Blurring the line between fluid and solid simulation methods:  A fully Eulerian-frame approach to fluid-structure interaction

Fluids and solids tend to be addressed using distinct solution approaches.  Solid deformation is most commonly simulated with Lagrangian-frame methods, typically finite elements, whereas fluid flow is amenable to Eulerian-frame approaches such as finite difference and finite volume methods.  This dichotomy makes it difficult to model Fluid-Structure Interaction (FSI) problems, where solids and fluids both coexist and must interact with each other along interfaces.  Here, we discuss the development of a method called the Reference Map Technique, which allows us to simulate arbitrary deformable solids on a fixed Eulerian grid.  This allows us to exploit many of the benefits of fixed-grid techniques but in the context of solid mechanics.  The key is to store and update the reference map field on the grid, which tracks the inverse motion.  Using this technique to represent the solid phase, we can solve entire FSI problems on a single fixed grid using fast update procedures very similar to those used in two-phase fluid simulations.  Various solid constitutive behaviors can be used, including nonlinear elasticity and plasticity.  Systems of many submerged and interacting solids can be simulated, and, by activating the solids internally, we can simulate systems of soft active media.  Incompressibility and/or rigidity constraints can also be applied by adopting Eulerian projection approaches commonly used in CFD.  The addition of the reference map field to the grid also presents certain benefits when computing level-set interface advection, including a procedure to guarantee mass conservation.