## 2 Materials and Method
### AM Filter Concept  
In Additive Manufacturing (AM) methods like Selective Laser Melting (SLM) or Electron Beam Melting (EBM), one of the constraints involves maintaining a minimum threshold angle for the incline of a surface facing downward (referred to as an overhang) in relation to the baseplate. When this standard is fulfilled for a specific part direction, it is understood that the part is self-supporting in that direction [23]. This widely recognized constraint in Additive Manufacturing has been extensively studied in various processes. The critical angle that is often cited approximates 45 degrees [46; 47; 48]. If a piece possesses areas with overhang angles falling beneath this critical degree in the chosen manufacturing orientation, it will not be self-supporting and cannot be printed in its current form. The support region in this context refers to the three-square elements in the layer underneath, depicted in Figure 3.  
Figure 2: Representation of FORM and SORM approaches on a normalized design space. In the present study, the filter recommended by [15], tailored to suit the requirements of the AM process, was utilized in the topology optimization of two-dimensional structures. According to this method, an element in a specific location must be sufficiently supported by the elements on its underlying layer to be printable. The supporting region is designated to include the element directly beneath or the immediate neighboring elements. The mathematical expression for the AM filter in the 2D scenario is provided as follows:  
\[ \begin{split}\xi_{(m,n)}=\min\Bigl{(}x_{(m,n)},\Xi_{(m,n)}\Bigr{)} \\ \Xi_{(m,n)}=\max\Bigl{(}\xi_{(m-1,n-1)},\xi_{(m-1,n)},\xi_{(m-1,n+1) }\Bigr{)}.\end{split}  \tag{11}\]  
Here, \(i\) and \(j\) denote the vertical and horizontal location of the element, \(\xi_{(m,n)}\) shows the printed density of the element at a given location, and \(\Xi_{(m,n)}\) shows the density of element \((m,n)\). Derivatives are taken using smin and smax smooth approximations for precision calculation in the optimization process. The mentioned approaches are expressed below:  
\[ \begin{split}\text{smin}(\text{x},\Xi)\equiv\frac{1}{2}(x+\Xi- \bigl{(}(x-\Xi)^{2}+\varepsilon\bigr{)}^{1/2}+\sqrt{\varepsilon})\\ \text{smax}(\xi_{1},\xi_{2},\xi_{3})\equiv\left(\sum_{k=1}^{n_{ \text{S}}}\xi_{k}^{p}\right)^{1/p}.\end{split}  \tag{12}\]  
Here, the \(\varepsilon\) and \(P\) parameters control the accuracy and smoothness of the approximations. The exact minimum and maximum operators for \(\varepsilon\to 0\) and \(P\to\infty\) are obtained, but the smoothness is lost. In this study, \(\xi_{0}=0.5\) was chosen as the default value. In this study, reliability analysis was adapted to the existing topology optimization approach and an AM constraint was added to consider the orientation of the printed part. Langelaar's suggested AM filter is incorporated into the renowned 88-line topology optimization code created by Andreassen et al. [15; 40]. By default, this code employs the OC optimizer. Within this procedure, an inner loop continually evaluates the volume constraint, meaning the AM filter is invoked several times, subsequently increasing computational time. As a result, modifications were made to the codes in alignment with the MMA [49] and the interior-point method. We compared the Optimality Criterion (OC), Method of Moving Asymptotes (MMA), and interior-point methods during the design iteration  
Figure 3: Definition of supporting regions for element \(m,n\)[18]. process. This assessment of the AM filter played a key role in shaping our final conclusions. The modification of the AM filter to fit the RBTO problem is depicted in Equation (13):  
\[ \begin{split}\min& C(\rho)=\mathbf{d}^{T}\mathbf{Kd}( \mathbf{p})\mathbf{d}\\ \text{subject to}&\beta(\mathbf{u})\geq\beta_{t}\\ \frac{V(\rho)}{V_{0}}&\leq f\\ \mathbf{K}(\rho,\mathbf{u},\mathbf{y})\cdot\mathbf{d}=\mathbf{ f}\\ 0<\rho_{min}\leq\rho_{e}&\leq 1\end{split} . \tag{13}\]  
In this formulation, \(\rho\) is the primary design variable.