## 2 Methodology
Therefore, all TE in the contour and interface areas, which lie exactly in the respective layer planes (\(x\)-\(z\) plane), must be detected and assigned to the hatching area. To make this possible, all surface elements in the contour and interface area whose normal vectors point in the print direction (\(y\)-axis) are determined. Subsequently, the corresponding TE are derived and clustered. Finally, the material properties of the hatching area from the database are assigned to these clustered TE (see Figure 5 top). In the _Transfer of data_ step, the material properties are now transferred to the initial FE mesh of the topology optimization which is used for the FE analysis in each iteration. Therefore, the TE mesh generated in the step _Determination of the material areas_ with the material properties assigned in the step _Material assignment_ is transferred to the initial FE mesh of the topology optimization. The mapper software MapLib from Fraunhofer SCAI is employed for this purpose [32]. Herein, a weighted element mapping algorithm is used (see Figure 5 bottom). Figure 4: Surface meshes (**top**) as well as meshed areas including assigned material properties of the interim result (**bottom**). Having determined the material properties in each element \(i\) of the initial FE mesh of the topology optimization, the fictive densities \(\rho[i]\) can be adjusted in the _Density distribution_ step. This adjustment is necessary as the optimization algorithm employs the fictive densities which Abaqus needs to calculate the sensitivities for the next iteration. Therefore, the Young modulus of the hatching area \(E_{Hatching}\) is used as a reference value, since it has been assigned to all elements in the _Initial FE model_ step. If an element in the initial FE mesh of the topology optimization has a Young modulus \(E[i]\) lower than that of the hatching area due to the transferred material properties, the element is penalized by a reduction in the fictive density. If, on the other hand, the Young modulus is higher, the fictive density is increased. This results in the following equation for the fictive density \(\rho_{A}[i]\) adjusted per element:  
\[\rho_{A}[i]=\rho[i]*\frac{E[i]}{E_{Hatching}} \tag{2}\]  
Using the SIMP approach, the scaled Young modulus \(E_{A}[i]\) for each adjusted element is given as follows:  
\[E_{A}[i]=\rho_{A}[i]^{3}*E_{Hatching}=\left(\rho[i]*\frac{E[i]}{E_{Hatching}} \right)^{3}*E_{Hatching} \tag{3}\]  
This scaled Young modulus is now used to perform the FE analysis, based on which the sensitivities are derived by Abaqus. Subsequently, the MMA algorithm implemented in Tosca carries out the adjustment of the fictive densities for the next iteration. These four steps consisting of _Determination of the material areas_, _Material assignment_, _Transfer of data_ and _Density distribution_ are applied in the topology optimization in each iteration until a convergence criterion is reached. The standard procedure used by Tosca is employed to determine convergence [33]. Thereby, the change in the objective function and the element densities is evaluated after each iteration. If the change in the objective function is less than 0.001 and that in the element densities is less than 0.005 compared to the last iteration, convergence can be assumed. The result of the developed optimization method is an optimized product design (_Optimized part_) which considers the three porous areas of the SLM. Figure 5: Porous areas including assigned material properties of the interim result considering the print direction (**top**) and material properties transferred to the initial topology optimization mesh (**bottom**).