## 5 Numerical Analyses and Qualification
### Finite Element Results  
Structural FE analyses were carried out using a 3D FEM model (in Altair Hypermesh environment). The 3D FE model of the manifold body and its internal ducts are detailed in Figure 9a,b. A high density of nodes (around 3.5 M) and elements (solid mesh _ctet10_[74], approximately 2.5 M) allowed for the identification of the stress distributions close to that of the crucial design details (i.e., hydraulic pipes and high curvature sections), which is, generally, a potential source of crack initiation. The FE analyses were actually performed considering the hydraulic manifold pinned (\(t_{x}=t_{y}=t_{z}=0\)) at the three interface holes (Figure 9c). The load was a pressure field (_pload4_[74]) applied uniformly on the internal wall's surface (shell elements _ctri3_[74], at about 72 k) (Figure 9d). The interesting stress contours are given in Figure 10. The [PERSON] stress peak (158 [MPa]) at the limit load occurred at the internal ducts, usually representing the critical areas of these components (Figure 10a,c). The max principal stress map used in the impulse fatigue analysis is provided in Figure 10b, and the pertinent stress peak (157.0 [MPa]) is highlighted in Figure 10d and was used for the extrapolation of the fatigue cycles in Figure 11. The safety margins (MS) at the static and cyclic loads are indicated in Table 4.  
Figure 8: Manifold after printing: (**a**) before drilling; manifold after machining and postprocessing heat treatment; (**b**) thread detail (top); (**c**) plugs threads detail (front). \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Test**} & **Pressure** & **No. of Cycles** & **Reference** \\ \cline{2-4}  & **[bar]** & **[-l]** & **[-l]** \\ \hline
0. Supply Pressure & DOP = 207 & - & - \\ \hline
1. Proof Pressure & 1.5 \(\times\) DOP = 310.5 & - & SAE AS8775 [72] \\ \hline
2. Endurance Cycles & 5-207–185–195–207–5 & 5,000 & Legacy spectrum \\ \hline
3. Hydraulic Fatigue & 0.5-310.5–0.5 & 100,000 & SAE ARP1383 Rev. C [73] \\ \hline
4. Burst Pressure & 2.5 \(\times\) DOP = 517.5 & - & SAE AS8775 [72] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Hydraulic loading conditions. Figure 10: Stress maps (limit pressure load, PP = 31.05 [MPa]): (**a**) [PERSON]; (**b**) max principal; (**c**) Von Mises peak (inner flow channels); (**d**) max principal peak (inner flow channels). Figure 9: Details of the three-dimensional FE model: (**a**) 3D model; (**b**) inner walls; (**c**) constraints; (**d**) internal pressure. The next relationships were used for the static safety margins at the limit (1), ultimate (2) loads, and fatigue damage calculations (3). These resistance criteria were based on a consideration of the structural parts as safe from failure if the MS were above 0.0 while the cumulated damage was less than 1.0. \[MS_{lim}=\left(\frac{Limit\;[PERSON]\;[PERSON]\;stress}{Yield\;strength}\right)-1=\left( \frac{fy}{f_{ty}}\right)-1 \tag{1}\]  
\[MS_{ult}=\left(\frac{Ultimate\;[PERSON]\;[PERSON]\;stress}{Tensile\;strength}\right)-1= \left(\frac{fu}{f_{tu}}\right)-1 \tag{2}\]  
\[D=\sum_{i}\frac{i-th\;cycles\;required}{cycles\;to\;failure}=\sum_{i}\frac{n_{i}}{N_ {f}} \tag{3}\]
as follows: the first one (in already-verified AlSi10 Mg alloy, Figure 12a) and the one made with the Al-HS1 alloy (Figure 12b). All experimental activities were carried out at the Magnaghi Aeronautica Qualification Laboratory, Naples, Italy. A proof pressure equal to PP = 1.5 \(\times\) DOP (DOP: design operative pressure) was applied at the temperature specified at least two successive times and held for 2 min for each pressure application. The rate of the increase in pressure did not exceed 25,000 [psi] per minute. No evidence of external leakage or permanent was allowed. In the case of the burst pressure load (BP = 2.5 \(\times\) DOP), the component did not rupture; because of its destructive nature, this test was performed last [72]. The component was subjected to cyclic operations (i.e., endurance) and to other fatigue tests, such as hydraulic impulse.