Numerical issues and turnpike phenomenon in optimal shape design

International audience; This article follows and complements where we have established the turnpike property for some optimal shape design problems. Considering linear parabolic partial differential equations where the shapes to be optimized acts as a source term, we want to minimize a quadratic criterion. Existence of optimal shapes is proved under some appropriate assumptions. We prove and provide numerical evidence of the turnpike phenomenon for those optimal shapes, meaning that the extremal time-varying optimal solution remains essentially stationary; actually, it remains essentially close to the optimal solution of an associated static problem.