## 3 Model Parameterization and Multidisciplinary Optimization  
### Parameterization of Nozzle and Rotor  
The parameterization of the turbine nozzle and rotor is based on Bezier and BSpline curves. The blade profile is defined by the blade thickness and beta-angle (the angle between the camber line and the meridional direction) distributions (see Figures 2 and 3). The nozzle blade is controlled by one section, thus the nozzle blade is a straight blade without lean and twist. The thickness and beta-angle distributions are parameterized by five control points, respectively. The first control point of the beta-angle distribution is fixed at 60\({}^{\circ}\) and the last control point of the thickness distribution is fixed at 0.3 mm (0.6 mm for the trailing edge diameter). Therefore, eight variables are involved in the nozzle optimization. Figure 2: Blade angle definition. The variation range for the beta-angle on each variable control point is set to \(\pm\)10% of its complement angle. The variation range for the thickness on each variable control point is also set to \(\pm\)10% of the local thickness. The scope that the nozzle blade varies is examined to make sure that an adequate design space is provided and no interference happens. The rotor is defined by: (1) the meridional contour of both the solid and fluid domain (Figure 4); (2) the beta-angle distribution of the camber line at hub, middle and tip sections (same as the nozzle parameterization in Figure 2); and (3) the thickness distribution is normal to the camber line at the hub and tip sections (same to the nozzle parameterization in Figure 3). Figure 4 shows the definition of the meridional contours of both the solid and fluid domain. The solid hub firmly holds all the blades against the centrifugal force. An initial back face profile is given and parameterized with seven control points, the first and the last two control points are fixed, and the other four control points can be modified in a prescribed range as indicated by the arrows in Figure 4a. The blade patch corresponds to where the blade is located, and its hub and shro  
Figure 4: Definition of the meridional contour. (**a**) Backface profile parameterization; (**b**) hub and shroud parameterization. Figure 3: Blade thickness definition. with five control points, respectively. The first control points of both curves are fixed such that the inlet blade height is kept constant and no translation occurs for the entire geometry. The rest control points can be modified by the optimization program and the possible variation in axial and radial direction is indicated by arrows in Figure 4b. In Figure 2, the rotor blade camber line at the hub, middle, and tip section is defined through beta-angle distribution with five equally spaced control points, respectively. The first control point at the hub section is fixed and the rest can be modified in a prescribed range. The 3-D camber line can be obtained using the transformation according to Figure 7 in Part I [20]. The thickness distribution is added and normal to the camber line at the hub and tip sections in both directions, and the blade shape between the two sections is interpolated. The thickness distribution is parameterized using the B-Spline curve with five control points in Figure 3, and the thickness at each control point can also be modified by the optimization program in a specified range. The parameterization method mentioned above will provide the blade profiles at different layers. They are 3-D curves consisted of discrete points out there. The blade profile on each layer represents the blade geometry at a certain relative blade height. Then an open source 3-D modeling software called FreeCAD, which is released by Matra Datavision Inc in France, is used to perform a skinning to envelop the blade profile to form a solid blade entity. After that, the meridional profile including the hub profile, back profile and front line are loaded in to revolve a cylinder. At last, the intersection part of the blade and the cylinder is chamfered.