In the first part of the talk, we consider minor simplifications to the simulation of a wave scattering from a number of dielectrics positioned above a horn. The scattering problem can be modeled using Helmholtz equation in the half space above the horn. Traditionally, finite differences are used to solve the scattering problem, this requires meshing the entire domain. Additionally, to ensure physical properties of the wave, Perfectly Matched Layers (PMLs) are often used to artificially bound the computational domain. Both use of finite differences and PMLs limit the accuracy of the simulations and increase the computational time. Instead, we transform the equation into an integral equation that we solve using the boundary integral method. This approach allows us to avoid needing to mesh the entire computational domain. We also impose the Sommerfeld radiation condition at infinity which allows us to avoid the use of an artificial boundary.
In the second part of the talk, we present an optimization problem to find the configuration of the dielectrics that would yield a desired response over a target region. We use the boundary integrals as a constraint to the optimization problem and compute the gradients by forming the continuous adjoint for the steady state boundary integral equations.