# Optimize Rotating Wind Energy Rotor Blades Using the Adjoint Approach
## 2 Optimization Method
### Gradient Projection and Evaluation  
For an optimization procedure, it is an essential point to work out suitable parameters. The definition of the rotor blade's shape can be done in various ways. It was shown in [17] that the free parametrization, i.e., all numerical grid points are design parameters, is not suitable for optimizing rotor blades. This results from large flow field gradients in the chordwise direction and in the spanwise direction, which transfer to the computed gradient. In areas of large gradients, the deformation is much larger, which leads to bumps and uneven surfaces. Instead of the use of all grid points, one can select parameters that define the blade geometry, like the twist distribution, thickness of airfoils along the blade and so forth, which define the blade geometry. For the developed optimization tool, an automatized routine is implemented that allows the projection of the gradients at each numerical grid cell to a reduced design space. The flow around airfoils and rotor blades is highly sheared with large flow gradients both in velocity and pressure. Therefore, the numerical grid has to be appropriately fine at those areas in order to catch the flow phenomena. Typical values for the amount of cells in the chordwise direction vary between 250 and 350. The adjoint approach leads to gradient information on each of these cells. To use a parametrization with less design parameters, a projection from these 250-350 gradients on less design parameters is implemented. The main idea for the parametrization used in the optimization tool comes from FSI using a finite beam. The gradients are treated like forces acting on the blade and therefore projected on central nodes. The principle of the parametrization is shown in Figure 1.  
As a pre-processing step, the blade is split into a selected amount of elements. Each of these elements has a central node, which is represented by a big red dot in Figure 1. All big dots represent one final design parameter each. The amount of design parameters is chosen beforehand by the user. The number can be controlled by adapting a dictionary that is defined for the usage of the developed tool. Figure 1: Principle of gradient projection for a wind turbine rotor blade. In this exemplary case, there are nine design parameters. For one exemplary element, the structured grid is shown with thin blue lines. In the figure, the represented grid has been coarsened for the sake of clarity. Within each cell of this grid, the small blue dot represents the local gradient of each grid cell, computed via the adjoint approach. The adjoint gradients are then transferred onto the nodes depending on the distance of the local cell to the central node of the element. Evaluating the projected gradients then leads to a movement of the nodes, which act like beam nodes. The difference to FSI is that the structure is not solved, and it is assumed that any movement given by the gradients can be realized. This assumption is necessary for the presented optimization. The final, optimized geometries can be used within the blade design process as an input to the design of the structure, for example when passive control shall be applied. Therefore, the structure is not final when the optimization takes place. In addition, the movement of the blade due to a structural response might interfere with the movement given by the gradients. The pre-condition for this projection is that an elastic deformation shall be guaranteed, where the blade cross sections are constant. However, even this parametrization on a reduced design space takes into account the gradient information of each numerical grid cell, as each local gradient is used within the projection. This makes the usage of the adjoint approach convenient. The gradients of each numerical cell that belongs to one central node of an element are transferred to two vectors.