We present and examine a number of improvements to a gradient-based algorithm for aerodynamic optimization. A Newton-Krylov algorithm is used to solve the compressible Navier-Stokes equations, the gradient is computed using the discrete-adjoint method with a preconditioned Krylov solver, and the optimum is found through a quasi-Newton algorithm with a rank-two update formula. Constraints are imposed by penalizing the objective function. Improvements are made in three areas: 1) thickness constraints are generalized to permit the location of maximum thickness to be determined by the optimizer or alternatively to constrain the cross-sectional area; 2) new scalings of design variables and initial estimates of the inverse Hessian matrix in the quasi-Newton method are investigated; 3) the algebraic grid perturbation algorithm is replaced by an algorithm based on a spring analogy. In each case, the effect of the improvements on the performance of the algorithm is presented.

A Newton-Krylov algorithm for aerodynamic shape optimization in three dimensions is presented for both singlepoint and multipoint optimization. An inexact Newton method is used to solve the Euler equations, a discrete adjoint method is used to compute the gradient, and an optimizer based on a quasi-Newton method is used to find the optimal geometry. The flexible generalized minimal residual method is used with approximate Schur preconditioning to solve both the flow equation and the adjoint equation. The wing geometry is parameterized by B-spline surfaces, and a fast algebraic algorithm is used for grid movement at each iteration. An effective strategy is presented to enable simultaneous optimization of planform variables and section shapes. Optimization results are presented with up to 225 design variables to demonstrate the capabilities and efficiency of the approach.

A Newton-Krylov algorithm for aerodynamic shape optimization in three dimensions is presented for both single-point and multi-point optimization. An inexact-Newton method is used to solve the Euler equations, a discrete-adjoint method to compute the gradient, and a quasi-Newton method to find the optimum. The flexible generalized minimal residual method is used with approximate-Schur preconditioning to solve both the flow equation and the adjoint equation. The wing geometry is parameterized by a B-spline surface, and a fast algebraic algorithm is used for grid movement at each iteration. For multi-point optimization, a composite objective function is used. Optimization results are presented to demonstrate the capabilities and efficiency of the approach.

A Newton-Krylov algorithm is presented for aerodynamic shape optimization in three dimensions using the Euler equations. An inexact-Newton method is used in the flow solver, a discrete-adjoint method to compute the gradient, and a quasi-Newton method to find the optimum. The Krylov subspace method flexible generalized minimal residual is used with approximate-Schur preconditioning to solve both the flow equation and the adjoint equation in a parallel computing environment. The wing geometry is parameterized by a B-spline control net, and a fast algebraic algorithm is used for grid movement. The discrete-adjoint gradient can be obtained in approximately one-fourth the time required for a converged flow solution. The accuracy of the gradient is compared against finite differencing and is found to be comparably accurate. A single-point test case is presented for a cruise configuration optimization at transonic speed. This example as well as an inverse design demonstrate that the optimizer is able to decrease the objective function and gradient by several orders of magnitude efficiently for problems with over 170 design variables.

An efficient Newton-Krylov algorithm for high-fidelity aerodynamic shape optimization is used to design low-sweep wings for maximum range at transonic speeds. In this approach, the steady flow solution is obtained using the Newton method with pseudo-transient continuation. The objective function gradient is computed using the discrete-adjoint method. Linear systems from both the flow and adjoint systems are solved using a preconditioned Krylov method. A quasi-Newton optimizer is used to find the search direction. It is coupled with a line-search algorithm. Our single-point optimization results show that it is possible to design shock-free unswept wings at Mach numbers and lift coefficients comparable to the operating conditions of modern transonic transport aircraft. Robust wing designs for low-sweep and unswept wings under the same operating conditions are obtained through multi-point optimization.