Projects

The broad goal is to provide a consistent method for the solution of dynamic boundary value problems (BVP) on finite domains of acoustic and elastic metamaterials. Current approaches either involve micromorphic theories (with a very high number of material parameters) or exact but nonlocal boundary conditions (formulated with Fredholm integral equations). The former is incompatible with much of the machinery of metamaterial device design (such as transformation methods) and the latter is complex enough to make its use with current metamaterial research nearly untenable. As a consequence, ideal metamaterial designs, when realized in practice, demonstrate significant and poorly understood performance degradations. We are interested in two ideas - utilizing and focusing on local metamaterials, and coupling them with Drude transition layers to come up with a consistent technique of dealing with boundaries.

The first notions of causality can be tracked back in to the work of Sokhotski (1873), Plemelj (1908), and Sommerfeld (1914). These ideas applied to linear systems and naturally connected to the Hilbert transform. In this form, causality has form novel applications in the scattering theory of phoXonic crystals as well as metamaterials. We are interested in what limits causality places on the scattering performance of such systems. For time-series analysis, causality was then quantified by Wiener (1956) and Granger (1969) as a statistical test for evaluating the ability of one time series to predict another. This was extended recently to deal with nonlinear time series (Schreiber 2000, Barnett 2009). In this form, causality analysis can be conducted to determine cause and effect relationships in highly nonlinear systems such as turbulence. We are interested in various notions of causality, perhaps bringing them together, and their applications to various areas of physics and engineering.

We are interested in the problem of analyzing the dynamics of a finite system in connection with an infinite system (sink). The finite system may be of a usual nature, such as a beam capable of vibration, or of a more exotic nature, such as a finite sample of a metamaterial (conservative or non-conservative). The infinite system (bulk) is the environment in which the finite system is embedded.

When the finite system is coupled with the infinite system, it sets up some additional problems of dynamics. First, it becomes possible for the energy in the finite system to leak out into the environment. This leakage of energy from the finite system may be seen as the emergence of a non-conservative effect, even though there may not be any non-conservative sources in the finite system to begin with. This non-conservative effect is also captured in the dynamical eigenvalues of the finite system, which necessarily become complex (with the imaginary part connected to this fictitious damping effect) upon interaction with the environment. We are interested in answering these questions through the formal machinery of open systems.