Fluid-structure interactions (FSI) are abundantly observed in contexts ranging from swimming in the pool, to industrial-level manufacturing, to bacteria collective motion on a petri dish. However, the governing equations are only analytically trackable in the simple cases, making simulations key to understanding this fantastic class of problems. Conventional computational methods often create a dilemma for FSI problems. Typically, solids are simulated using a Lagrangian approach with a grid that moves with the material, whereas fluids are simulated using an Eulerian approach with a fixed spatial grid. FSI methods often require some type of interfacial coupling between the two different perspectives. We present a fully Eulerian FSI method that addresses these challenges. The method makes use of a reference map, which maps the solid in the current space to the reference space. A reference map is a common concept in finite strain theory, but it has been under-utilized as a primary variable for solid and FSI simulations. There are computational challenges that come with simulating complex physical systems in three dimensions that can only be adequately addressed with high-performance computing. In particular, a key challenge in applying the reference map technique (RMT) in FSI is to extrapolate reference map values from grid cells occupied by the solids to unoccupied grid cells in order to calculate derivatives using finite difference schemes. We develop an extrapolation algorithm based on least-square linear regression that is suitable for parallelization. We show 3-D examples to demonstrate RMT in simulating soft, highly deformable materials and many-body contact problems.
