Optimization for scattering and radiation problems based on a stable FEM-FDTD hybrid method

Doctoral thesis, 2007

In this thesis, a stable hybrid method combining the finite-element method (FEM) and the finite-difference time-domain (FDTD) scheme for Maxwell's equations in two dimensions with both electric and magnetic losses is presented. It combines the flexibility of the FEM with the efficiency of the FDTD scheme. The hybrid method time-step the Ampère's and Faraday's law and it is stable up to the Courant stability limit of the FDTD scheme. Numerical computations for smooth scatterers demonstrate second order convergence for the hybrid scheme. We apply the FEM-FDTD hybrid to shape and material optimization problems with the aim of reducing the radar cross section of an airfoil over a range of frequencies and incident angles. Efficient gradient optimization based on the adjoint formulation is employed. It is shown that optimization over a wide frequency band effectively suppresses corrugations and allows for a reduction in angular resolution and hence a reduced overall computational cost. Finally, the hybrid method is exploited in the pattern synthesis for conformal array antennas whereby effects such as mutual coupling are included. The objective function is a linear combination of two terms, expressing the deviation from a desired radiation pattern and the reflected power, respectively. The array excitation is expressed in terms of phase modes, which allow for convenient parameterization and control of the radiating modes. A gradient based optimization method is employed, which yields a radiation pattern similar to that obtained by Dolph-Chebyshev synthesis, but with a significantly reduced reflected power.