Jonas Kusch, Karlsruhe Institute of Technology
An approximate Newton Smoothing Method for Shape Optimization
In this talk, we derive a smoothing method for shape optimization in Stokes and Navier-Stokes flows. The smoothing routine automatically picks a spatially dependent smoothing parameter in such a way that the optimization process is accelerated, turning the smoothing routine into an approximate Newton method.
This task is achieved by analytically deriving the symbol of the Hessian for the Stokes equations. We numerically investigate the Hessian symbol for convective flows and demonstrate the applicability of the symbol for the Navier-Stokes equations.
The constructed preconditioner approximates the derived symbol using windowed Fourier transform and thereby accelerates the optimization process while yielding a smooth search direction. Due to the fact that the smoothing is performed locally, the method will identify areas in which a non-smooth design is physically meaningful and will automatically turn off smoothing in these regions.