Air-Gap Shape Optimization for Minimizing Proximity Losses in an Inductor

In air-gapped inductors, the magnetic flux bulges outward from the gap - a phenomenon called the fringing effect. When these flux lines cut across the conductor windings, they generate undesired eddy currents that lead to substantial AC losses and reduced efficiency. To mitigate these losses, we develop a constrained shape optimization, aiming at finding an optimal air gap profile that minimizes conductor losses and keeps the inductance at a desired value. Our framework leverages established techniques such as complex permeability homogenization for loss modeling, shape sensitivity analysis coupling the adjoint method with the so-called Hilbertian regularization-extension method. The optimized air gap profile is curved to concentrate the flux lines away from the (fixed) conductors. Our implementation uses the open-source finite-element toolbox NGSolve and the Mmg remeshing library; all the codes are freely available.