At micrometer and sub-micrometer length scales, Stefan-Boltzmann law fails in describing radiative heat transfer adequately and measured heat fluxes exceed the black body “limit” by orders of magnitude. This phenomenon is caused by a complex combination of coupled surface waves, interference and tunneling effects and can be adequately modelled using a fluctuating-surface-current formulation of radiative heat transfer (Phys. Rev. B., 88(5):054305, 2013). The resulting equations are very similar to those used in the boundary element method/method of moments and can be assembled using the same tools. The solution method unfortunately involves inverting a fully dense matrix and this limits the tractability of high resolution simulations. The ability to model the heat flow and the flux patterns accurately enables the development of contactless scanning probe thermal microscopy and thermal lithography. Every application requires a specific probe shape that can be found through shape optimization. The results of a crude first attempt at shape optimization can be seen in the video below (Figure 2).
The current method is too slow for high resolution shape optimization. You will be investigating ways of obtaining sensitivity information to minimize the number of iterations, and new solution methods that minimize the amount of degrees-of-freedom. You will demonstrate these methods on several practical cases related and help us improve the lateral resolution of scanning probe thermal microscopes.