UC Santa Barbara

Continuum formulation of interfacial transport phenomena views interfaces as sharp 2D surfaces separating contiguous 3D bulk phases. In this view, interfacial processes are described by phenomenological models for the underlying microscopic processes or in analogy to conservation/constitutive laws that regulate transport in surrounding bulk phases. Mathematically, these surfaces act as boundary conditions for the governing partial differential transport equations in the bulk and determine discontinuities in solution of the bulk phase as well as the solution gradient across the interfaces. From a computational perspective, accurate numerical solution of these equations poses several algorithmic and computational challenges that require application of advanced numerical algorithms as well as high performance computing techniques. In the first part of my dissertation I will present our design and implementation of two parallel simulation engines based on the level-set method and using finite difference and finite volume discretization techniques on adaptive Voronoi and Cartesian (Quad-/Oc-tree) grids. These simulations are capable to realize large enough computational domains at high resolutions that allow for realization of the mound formation phenomenon in epitaxial growth of materials systems, as well as detailed membrane permeabilization statistics emerging in electroporation of multicellular systems. In the second part of my dissertation, I will derive a general reduced-order state-space mathematical model for dynamics of probability density of interfacial polarizations in heterogeneous systems and compare its predictions with that of my direct numerical simulations.