This article describes the development of a method for optimization of the geometry of three-dimensional turbine blades within a stage configuration. The method is based on flow simulations and gradient-based optimization techniques. This approach uses the fully parameterized blade geometry as variables for the optimization problem. Physical parameters such as stagger angle, stacking line, and chord length are part of the model. Constraints guarantee the requirements for cooling, casting, and machining of the blades.The fluid physics of the turbomachine and hence the objective function of the optimization problem are calculated by means of a three-dimensional Navier-Stokes solver especially designed for turbomachinery applications. The gradients required for the optimization algorithm are computed by numerically solving the sensitivity equations. Therefore, the explicitly differentiated Navier-Stokes equations are incorporated into the numerical method of the flow solver, enabling the computation of the sensitivity equations with the same numerical scheme as used for the flow field solution.This article introduces the components of the fully automated optimization loop and their interactions. Furthermore, the sensitivity equation method is discussed and several aspects of the implementation into a flow solver are
879efficient 3D optimization requires, on the one hand, a stable grid generator and flow solver, and on the other hand, a blade parametrization that needs just a few parameters to describe the whole range of technical relevant 3D blade geometries.Many optimization concepts have already been introduced. Demeulenaere and Van der Braembussche [2] present fast inverse design methods for the Euler equations. These methods may be very powerful if one knows the optimal pressure distribution. Genetic algorithms are discussed in [15] and [34]. They are able to achieve the global minimum, but the convergence rate is very slow. Burns and Borggard [3,4] consider the sensitivity method. They differentiate the governing conservation law and then solve the sensitivity equations by a standard (Enquist-Osher) discretization [7]. The most efficient method for providing gradient information for a large number of parameters is the adjoint method. It is discussed by Elliot [6], Giles et al. [10], Iollo et al. [17], and Jameson [19].The geometry optimization method in this paper, in its 2D aspects, is based upon [27] and [1]. Physical and technological aspects are discussed in [23,24]. To our knowledge, we present one of the first complete 3D-optimization loops for the optimization of turbine blades.Sect. 2 starts with a short overview of the underlying fluid solver ITSM3D, see [22,26]. Then we discuss the components needed for a complete optimization loop. Due to the very expensive function evaluation, gradient based methods are preferred. A short flow chart of the SQP algorithm is given in the first part, see also [11]. The second part gives some hints on the parametrization of the turbine blade. The choice of parametrization of the blade is very sensitive with respect to performance of the optimization loop. Providing gradient information is the topic of the third part. Here, different methods for computing the gradients are discussed. Due to reasons of efficiency, the sensitivity analysis must be integrated into the solver ITSM3D and thus exploiting its infrastructure.The sensitivity analysis itself is discussed in Sect. 3. First, a theoretical result justifies the direct differentiation of the fluxes. The computation of these derivatives is discussed, with main focus on the treatment of boundary conditions. The requirement of changing blade geometries leads to an inhomogeneous wall condition for the velocity. Numerical compatibility conditions complete the condition for the wall. Furthermore the derivatives of the input and output boundary conditions are discussed.Sect. 4 considers the discretization of the flow solver and of the sensitivity equation. Here, the strong interaction of both discretizations are exposed.Sect. 5 deals with implementation aspects, namely the verification of fluxes and an alternative formulation of the sensitivity equations which is computationally more advantageous with respect to the inhomogeneous boundary conditions. Finally, in Sect. 6, the performance of the method is demonstrated on a complex stato...
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