High Order Aerodynamic Optimization Using New Hybrid Sequential Quadratic Programing-Particle Swarm Intelligence Technique

Abstract

In this paper we present results of aerodynamically and geometrically constrained drag minimization using both second order and high order CFD simulations. We present a new strategy for finding the global optimal solution of an objective function using a hybrid scheme of gradient based and gradient independent optimization techniques. The gradient based phase is a Quasi-Newton line search minimization with Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) approximation of the Hessian matrix. We use Regrouped Particle Swarm Optimization (RegPSO) for the gradient independent phase. We parameterized the airfoil geometry using our own parameterization technique and presented a straight forward method to add a thickness constraint equation as a linear constraint equation to the optimization problem. The semi-torsional spring analogy is used to adapt the grid in the entire field as the airfoil shape changes with shape optimization iterations. We present two transonic drag reduction test cases, each with two starting geometries, NACA 0012 and NACA 00083. The first test case is transonic drag reduction wi th lift constraint. The second test case is a transonic drag reduction with both lift and thickness constraint. We present the optimization convergence history of second and fourth order schemes as well as the final optimal geometries. For the first case, we present a plot of the objective function value when the airfoil geometry varies linearly from the optimal solution found by SQP and the optimal profile found by the hybrid scheme; this pl ot shows that there may be a descent direction but the gradient based optimizer can not find it because it is a lmost orthogonal to the gradient vector computed using the SQP optimizer. The proposed hybrid scheme is 4 to 9 times expensive than SQP optimization alone, but shock free profiles are obtained in both test cases and wit h both starting geometries. Aerodynamic design used to rely on CFD simulations in conjunction with experimental testing and the engineering intuition of the designer. With the growth of high speed computers, integrating numerical optimization schemes with CFD simulations has become possible and is now used for aerodynamic design and optimization. Gradient-based optimization techniques are widely used because they reach an optimized shape after a reasonable execution time; however, the final optimal shape is a local minimum near the op timization starting point. Non-gradient-based methods like genetic algorithm (GA) or particle swarm (PS) are slower to find an optimum but can find the global minimum regardless of the starting point; their drawback is the larg e number of iterations required to reach this minimum compared to gradient-based schemes. Gradient-based optimization depends on evaluating the gradient of the objective function with respect to the geometric design variables and using it in a linear model (steepest descent) or a quadratic model (Newton’s or Quasi Newton’s model) to find a search direction; this search direc tion is the direction in which the design variables should change their values to minimize the objective function 25 . Gradient-based techniques have been widely used in aerodynamic optimization due their fast convergence to the nearest local minimum point. The obtained optimal shape is biased by the optimization starting point (initial aerodyn amic shape), and there is no guarantee that gradient-based methods can find the global best optimal shape in the design sp ace 1 . The use of the adjoint method, which was originally applied to aerodynamics problems by Jameson 17 , to compute the gradient reduced the computational cost of