# Mathematical and Algorithmic Methodologies for Computing Multi-phase Multiphysics: Industrial Foams, Materials, Manufacturing, and Cell Mechanics

Many problems involve the physics of multiply-connected moving interfaces. Examples include liquid foams (e.g. soap bubbles, polyurethane and colloidal mixtures), and solid foams, such as wood and bone. Manufactured solid foams lead to lightweight cellular engineering materials, including crash absorbent aluminum foams, and controlling foams is critical in chemical processing.

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. As an example, the interaction of multiple phases in fluid simulations requires a robust treatment of the interplay between membrane dynamics, fluid mechanics, surface tension, and intricate jump conditions.

Producing good mathematical models and numerical algorithms that capture the motion of these interfaces is challenging, especially at junctions where multiple interfaces meet, and when topological connections change. Methods have been proposed, including front tracking, volume of fluid, variational, and level set methods. It has remained a challenge to robustly and accurately handle the wide range of possible motions of an evolving, highly complex, multiply-connected interface separating a large number of phases under time-resolved physics.

CRD Math researchers have recently built a computational methodology, known as the *Voronoi Implicit Interface Method*, to track multiple coupled interfaces moving under complex physics constraints. Mathematically, they provide a consistent formulation of 3D multiphase physics as a single time-dependent PDE, which is solved using intertwined steps of initial value upwind solvers and fast Eikonal solvers for Voronoi reconstructions. The results are algorithms which efficiently and accurately compute topological change while coupling to fluid and solid mechanics, surface tension, and geometrical constraints. The methods are highly efficient and parallelizable. The methods are now being used to compute industrial mixing and materials manufacturing. The original paper was awarded the 2011 Proceedings of the National Academy Sciences Cozzarelli Prize for Best Paper in Applied Sciences and Engineering.

Such algorithms allow for a more accurate understanding of a vast array of science problems and applications. The algorithms are applicable to many computing problems in both industry and research, including fluid mixing of multiple species in combustion and reactor designs, grain metal and materials growth in industrial manufacturing, semiconductor failure analysis of metal interconnect lines, liquid foam drainage in chemical and biological devices, and cell mechanics and structural stability analyses in biofuels.