  Our paper proposes an innovative approach for modeling Fluid-Structure
Interaction (FSI). Our method combines both traditional monolithic and
partitioned approaches, creating a hybrid solution that facilitates FSI. At
each time iteration, the solid mesh is immersed within a fluid-solid mesh, all
while maintaining its independent Lagrangian hyperelastic solver. The Eulerian
mesh encompasses both the fluid and solid components and accommodates various
physical phenomena. We enhance the interaction between solid and fluid through
anisotropic mesh adaptation and the Level-Set methods. This enables a more
accurate representation of their interaction. Together, these components
constitute the Adaptive Immersed Mesh Method (AIMM). For both solvers, we
utilize the Variational Multi-Scale (VMS) method, mitigating potential spurious
oscillations common with piecewise linear tetrahedral elements. The framework
operates in 3D with parallel computing capabilities. Our methods accuracy,
robustness, and capabilities are assessed through a series of 2D numerical
problems. Furthermore, we present various three-dimensional test cases and
compare their results to experimental data.
