Multilevel Subdivision Parameterization Scheme for Aerodynamic Shape Optimization

Subdivision curves are defined as the limit of a recursive application of a subdivision rule to an initial set of control points. This intrinsically provides a hierarchical set of control polygons that can be used to provide surface control at varying levels of fidelity. This work presents a shape parameterization method based on this principle and investigates its application to aerodynamic optimization. The subdivision curves are used to construct a multilevel aerofoil parameterization that allows an optimization to be initialized with a small number of design variables, and then be periodically increased in resolution throughout. This brings the benefits of a low-fidelity optimization (high convergence rate, increased robustness, low-cost finite difference gradients) while still allowing the final results to be from a high-dimensional design space. In this work, the multilevel subdivision parameterization is tested on a variety of optimization problems and compared with a control group of single-level subdivision schemes. For all the optimization cases, the multilevel schemes provided robust and reliable results in contrast to the single-level methods that often experienced difficulties with large numbers of design variables. As a result of this, the multilevel methods exploited the high-dimensional design spaces better and consequently produced better overall results.