# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 2 Modeling and Analysis
### Design Parameterizations  
In the present work, a particular uCRM design can be described by several sets of design variables, including shape (\(\mathbf{x_{s}}\)), structural (\(\mathbf{x_{t}}\), i.e., shell thicknesses), and aerodynamic (\(\mathbf{x_{a}}\), e.g., cruise angle of attack). Different combinations of these parameter sets, which are all normalized by bounds to range from 0 and 1, were considered so that optimization cost trends can be assessed. Figure 1: Baseline ESP representation of the uCRM geometry, with grey and red showing outer mold line (OML) and inner mold line (IML) geometries, respectively. Two optional sets of shape design variables may be optimized in this work. Each set controls the sectional twist angles and airfoil shapes at the wing Yehudi and tip (where Yehudi refers to where the trailing edge sweep angle changes near mid-span), with one having 7 airfoil parameters per section, while the other has 17. The sets of initial parameters were both fit to UM's reference geometry, with the higher-dimensional parameterization (and lower-dimensional one to a lesser extent) matching the reference geometry well at the two sections. Figure 2a illustrates these sections on the low-fidelity ESP geometry (which excludes the fuselage), with the inner mold line (IML) geometry shown for reference. In this figure, the two outboard wing cross-sections may be varied, while the two wing root sections and tail geometries remain fixed. Intermediate sectional shapes elsewhere in the wing are formed based on linear interpolation of the centerline, wing-fuselage junction, Yehudi, and tip airfoil sections. While an ESP model with many more airfoil sections has been developed and tested (containing ten additional cross-sections with around 200 design variables), it is not demonstrated in the present work. Furthermore, while the high-level planform parameters described in [18] may also be varied, they were held fixed in the present work. In each airfoil section, the design variables comprise a chordwise row of control point z-coordinates above the airfoil, distances to a corresponding row of points below the airfoil, and one variable that influences the trailing edge angle (all of which are normalized by chord length). The lower bounds for the distance variables were set to \(-0.5\), while the upper bounds were set to their initial values, thus avoiding designs that are thinner than the baseline design (approximately). When planform remains fixed, this approximately ensures that the wing volume does not decrease, allowing the same amount of room for fuel and other components (as well as the structure for rigid cases). Lower bounds for the upper set of points were set to 0.05. The upper bound for the top leading edge point was set to \(-0.05\) minus the lower leading edge point's initial value. This ensures that the lower leading edge point cannot go above \(-0.05\), thus avoiding designs with sharp leading edges (which would presumably yield poor performance in off-design conditions). All other upper bounds for the top points were set to 0.5. Bounds for the trailing edge variable were \(-0.005\) and 0.005. The sectional twist design variables at these two sections were allowed to vary from \(-3\) to 6 degrees (initial values being 0.837972 and 4.612053 degrees, respectively). The optional set of structural design variables includes the shell thickness of each rib, skin, and spar face, each comprising aluminum with a Young's modulus of \(6.9\times 10^{10}\) Pa a Poisson's ratio of 0.32, a density of 2700 kg/m\({}^{3}\), and a yield stress of \(2.76\times 10^{8}\) Pa. Figure 2b illustrates these different faces in the IML geometry. Nodes at the centerline are assumed to be clamped ([PERSON] with zero displacement and rotation), while those at the wing-fuselage junction are pinned ([PERSON] with zero displacement). Consisting of 114 top and bottom skins, 58 ribs, and 111 spars, with arbitrary initial thicknesses of 0.02, 0.03, and 0.04 m, this set of design variables adds 283 design variables in total.