# Optimize Rotating Wind Energy Rotor Blades Using the Adjoint Approach
## 4 Results
### Convergence Behavior of the Optimizations  
In this work, different types of convergence are used. Numerical convergence of the computed fields is investigated as well as convergence of the optimization tool. Convergence criteria for the flow field are mentioned in Section 3. The latter is defined depending on the ratio between current cost function value and aimed cost function value. The convergence behavior of the various optimizations  
Figure 4: Field visualisation based on the Q-criterion in terms of vorticity. The full rotor for the flow field in (**a**) and for the adjoint field in (**b**,**c**) is shown. For good comparability, the adjoint field is shown in (**b**) in the same direction, as the flow field, i.e., downstream. As it moves upstream, it is secondly shown in the direction of the propagation of the adjoint field in (**c**). for different aimed values and varying lengths of the twistable part of the blade is compared in Figure 5a-d.  
For all plots, the linear lines between the points are added to ease the following of the trend, whereas only the dots represent computed values of the actual geometries of the blade during the optimization. The thrust is normalized by the initial value of 958.9 N to ensure comparability of the different aimed values and convergence ranges. The aimed values equal 925 N for the upper plots (a) and (b) and 900 N for the lower plots (c) and (d). The left plots are for the wider allowed range of 2.5% around the aimed value, whereas the right plots are for a range of 1.5% around the aimed value. All optimizations, besides the short and long blade parts for an aimed value of \(900\pm 1.5\%\) N, show a similar behavior. With each optimization step, the thrust is reduced until the upper limit of the allowed aim range is met. The two named exceptions show a zig-zagging behavior close to the bound, which is a known problem of steepest descent. Comparing all optimizations, it can be seen that more optimization steps are needed when the aim is further away from the initial value and when the bounds are closer. Furthermore, longer movable blade parts lead to faster convergence, except for the case shown in Figure 5d. In this case, the middle length converges faster than the long length case. It was expected that longer blade parts lead to faster convergence of the optimization as larger parts of the blade are moved relatively to the inflow angle, which then has larger influence on the forces, respectively. For the case shown in Figure 5d, the sensitivity of the optimization process with respect to the user's input, like initial step-size and pre-convergence of the flow and adjoint field, was found to be the largest. This explains the outlier of the long radial section \(r_{1}\) compared to \(r_{2}\). Figure 5: Development of the normalised thrust force over the optimization steps for design parameters within varying parts of the blade. Long: 2 m–4.5 m radial position, middle: 3 m–4.5 m radial position and short: 3.75 m–4.75 m radial position. The aimed values and the limits of the allowed range are shown with black and grey lines, respectively. It has to be remembered that each optimization step represents a larger amount of CFD iterations, which are necessary to compute reliable gradients via the adjoint approach. For a convergence after four optimization steps, e.g., shown for the short and middle length in Figure 5c and the given pre-convergence iterations and iterations between two optimization steps specified in Section 3, this sums up to 54,500 CFD iterations per optimization. Roughly 35,000 of these iterations include solving two fields. These numbers are still very convincing for gradient based optimization based on CFD. When the optimizations for these cases were conducted, a dependence of the convergence, appearance of zig-zagging and initial step-size were observed. As mentioned in Section 2.2, the projected gradients are normalized by the current maximal value.