Shape optimization of planar antennas using Level Set Method

The inverse scattering problem of finding the optimal shape of 2D or 3D metallic or dielectric objects has received a growing attention in the last decade using different strategies either deterministic or stochastic approaches. Here we describe the development of an inverse scattering method based on a surface integral formulation of the EM problem for finding the solution of an inverse scattering problem, which is the optimal shape of planar antennas illuminated by plane waves or dipoles from the minimization of a functional involving the desired imposed constraints (e.g. an ideal radiation pattern known at a single frequency or for a wide frequency band). The state problem is solved numerically using a moment method (or variational principle) with triangular meshes. Many inverse scattering algorithms based on gradient methods are using forward and adjoint problems for calculating the cost function derivative. Here, in order to avoid the use of an adjoint problem and to deal with a more general method, we are calculating directly the derivative of the cost functional. In this way, we have directly access to the sensitivity of the cost functional versus parameters we are interested in. After calculating the shape derivative of the cost functional, we develop an optimization procedure according to the Level Set method. This method based on an implicit representation of the antenna boundary and the speed allows reconstructing a new planar triangular mesh structure at each iteration to decrease the value of the cost functional and performing possible several topological changes. A quasi-Newton method for optimizing the parameters of specific geometries has been also developed and applied to the design of a 2D period structure such as a Frequency Selective Surface (FSS).