We propose an FDTD scheme based on Generalized Sheet Transition Conditions (GSTCs) for the simulation of polychromatic, nonlinear and space-time varying metasurfaces. This scheme consists in placing the metasurface at virtual nodal plane introduced between regular nodes of the staggered Yee grid and inserting fields determined by GSTCs in this plane in the standard FDTD algorithm. The resulting update equations are an elegant generalization of the standard FDTD equations. Indeed, in the limiting case of a null surface susceptibility ($\chi_\text{surf}=0$), they reduce to the latter, while in the next limiting case of a time-invariant metasurface $[\chi_\text{surf}\neq\chi_\text{surf}(t)]$, they split in two terms, one corresponding to the standard equations for a one-cell ($\Delta x$) thick slab with volume susceptibility ($\chi$), corresponding to a diluted approximation ($\chi=\chi_\text{surf}/(2\Delta x)$) of the zero-thickness target metasurface, and the other transforming this slab in a real (zero-thickness) metasurface. The proposed scheme is fully numerical and very easy to implement. Although it is explicitly derived for a monoisotropic metasurface, it may be straightforwardly extended to the bianisotropic case. Except for some particular case, it is not applicable to dispersive metasurfaces, for which an efficient Auxiliary Different Equation (ADE) extension of the scheme is currently being developed by the authors. The scheme is validated and illustrated by five representative examples.

The recently defined class of integer programming games (IPG) models situations where multiple self-interested decision makers interact, with their strategy sets represented by a finite set of linear constraints together with integer requirements. Many real-world problems can suitably be fit in this class, and hence anticipating IPG outcomes is of crucial value for policy makers and regulators. Nash equilibria have been widely accepted as the solution concept of a game. Consequently, their computation provides a reasonable prediction of the games outcome. In this paper, we start by showing the computational complexity of deciding the existence of a Nash equilibrium for an IPG. Then, using sufficient conditions for their existence, we develop two general algorithmic approaches that are guaranteed to approximate an equilibrium under mild conditions. We also showcase how our methodology can be changed to determine other equilibria definitions. The performance of our methods is analyzed through computational experiments in a knapsack game, a competitive lot-sizing game, and a kidney exchange game. To the best of our knowledge, this is the first time that equilibria computation methods for general integer programming games have been designed and computationally tested.