Airfoil optimization design of drag minimization with lift constraint using adjoint equation method

Abstract

This paper presents an application of the adjoint equation method in drag minimization of transonic airfoils with lift constraint. The flow field is calculated by solving the Euler equations with finite volume space discretization and a multi-stage explicit time-stepping scheme. An adjoint equation, which the Lagrange multipliers (or costate vector) must satisfy, is created in simplifying the expression of the cost function variation and has to be solved before calculating the gradient of the cost function. To solve the adjoint equation, its equivalent version in physical space is used in integral form and discretized by finite volume method, and then is iterated by the same multi-stage explicit time-stepping scheme as in the Euler equations. The far-field boundary conditions for the adjoint equation is specified by diagonalizing the equation on the far-field boundary and assigning values to the Lagrange multipliers according to the sign of the eigenvalues of the coefficient matrix of the equation. A target shape and its corresponding pressure distribution are used to examine whether the adjoint equation is correctly solved and whether the gradient is correctly calculated. The results show that the adjoint equation converges well and as the cost decreases, the designed shape approaches the target shape. This means that the adjoint equation is correctly solved and the gradient is correctly evaluated. The results of the drag minimization with lift constraint show that the lift constraint can be guaranteed in minimizing the drag coefficient.