Shape gradients of PDE constrained shape functionals can be stated in two equivalent ways. Both rely on the solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used with finite element solutions, the equivalence of the two formulas breaks down. By means of a comprehensive convergence analysis, we establish that the volume based expression for the shape gradient generally offers better accuracy in a finite element setting. The results are confirmed by several numerical experiments.

Microlenses are highly attractive for optical applications such as super resolution and photonic nanojets, but their design is more demanding than the one of larger lenses because resonance effects play an important role, which forces one to perform a full wave analysis. Although mostly spherical microlenses were studied in the past, they may have various shapes and their optimization is highly demanding, especially, when the shape is described with many parameters. We first outline a very powerful mathematical tool: shape optimization based on shape gradient computations. This procedure may be applied with much less numerical cost than traditional optimizers, especially when the number of parameters describing the shape goes to infinity. In order to demonstrate the concept, we optimize microlenses using shape optimization starting from more or less reasonable elliptical and semi-circular shapes. We show that strong increases of the performance of the lenses may be obtained for any reasonable value of the refraction index.

Shape sensitivity measures the impact of small perturbations of geometric features of a problem on certain quantities of interest. The shape sensitivity of PDE (partial differential equation) constrained output functionals is derived with the help of shape gradients. In electromagnetic scattering problems, the standard procedure of the Lagrangian approach cannot be applied because of solution of the state problem is complex valued. We derive a closed‐form formula of the shape gradient of a generic output functional constrained by Maxwell's equations. We employ cubic B‐splines to model local deformations of the geometry and derive sensitivity probings over the surface of the scatterer. We also define a sensitivity representative function over the surface of the scatterer on the basis of local sensitivity measurements. Several numerical experiments are conducted to investigate the shape sensitivity of different output functionals for different geometric settings.

No abstract

No abstract