Although materials exhibiting negative permeability and permittivity, referred to as metamaterials or left-handed materials (LHMs), have been postulated for many years, it is only recently that such materials have been physically realized. Previous attempts to model these structures embedded a material with a negative index of refraction as a black box and utilized the effective index approximation. While such attempts are sufficient for demonstrating the concepts of negative refraction, they do not utilize physically realizable elements to build LHMs and, hence, cannot account for various spectral and temporal material properties. Thus, in order to determine the frequency range over which such structures exhibit a negative refraction, a rigorous numerical modeling of the LHM is necessary. However, this is a non-trivial task, given the level of detail that is required in the model, which translates into massive computational size and time. Thus, there is a clear need for a numerical platform capable of handling such computationally intense problems. To this end, we have developed a novel, hardware-based platform to analyze LHM structures. This platform has demonstrated performance comparable to a 100-node PC cluster at a fraction of the power, area, and cost. In this paper, we describe this platform and its application to LHM structures. Specifically, the hardware accelerator is used to calculate the transmission spectra of an LHM structure, too large to be modeled using standard software simulation tools, in order to identify the frequency regions where the permittivity and permeability are negative. The hardware platform is then used to demonstrate negative refraction.