# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 2 Modeling and Analysis
### Multi-Fidelity Design Benchmark Problems
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Description** & **Value** & **Units** \\ \hline Design range & 7725 & nmi \\ Specific fuel consumption & 0.53 & \(\text{lb}_{\text{m}}/(\text{lb}_{\text{r}}\text{th})\) \\ Engine diameter & 2.85 & m \\ Takeoff fuel burn & 2.5 & \% MTOW \\ Landing fuel burn & 1.0 & \% Mass at end of cruise \\ Fixed mass & 91,250 & kg \\ Payload mass & 34,000 & kg \\ Reserve fuel mass & 15,000 & kg \\ Engine mass & 16,564 & kg \\ Structural mass for rigid cases & 27,250 & kg \\ Fixed mass center of gravity x-coordinate & 0.569 & Fraction of fuselage length \\ Payload mass center of gravity x-coordinate & 0.569 & Fraction of fuselage length \\ Maneuver load factor & 2.5 & g \\ KS stress factor of safety & 1.5 & - \\ Cruise Mach number & 0.85 & - \\ Cruise altitude & 37,000 & ft \\ Maneuver Mach number & 0.64 & - \\ Maneuver altitude & Sea level & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fixed parameters used for the objective and constraints. Note that engine diameter and mass (based on Boeing 777-200 ER/GE90) are only used in the pitching moment constraints (by means of thrust and mass). with the engine's center of thrust (\(z_{C_{g},\text{engine}}\)) being positioned one-half the nacelle's diameter below the wing's Yehudi leading edge point. The moment from fuel weight \(M_{\text{fuel mass}}\) is computed as  
\[M_{\text{fuel mass}}=\eta_{\text{criuse}}\times(MTOW-FB_{\text{takeoff}})\times \Big{(}x_{C_{g},\text{fuel}}-x_{\text{nose}}\Big{)}, \tag{19}\]  
with the fuel's center of mass (\(x_{C_{g},\text{fuel}}\)) presumed to reside at the centroid of the wing's inboard section. The moment from structural weight is computed based on the cruise load factor and the structure's mass and center of gravity (which remain fixed for rigid optimization cases and vary for aeroelastic cases). The moment from the engine's mass is computed using the cruise load factor and a presumed engine mass, with the engine's center of gravity assumed to reside at the wing Yehudi's leading edge x-coordinate. The moment from the remaining mass (\(M_{\text{fixed mass}}\)), including fixed and payload masses, is computed based on the cruise load factor, presumed masses, and a somewhat arbitrarily chosen center of gravity coordinate, which was chosen such that the rigid baseline geometry (with arbitrary structural thicknesses) was trimmed. Finally, when designing structural gauges, an aggregated stress constraint is computed by TACS using the discrete Kreisselmeier-Steinhauser (KS) [35; 36] formulation with a weighting of 50. For design optimization, the objective function is total fuel burn normalized by 80,000 kg,  
\[f=\frac{FB_{\text{total}}}{80,000} \tag{20}\]  
so that it is roughly on the order of 1. Among the inequality constraints \(g\leq 0\), the first constraint is the cruise load factor constraint,  
\[g_{1}=1-\eta_{\text{criuse}}, \tag{21}\]  
which ensures that lift is greater than or equal to weight at the beginning of cruise. To ensure a total pitching moment of approximately zero at the start of cruise, two additional constraints are enforced,  
\[g_{2}=\frac{M_{\text{total}}}{Length_{\text{fuselage}}}-0.01 \tag{22}\]  
and  
\[g_{3}=-\frac{M_{\text{total}}}{Length_{\text{fuselage}}}-0.01, \tag{23}\]  
with the total moment being (somewhat arbitrarily) normalized by fuselage length to make it on the order of 10 and a downward shift of 0.01 to help avoid optimization convergence issues. When including the structural sizing variables, maneuver load factor, and KS stress constraints,  
\[g_{4}=2.5-\eta_{\text{maneuver}} \tag{24}\]  
and  
\[g_{5}=KS-1/1.5, \tag{25}\]  
are included so that the aggregated function (with a factor of safety of 1.5) is enforced at a 2.5 g pull-up maneuver.