Incorporating topological derivatives into shape derivatives based level set methods

Shape derivatives and topological derivatives have been incorporated into level set methods to investigate shape optimization problems. The shape derivative measures the sensitivity of boundary perturbations while the topological derivative measures the sensitivity of creating a small hole in the interior domain. The combination of these two derivatives yields an efficient algorithm which has more flexibility in shape changing and may escape from a local optimal. Examples on finding the optimal shapes for maximal band gaps in photonic crystal and acoustic drum problems are demonstrated.