## 2 Method
### Evaluator: Compliance  
The task of the compliance evaluator is the computation of the global mean compliance. For this purpose, an algorithm based on finite element method (FEM) [26] is used. The global mean compliance  
\[c=f_{\text{eva},c}(\mathbf{x},\mathbf{r}_{k},\mathbf{r}_{s}) \tag{11}\]  
Figure 5: Predictor’s artificial neural network (ANN) topology (simplified).  
Figure 6: Sigmoid, ReLU and PReLU activation functions.  
is defined according to [26] as follows:  
\[c=\mathbf{u}^{\mathsf{T}}\mathbf{K}\mathbf{u}=\mathbf{u}^{\mathsf{T}}\mathbf{f} \tag{12}\]  
where \(\mathbf{K}\) is the stiffness matrix, \(\mathbf{f}\) is the force vector and \(\mathbf{u}\) is the displacement vector. The compliance has the dimension of energy. As is usual in [26; 29], in the following, the units will be omitted for the sake of simplicity.  
As already explained, the static boundary conditions vector \(\mathbf{r}_{\mathrm{s}}\) consists first of x-entries and then \(\mathrm{y}\) -entries. Since the degrees of freedom of the stiffness matrix are arranged in an alternate way (one x-entry and one y-entry), the force vector is to be built accordingly. In order to transform the static boundary condition vector \(\mathbf{r}_{\mathrm{s}}\) into the force vector \(\mathbf{f}\), the number of nodes  
\[l=(d+1)^{2} \tag{13}\]  
and a collocation matrix \(\mathbf{I}_{\mathrm{R}}\)  
\[\mathbf{I}_{\mathrm{R}}=\begin{cases}i_{\mathrm{R}_{2;l}}=1&\text{where }i=\{0, \ldots,l-1\},\\ i_{\mathrm{R}_{2(j-l)+1,j}}=1&\text{where }j=\{l,\ldots,2l-1\},\\ i_{\mathrm{R}}=0&\text{otherwise}\end{cases} \tag{14}\]  
are required. The force vector is then obtained as follows:  
\[\mathbf{f}=\mathbf{I}_{\mathrm{R}}\mathbf{r}_{\mathrm{s}}. \tag{15}\]  
The system's equation is as follows:  
\[\mathbf{f}=\mathbf{K}\mathbf{u}. \tag{16}\]  
The stiffness matrix \(\mathbf{K}\) depends linearly on the geometry \(\mathbf{x}\) and is expressed by the following:  
\[\mathbf{K}=\sum_{i=1}^{d^{2}}x_{i}^{p}\mathbf{K}_{i} \tag{17}\]  
where the matrices \(\mathbf{K}_{i}\) are the unscaled contributions of the single elements to the stiffness matrix. The penalization exponent \(p\) achieves the desired focusing of the geometry toward the limits of values \(x_{\mathrm{min}}\) and \(1\) as described in Section 2.1.  
The stiffness matrix \(\mathbf{K}\) is then reduced by removing the columns and rows corresponding to the fixed degrees of freedom according to the kinematic boundary conditions. The result is the reduced stiffness matrix \(\mathbf{K}_{\mathrm{red}}\), which then can be inverted. The reduced force vector \(\mathbf{f}_{\mathrm{red}}\) is determined according to the same principle. From the reduced equation,  
\[\mathbf{f}_{\mathrm{red}}=\mathbf{K}_{\mathrm{red}}\mathbf{u}_{\mathrm{red}} \tag{18}\]  
the reduced displacement vector is obtained as follows:  
\[\mathbf{u}_{\mathrm{red}}=\mathbf{K}_{\mathrm{red}}{}^{-1}\mathbf{f}_{ \mathrm{red}}. \tag{19}\]  
The reduced global mean compliance \(c_{\mathrm{red}}\) is finally computed as follows:  
\[c_{\mathrm{red}}=c=\mathbf{u}_{\mathrm{red}}{}^{\mathsf{T}}\mathbf{K}_{ \mathrm{red}}\mathbf{u}_{\mathrm{red}}. \tag{20}\]  
where the calculation of the mean global compliance \(c\) according to (12) or \(c_{\mathrm{red}}\) according to (20) leads to the same result since \(\mathbf{u}\) at the fixed degrees of freedom vanishes and, therefore, has no effect on \(c\).