## 2 Materials and Methods
### Numerical Solver  
For any given \(c(\theta)\) and control mechanism, \(\varepsilon_{T}\) can be computed with the aid of aerodynamic tools and numerical optimization algorithms. Here, we use MachUp, a modern numerical lifting-line algorithm to predict the aerodynamics for a given planform and control configuration [30; 31]. MachUp differs from what is commonly referred to as the numerical lifting-line algorithm [34] in that it solves for the vorticity strength at each wing section by relating the lift predicted from the section airfoil properties to the lift predicted by the three-dimensional vortex-lifting law [35], instead of enforcing a Neumann condition at the three-quarter-chord location [34]. In this regard, MachUp is more analogous of a numerical lifting-line method than that from [PERSON] and [PERSON] [34] since it uses the same fundamental relation as was used by [PERSON] in classical lifting-line theory [24; 25]. MachUp extends past the applicable scope of classical lifting-line by being able to model multiple lifting surfaces, each with arbitrary sweep and dihedral, even though this work considers only single wings with no sweep or dihedral. This method has shown good agreement with computational fluid dynamics solutions and experimental results for wings with aspect ratios greater than about 4 [30]. MachUp is used in conjunction with the open-source Sequential Least Squares Programming optimization algorithm [36]. The optimization algorithm calls MachUp as its objective function to determine the induced drag of the wing and allows the twist deflections to vary at the control actuation locations to minimize induced drag and maintain a given lift coefficient. For a given wing, the optimization algorithm runs multiple times for \(C_{L}\) values 0.5, 0.8, 1.1, and 1.4 to give a range of drag versus lift data. Then, using the \(\kappa_{P}\) values from Figure 2, Equation (21) is fit to the optimization data by varying \(\varepsilon_{T}\) such that the root mean squared (RMS) of the percent error is minimized. This provides an accurate estimate of \(\varepsilon_{T}\) for the given wing planform and control design. For all solutions shown here, a section lift slope of \(2\tau\) was used, and an RMS error less than \(1.35\times 10^{-6}\) was obtained. Since the airfoil properties are modeled as linear functions without stall and since \(C_{D_{i}}\) is exactly proportional to \(C_{L}^{2}\) according to lifting-line theory, the choice of \(C_{L}\) values for this process is somewhat arbitrary. It was found that the optimization algorithm could find solutions more effectively at higher lift coefficients where gradients are stronger. Therefore, the high lift coefficient values used here are merely a means to compute a more accurate twist effectiveness term at low angles of attack. The choice of this range of lift coefficients does not imply that anything in this work applies near or beyond stall conditions. The analysis was only applied to a specific class of control mechanisms, even though it could be applied to others. For this class of control mechanisms, the actuators were assumed to be spaced evenly in the spanwise direction. The continuous control surfaces are modeled by having a linear variation in geometric twist between control actuators, with one always at the wing root and another always at the wing tip. Since twist is defined relative to the root, the actuator at the root determines the angle of attack of the wing. The discrete control surfaces are modeled by equally sized discrete sections of constant geometric twist. The number of sections is determined by the number of control actuators per semispan, where again the actuator at the root determines the angle of attack of the wing. Figure 1 shows an example of a four-actuator discrete and continuous control mechanism using trailing-edge flaps instead of geometric twist for clarity. The actuator deflections are equivalent between the two mechanisms shown. This class of control mechanisms has no limits placed on the magnitudes of the control deflections.