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 methods 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.
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