Shape optimization for elliptic eigenvalue problems in an imhomogeneous media
Department of Mathematics, The Ohio State University
(November 14, 2006 3:30 PM - 4:30 PM)
In this talk, we will introduce numerical approaches to find the optimal shape and topology for elliptic eigenvalue problems in an inhomogeneous medium by using shape derivatives and topological derivatives in combination with level set methods. The common numerical approach for these problems is to start with an initial guess for the shape and then gradually evolve it, until it morphs into the optimal shape. One of the difficulties is that the topology of the optimal shape is unknown. The level set approach based on shape derivatives has been well known for its ability to handle topology changes but it may get stuck at shapes with fewer holes than the optimal geometry. By incorporating topological derivatives into the level set method one can provide an alternative way to create holes efficiently and escape from the shapes with fewer holes. We will show results in applications including resonant frequency control and photonic devices design.