Shape derivative for obstacles in crowd motion

We consider different PDE modeling for crowd motion scenarios, or other sort of fluid flows, and we insert in the given domain $ R $ an obstacle $ O $. We then compute the shape derivatives of a cost functional, the average exit time, in order to be able to optimize the geometry of the obstacle $ O $ and so to minimize the average exit time of particles in the domain $ R $. This computation can be used to derive numerical simulations and understand whether the presence of an obstacle is or not profitable for the evacuation, or to optimize its shape and position, for instance when the presence of a structure (column, …) is already necessary in the building plan of a public space.