In this thesis, the Voronoi Implicit Interface Method (VIIM) is presented together with several applications in multiphase curvature flow, multiphase incompressible fluid flow, mesh generation for interconnected surfaces, and multiscale modelling of foam dynamics. The VIIM tracks the evolution of multiple interacting regions ("phases") whose motion may be determined by geometry, complex physics, intricate jump conditions, internal constraints, and boundary conditions. From a mathematical point of view, the method provides a theoretical framework to evolve interconnected interfaces with junctions. Discretising this theoretical framework leads to an efficient Eulerian-based numerical method that uses a single unsigned distance function, together with a region indicator function, to represent a multiphase system. The VIIM works in any number of spatial dimensions, accurately represents complex geometries involving triple and higher-order junctions, and automatically handles topological changes in the evolving interface, including creation and destruction of phases. Here, the central ideas behind the method are presented, implementation is discussed, and convergence tests are performed to illustrate the accuracy of the method. Several applications of the VIIM are shown, including in constant speed normal driven flow; multiphase curvature flow with constraints; and multiphase incompressible fluid flow in which density, viscosity, and surface tension can be defined on a per-phase basis and membranes can be permeable.
An efficient and robust mesh generation algorithm for interconnected surfaces is also presented. The algorithm capitalises on a geometric construction used in the VIIM, known as the "Voronoi interface", to generate high-quality triangulated meshes that are topologically consistent, such that mesh elements meet precisely at junctions without gaps, overlaps, or hanging nodes. The generated meshes can be used in finite element methods for solving partial differential equations on a network of evolving interconnected curved surfaces.
Finally, a scale-separated, multiscale model for the dynamics of a soap bubble foam is presented. The model leads to a computational framework for studying the interlinked effects of drainage, rupture, and rearrangement in a foam of bubbles, coupling microscale fluid flow in a network of thin-film membranes ("lamellae") and junctions ("Plateau borders") to macroscale gas dynamics driven by surface tension. Here, thin-film equations for fluid flow inside curved lamellae and Plateau borders are derived, flux boundary conditions which conserve liquid mass are developed, and local conservation laws for transport of film thickness during rearrangement are designed. From a numerical perspective, several new numerical methods are developed, including Lagrangian-based schemes for conserving liquid in the membranes during rearrangement, finite element methods to solve fourth-order nonlinear partial different equations on curved surfaces, methods to accurately solve coupled flux boundary conditions at Plateau borders and quadruple points, and projection methods to couple gas dynamics to the VIIM. Convergence tests are performed to demonstrate the accuracy of the numerical methods, and results of the multiscale model are shown for a variety of problems, including collapsing foam clusters displaying thin-film interference effects.