# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 2 Modeling and Analysis  
To serve as a benchmark aircraft design problem, the present work considers the undeflected high-aspect-ratio variant [5] (developed by the University of Michigan's MDO lab) of NASA's Common Research Model (CRM). Beginning with an IGES CAD file (Available at [[https://mdolab.engin.umich.edu/wiki/ucrm](https://mdolab.engin.umich.edu/wiki/ucrm)]([https://mdolab.engin.umich.edu/wiki/ucrm](https://mdolab.engin.umich.edu/wiki/ucrm)), accessed on 17 January 2020), the fuselage was modeled in Engineering Sketch Pad (ESP) [7] as a series of ellipses along with several rows of spline coordinates for the wing hump region (which are normalized based on fuselage width and height and wing root chord and position). The tail is formed by lofting through NACA airfoil sections. To generate the wing's planform shape, six high-level parameters (area, Yehudi break location, aspect ratio, leading edge sweep angle, inboard trailing edge angle, and taper ratio) are used to compute lower-level parameters such as chord lengths at the centerline, Yehudi break, and tip sections. Readers are referred to [18] for baseline values of the high-level wing parameters and the equations used to compute lower-level parameters.  
Wing sectional shapes are parameterized using either NACA four-series airfoils or Kulfan [19; 20] airfoils. Kulfan airfoil parameterizations, also known as Class-Shape Transformations (CSTs), unify shape and class functions so that critical parameters of an existing airfoil class (e.g., leading edge radius or trailing edge angle) can be controlled, while Bernstein polynomial shape functions provide more general geometric variations. To obtain the baseline geometry, each of the Kulfan parameterizations were fit to airfoil points from UM's surface geometry via the Parametric Legacy Unstructured Geometry System (PLUGS) [21], a parametric geometry fitting capability in ESP. While any number of shape functions can be used, this work uses either 7 or 17 design variables per spanwise section, which produced maximum distances of the reference airfoil points to the generated geometry on the order of 0.5% and 0.1% of chord, respectively. Meanwhile, the inner mold line geometry is formed based on fractional spar locations at the wing-fuselage junction, Yehudi break, and wing tip, as well as the numbers of ribs in the wing and fuselage (5 from the centerline to wing-fuselage junction and 52 in the inboard and outboard sections). These inner and outer mold line ESP geometries, illustrated in Figure 1, are built within a pyCAPS [22] script, which can configure various analyses of rigid and aeroelastic physics at different levels of fidelity.  
To assess a particular ESP geometry, pyCAPS provides an assortment of Analysis Interface Modules (AIMs), which prepare various aerodynamic or aeroelastic analyses based on the attributed geometry. This includes triangular, quadrilateral, or mixed mesh generation for shell structures, unstructured surface and volume meshing for nonlinear aerodynamic analyses (including surface grid sensitivities), and other input file generation for analyses that utilize linear aerodynamics (e.g., Nastran [23]). Through the use of several AIMs, basic analysis meshes and input files are generated for the parameterized ESP geometry. Pre- and post-processing external to pyCAPS are used for additional functionality. Details on particular aerodynamic or aeroelastic analyses are provided in Section 2.2.