### Two-level optimization predicts designs with the least contact pressure and stress.

The reliability of a gas or steam turbine is strongly dependent on the structural design of its blades. In land-based power generation units, turbine blades are connected to the turbine disk by a dovetail joint or fastener. As the turbine rotates, the dovetail experiences centrifugal loading including contact pressure and Mises stress. Efforts to improve turbine system efficiency increase the severity of temperature and pressure operating conditions, putting greater structural demands on the blades.

Computational simulation with finite element analysis (FEA) provides a powerful method for predicting the stresses and failure conditions for a given blade design. The addition of process automation and optimization tools helps engineers arrive at optimal designs in a reasonable amount of time.For this study, Abaqus FEA was used in conjunction with Isight process automation and optimization software to determine a design that minimized the contact pressure and stress in the dovetail region of the blade. Instead of using a single optimization process that considered all design variables at once, a two-level approach was employed. First, a Design of Experiments (DOE) analysis was undertaken to identify the most critical variables. In the second level, those variables were then more fully explored and optimized.

To construct the finite element model for the analysis (Figure 1), the geometry of the turbine blade airfoil was created using Eblade, an airfoil aerodynamic design optimization software package. Coordinate data points describing the 3D airfoil shape of the blade were imported into Abaqus/CAE, which was also used to construct the geometries of the remaining parts (shroud, platform, and disk). It was assumed that the geometry of the cross-section of the contact region in the dovetail was the same in the blade platform and the disk.

For this investigation, only centrifugal loading was considered, cyclic symmetry conditions were enforced, and the bottom of the disk was held fixed. A nonlinear static procedure was used for stress analyses. The construction of the FEA model was automated in order to parameterize the geometric dimensions and apply the contact and boundary conditions as the geometry changed during optimization.

An initial cross-sectional shape for the dovetail with nine design variables (DVs) was chosen to demonstrate a typical parameterization method and to investigate the optimization processes. Assuming symmetry, the dimensions of only one side of the dovetail were considered.

In the first level of the analysis, the optimization software used all nine variables to perform a DOE that sampled the design space and created approximate representations of the blade’s response to pressure and stress loads around the dovetail. Approximations were calculated from a set of finite element results obtained by sampling the design variables in specified ranges. Various geometries of the dovetail generated by the DOE runs were used to build approximate models.

Once the approximation models were created, an optimization analysis was performed on the dovetail for contact pressure and Mises stress using a variety of different analysis methods. The optimization software also provided tools for assessing the sensitivity of individual design variables for the different approximate models (Figure 2).For the second-level optimization, in which targeted variables were more fully explored, the values of four variables (which changed little between various methods in the firstlevel optimization) were treated as constant, and other values (obtained from the optimal method in the first-level analyses) were chosen as the starting points for the other design variables. The analysis options were modified to obtain about 100 total iterations for the design.

Comparing the results of first- and second-level optimizations showed that gradient-based methods did not provide a better solution in the second level, but that direct and exploratory methods provided improved results over those in the first level.

Following this two-level optimization, a more traditional single-level optimization was performed for comparison. Fewer optimization methods were chosen for this analysis, and all design variables were allowed to change. In this single-level approach, each optimization was defined so as to include approximately 250 total design evaluations, which was about the same as the combined number of evaluations for the two-level approach.

The study illustrated that a two-level optimization approach using design process automation in combination with FEA provided a more robust design for a turbine dovetail than using a single-level optimization of all design variables. By initially exploring the design space with an approximate model, the dominant design variables were first identified. The secondlevel optimization was then able to focus on a smaller number of parameters, allowing for faster analysis and/or the use of a larger number of optimization techniques for the same number of total iterations.

*This work was done by Youngwon Hahn, Engineering Specialist, and John I. Cofer, Senior Technical Marketing Specialist at SIMULIA, Dassault Systèmes. For more information, visit http://info.hotims.com/45603-122.*