# Optimized One-Click Development for Topology- Optimized Structures +
## 2 Materials and Methods
### Finite Spheres as Manufacturing Constraints  
The concept of finite spheres means a useful manufacturing constraint for casting processes and AM as it describes and solves manufacturing conflicts [42]. For the implementation of the concept of finite spheres to the TO problem, we assumed that every element has a physically reasonable volume. Then, each element was abstracted by a sphere with an identical center of gravity and volume \(V_{i}\). Therefore, the effective radius \(r\) of the sphere for element \(i\) is:  
\[r_{i}\ =\sqrt[3]{\frac{3\ V_{i}}{4\ \pi}}\ \sin t \tag{9}\]  
with a tolerance angle \(t\ \in\ \left[0,\ \frac{\pi}{2}\right]\). The tolerance angle is a solely technical tool which was used later to describe a manufacturing conflict. Basically, the idea is not to evaluate the elements but the finite spheres if they have manufacturing conflicts. A manufacturing conflict is defined here as two spheres with overlapping radii. In the context of casting processes, this describes an undercut. Hence, the tolerance angle can be used to specify that an overlapping only occurs, if two spheres encounter in an angle below \(t\). Therefore, the effective radius of the spheres was adapted as shown in Figure 3.  
This abstracted concept was then applied to calculate manufacturing conflicts in a given manufacturing direction \(s\) (stacking direction AM; demolding direction, casting) in combination with a manufacturing angle \(\omega\) (critical printing angle AM; demolding angle, casting) as illustrated in Figure 4.  
Since we are generally interested in obtaining symmetric solutions, this consideration was performed in the positive and negative direction. This led to two cones for potential manufacturing conflicts as shown in Figure 5.  
Figure 3: Determination of the effective radii with the tolerance angle; S denotes the center of gravity: (**a**) the spheres with unadjusted radii collide in angle \(t\); (**b**) the radii are multiplied with \(\sin t\); so that (**c**) the spheres can pass without overlapping. Figure 4: Spheres with (red) and without (green) manufacturing conflict potential to the grey sphere in manufacturing direction \(s\) and manufacturing angle \(\omega\). Figure 5: Visualization of the conflict cones in the positive (\(+s\)) and negative (\(-s\)) direction. Conflict spheres in red and conflict-free spheres in green. The conflict spheres in the positive direction are stored in \(K^{+}\), while the conflict spheres in negative direction are stored in \(K^{-}\). These simplifications allowed to mathematically characterize the manufacturing conflicts. In this context, a casting part was manufacturable, if the geometry can be demolded in a given angle and direction. An AM part, exemplarily a fused filament fabrication part, could be fabricated without support structures in a given stacking direction, if the critical printing angle was below a certain value, commonly 45 degrees. The sets \(K_{i}^{\pm}\) of the potential conflict elements for each element is then given by  
\[K_{i}^{\pm}\ =\Big{\{}j\ :\ \pm\ s\cdot S_{ij}\ >0\ \land\ \Big{(}1-\sin^{2}\omega \Big{)}\Big{(}||s||_{2}^{2}||S_{ij}||_{2}^{2}-\big{(}s\cdot S_{ij}\big{)}^{2} \Big{)}\ <\ \big{(}||s||_{2}\big{(}r_{i}+r_{j}\big{)}\pm\ s\cdot S_{ij}\sin\omega \big{)}^{2}\Big{\}}\, \tag{10}\]  
with the center of gravity \(S_{ij}\). Thereby two sets, which contain all potential positive conflict elements, respectively, in the negative manufacturing direction, were obtained for each element. In order to consider also previously defined passive regions, their elements were gathered in fix sets which are defined as follows:  
\[ \begin{array}{l}F^{0}\ :\text{deactivated Elements},\ \ \rho_{i}=0\\ F^{1}\ :\ \text{activated Elements},\ \ \rho_{i}=1\end{array}  \tag{11}\]  
Hence, at the beginning of the optimization, no areas with \(\rho_{i}=0\) existed, the set \(F^{0}\) is a priori empty.