Comparison between topological and surface sensitivities for shape optimization

The two most common shape-optimization methods for fluid mechanics problems are based on topology modifications and surface modifications. When the number of design parameters is large compared to the number of objective functions the most efficient way to evaluate the sensitivity derivatives is using the solution from adjoint equations. In external aerodynamics such as aircraft wings, cars, and trains, surface sensitivities are commonly applied since the topology remains the same and the surface quality and precision are important factors. In internal flows, such as ducts and tubes, the choice between topology and surface modifications is not trivial. Both methods can lead to useful optimal solutions, but either possesses its own pros and cons. Changing the topology might be admissible, and even adding material (duct thickness) can lead to unexpected topologically different solutions. This is also true in many bio-mechanical applications such as surgery of the upper airways (UA). In this paper, topological and surface sensitivities are evaluated and compared in OpenFOAM by solving the adjoint equations for a simple geometry first, and and then for the upper airways. Two different geometries of the UA are investigated: the first consists of only the nasal cavity and the sensitivity analysis is applied to the inner geometry and the surrounding walls. In the second case, a tissue of a certain thickness is added to the first to simulate a tissue removal around the existing airways. The different geometries are analyzed and discussed, evidencing also pros and cons of the different processes.