Weighted Sobolev spaces for the advection operator: a variational method for computing shape derivatives of geometric constraints along rays. 

 

In the formulation of shape optimization problems, multiple geometric constraint functionals involve the signed distance function to the optimized shape Ω. The numerical evaluation of their shape derivatives requires to integrate some quantities along the normal rays to Ω, a task that is usually achieved thanks to the method of characteristics. We present a variational problem for this purpose, which can be solved conveniently by the finite element method. The well-posedness of this problem is established thanks to the analysis of weighted graph spaces of the advection operator β · ∇ associated to a C 1 velocity fields β. The novelty of the approach is the ability to handle velocity fields with possibly unbounded divergence: we do not assume div(β) ∈ L ∞. We provide numerical comparisons of our variational method with respect to the direct numerical integration along the normal rays, and we demonstrate the potential of our method when it comes to enforcing maximum and minimum thickness constraints in structural shape optimization.