Notes on the connections between shape definition and the objective function landscape

The key to effective shape optimization is the selection of the appropriate mathematical formulation for the parametric description of the geometry of the artifact being optimized. It is widely understood that a good parameterization scheme is concise, mathematically well-posed, robust and flexible. What is less clear, however, is the way in which the choice of parameterization approach influences the features of the resulting objective function landscape. In this article we examine the issue through a simple, four-variable design problem. Key words: optimal design, shape optimization, geometry design, brachistochrone, modality