Many problems involve multiply-connected moving interfaces, including liquid and solid foams, coarsening in materials, complex mixing in fluids, and evolving cell structures in biology. These problems have multiple domains which share common walls meeting in multiple junctions. Boundaries move under forces which depend on both local and global geometric properties, such as surface tension and volume constraints, as well long-range physical forces, including incompressible flow, membrane permeability, and elastic forces.
This proposal is aimed toward developing, implementing, and applying new numerical methods for propagating multiphase interfaces in a complex physical settings. One of the central computational tools will be the recently developed Voronoi Implicit Interface Methods, which is a mathematical perspective and associated numerical methodology for tracking interfaces in general multiphase problems. These methods offer accurate, consistent, and efficient schemes for multi-dimensional coupled multiphase, which handle complex triple joints/junction, topological change, and naturally couple to complex physics. The investigator and his colleagues will develop new algorithmic tools for multiphase coupled transport problems, and will apply these techniques to computing solid and liquid foams, as well as manufacturing techniques for new materials, providing new methods to compute, refine, and optimize the design and performance cf complex materials.
