Gradient-Limiting Shape Control for Efficient Aerodynamic Optimization

Abstract

Local shape control methods, such as B-spline surfaces, are well-conditioned such that they allow high-fidelity design optimization; however, this comes at the cost of degraded optimization convergence rate as control fidelity is refined due to the resulting exponential increase in the size of the design space. Moreover, optimizations in higher-fidelity design spaces become ill-posed due to high-frequency shape components being insufficiently bounded; this can lead to nonsmooth and oscillatory geometries that are invalid in both physicality (shape) and discretization (mesh). This issue is addressed here by developing a geometrically meaningful constraint to reduce the effective degrees of freedom and improve the design space, thereby improving optimization convergence rate and final result. A new approach to shape control is presented using coordinate control ( $x,z$ ) to recover shape-relevant displacements and surface gradient constraints to ensure smooth and valid iterates. The new formulation transforms constraints directly onto design variables, and these bound the out-of-plane variations to ensure smooth shapes as well as the in-plane variations for mesh validity. Shape gradient constraints approximating a $C^2$ continuity condition are derived and demonstrated on a challenging test case: inviscid transonic drag minimization of a symmetric NACA0012 airfoil. Significantly, the regularized shape problem is shown to have an optimization convergence rate independent of both shape control fidelity and numerical mesh resolution, while still making use of increased control fidelity to achieve improved results. Consequently, a value of 1.6 drag counts is achieved on the test case, the lowest value achieved by any method.