Gradient-Limiting Shape Control for Efficient Aerodynamic Optimisation

Without addressing shape smoothness, gradient-based optimisation methods naturally amplify high-frequency shape components which can lead to poor convergence of the optimisation problem and a convergence rate exhibiting dependency on the fidelity of shape-control. In this paper it is demonstrated that this problem arises due to the discrete shape problem being ill-posed since it naturally includes geometries that are invalid both in physicality (shape) and discretisation (mesh). A new approach to shape control for optimisation is presented here using coordinate control to recover shape-relevant displacements and surface gradient constraints to ensure smooth and valid iterates. Shape gradient constraints approximating a C 2 continuity condition for discrete mesh points and cubic B-Spline curves are derived, and demonstrated on a challenging ADODG test case: inviscid drag minimisation of a non-lifting NACA 0012 aerofoil at Mach 0.85. The local control methods provide high-fidelity control whereas the surface constraints exclude non-physical shapes from the design space making the shape problem well-posed. Consequently, a value of 7 drag counts is achieved on the test case, the lowest value achieved for comparable mesh resolution. Significantly, the regularised shape problem is shown to have an optimisation convergence rate independent of shape control fidelity when using B-Splines, while still making use of increased control fidelity to achieve improved results. As a result, high-fidelity shape optimisation is possible at a reasonable computational cost.