Shape Optimization for Aerodynamic Problems

Abstract

Shape optimization problems arise in many engineering applications of fluid dynamics, acoustics, and other fields. In such problems, one aims to find a shape that optimizes a certain objective while satisfying a given set of constraints. A persistent challenge is maintaining mesh quality during the optimization process. Poor mesh quality, such as distorted or overlapping elements, can lead to numerical instability and ultimately failure of the algorithm. Hence, it is of great importance to develop a robust method that produces meshes of high quality. Most importantly, one wants to ensure that the solvability of the problem itself is not lost in the process.

I consider the problem of minimizing drag in a flow tunnel, which is a typical shape optimization problem with partial differential equation constraints. In order to increase the set of reachable shapes, an extension operator is used -- a mapping from the control variable on the boundary to the deformation field defined over the entire domain. Both linear and nonlinear formulations of the extension equation are investigated and their influence on the solution quality and robustness is assessed. The results demonstrate that appropriate extension strategies can significantly improve mesh quality without compromising the fidelity of the optimization process. It is shown that nonlinear extensions outperform linear ones in preserving element quality under large deformations.