Design sensitivities for scattering-matrix calculation with tetrahedral edge elements

The rates of change of the scattering matrix of a microwave device with respect to geometric or material parameters are of interest to designers, and useful in automatic optimization. They are most efficiently calculated by the adjoint variable method, but this usually entails the additional cost of funding the adjoint solutions. In the finite-element formulation presented here, no new solutions are needed beyond those required to calculate the complete scattering matrix. Explicit expressions are given for the calculation of the required matrix for a tetrahedral edge element. Results are presented for three rectangular waveguide problems: a uniform length terminated in a short-circuit; an H-plane miter bend; and a two-step waveguide impedance transformer.