Airfoil design using an optimization procedure and the implicit function theorem

Abstract

An airfoil shape design method is described for a compressor cascade. Two-dimensional compressible flow is dealt with. An objective function, which depends on the design variables for the shape, is introduced. The shape is optimized by minimizing an objective function. The sequential quadratic programming procedure is used. To reduce CPU time, at each iteration, the gradient of the objective function with respect, to the desugn variables is calculated by using the implicit function theorem. An optimization example is presented. Introduction This paper is concerned with airfoil shape optimization using the sequential quadratic programming (SQP) procedure. The shape is defined using HD design variables xp. An objective function F(XD) is introduced; it depends on the design variables. The objective function is minimized using the SQP procedure. The SQP procedure iterates a local minimization process. At each iteration, it requires the values of the objective function and its gradient g with respect to the design variables at current design variables Usually, the gradient g is calculated by executing as many flow simulations as the design variables. However, the calculation requires much CPU time. In this paper, the implicit function theorem is used for calculating the gradient g. The calculation requires as much CPU time as one flow simulation. Similar calculation methods have been reported for one-dimensional compressible flow inside a duct [Ij and two-dimensional imcompressible flow around an airfoil [2], In this paper, two-dimensional compressible flow around a compressor cascade is dealt, with. Firstly, the gradient calculation method is decribcd. Then, the method is numerically tested. Gradient, calculated by the method is compared with that calculated by executing flow simulations. Finally, an optimization example is presented using the method and the SQP procedure. Sequential Quadratic Programming procedure Let xD be current design variables. The objective function F(XD) is approximated for the design variables xp — x + p as F(xD); F(xD)+gp+\pHp (1) where g is the gradient of the objective function with respect to the design variables at s D, H is a. positive definite approximation of the Hessian matrix. By linearizing the nonlinear constraints and using the quadratic programming, the optimal vector pk of p is found such that the objective function F(XD) approximated by (1) attains its minimum value. Then, the design variables are updated by