5-8}  & & & & \(0.01\) & \(221.09332\) & \(-0.3842\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Analysis of the continuous maximum stress derivative determined by the Python code and the derivative obtained through finite differences.