  We present a numerical method specifically designed for simulating
three-dimensional fluid--structure interaction (FSI) problems based on the
reference map technique (RMT). The RMT is a fully Eulerian FSI numerical method
that allows fluids and large-deformation elastic solids to be represented on a
single fixed computational grid. This eliminates the need for meshing complex
geometries typical in other FSI approaches, and greatly simplifies the coupling
between fluid and solids. We develop the first three-dimensional implementation
of the RMT, parallelized using the distributed memory paradigm, to simulate
incompressible FSI with neo-Hookean solids. As part of our new method, we
develop a new field extrapolation scheme that works efficiently in parallel.
Through representative examples, we demonstrate the method's accuracy and
convergence, as well as its suitability in investigating many-body and active
systems. The examples include settling of a mixture of heavy and buoyant soft
ellipsoids, lid-driven cavity flow containing a soft sphere, and swimmers
actuated via active stress.
