## 2 Materials and Methods
### FEM Use Cases
However, in our approach, we evaluate the generalization over the perforation diameter and use material properties for steel and a top edge displacement of 5 mm in positive \(y\)-axis to consider a more challenging use case. We execute different simulation settings, where the generalization variable (diameter of perforation) is changed in each simulation; see Figure 2a and Table 2. In our second use case, we simulate a bending beam that end is displaced about 5 mm in the positive x-direction; see Figure 2b. We vary the yield stress generalization variable in each simulation setting; see Table 2. In our third use case, we simulate a quarter-symmetric block with four perforations under compressive load, which is compressed about 5 mm in the negative y-axis; see Figure 2c. In this use case, we vary two generalization variables (yield stress and width of the block) in each simulation; see Table 2.  
We evaluate our models on interpolation (i.e., that the generalization variables for testing are within the range of the generalization variables observed during training) and extrapolation (i.e., that the generalization variables for testing are outside the range of the generalization variables observed during training) tasks. In Table 2, we mark interpolation tasks with superscript (\(i\)) and extrapolation tasks with superscript (\(e\)). In Figure 3, we present the perfect nonlinear elastoplastic material behavior of our use cases. The Young's modulus is 210 GPa, Poisson's ratio 0.3 and the yield stress 900 MPa. In our first use case, the perforated plate, we use this setting in each simulation. In the other two use cases, the yield stress varies, while the remaining material parameters stay the same. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Use Case** & **Change** & **Deformation** \\ \hline Plate & Geometry & Elongation \\ Beam & Material Properties & Bending \\ Block & Geometry, Material Properties & Compression \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Classic FEM use cases. Overview of the three use cases and their main change and types of deformations. In the first two use cases, only a single change is conduced, while in the last use case, a combination of changes is studied. All parts are meshed, using plane strain 4-node bilinear quadrilateral elements with reduced integration and hourglass control. Please note that although [22] recommends the use of larger order elements for the approximation of body forces, we use bilinear elements since we do not use body forces in our surrogate modeling approaches. We create a finer mesh near additional geometric details (i.e., perforations in the plate and block use cases) and seed the perforation edge of the plate with an approximate size of 3.8 mm and the remaining edges with an approximate size of 5 mm. The perforation edges of the block are seeded with an approximate size of 3 mm and the remaining edges with an approximate size of 4 mm. The beam exhibits no comparable geometric details; thus, we seed all edges with an approximate size of 1.5 mm. Figure 2: The three use cases: (**a**) elongation of a plate (diameter = 100 mm) about 5 mm at the top end in positive y-direction, (**b**) bending of a beam by a displacement at the top end about 5 mm in positive x-direction, (**c**) compression of a block with four perforations in the center of the quarter-symmetric parts (width = 220 mm) about 5 mm in negative y-direction and (**d**) the considered coordinate system.