The shape Hessian is usually formulated as a boundary operator, which is even symmetric in the Riemannian variant. However, it might be advantageous to use its volume formulation for analytical and also numerical reasons. This has the drawback of a severe lack of definiteness. This talk discusses an approach to deal with the quite large null space of the volumic shape Hessian within an overall shape optimization algorithm utilizing second order information.
In this talk, we consider shape optimization problems constrained by variational inequalities (VI) in shape spaces. These problems are in particular highly challenging because of two main reasons: First, one needs to operate in inherently non-linear, non-convex and infinite-dimensional spaces. Second, one cannot expect the existence of the shape derivative, which imply that the adjoint state cannot be introduced and, thus, the problem cannot be solved directly without any regularization techniques. We investigate computationally a VI constrained shape optimization problem of the first kind.
joint work with Kevin Sturm
We present the application of design optimization algorithms based on the shape and topological derivative for 3D nonlinear magnetostatics to the optimization of an electric motor. The topological derivative for this quasilinear problem involves the solution of two transmission problems on the unbounded domain for each point of evaluation. We present a way to efficiently evaluate this quantity on the whole design domain. Moreover, we present optimization results obtained by a level set algorithm which is based on the topological derivative, as well as shape optimization results.