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We present a novel framework inspired by the Immersed Boundary Method for predicting the fluid-structure interaction of complex structures immersed in flows with moderate to high Reynolds numbers. The main novelties of the proposed fluid-structure interaction framework are 1) the use of elastodynamics equations for the structure, 2) the use of a high-order Navier-Stokes solver for the flow, and 3) the variational transfer (L2-projection) for coupling the solid and fluid subproblems. The dynamic behavior of a deformable structure is simulated in a finite element framework by adopting a fully implicit scheme for its temporal integration. It allows for mechanical constitutive laws including nonhomogeneous and fiber-reinforced materials. The Navier-Stokes equations for the incompressible flow are discretized with high-order finite differences which allow for the direct numerical simulation of laminar, transitional and turbulent flows. The structure and the flow solvers are coupled by using an L2-projection method for the transfer of velocities and forces between the fluid grid and the solid mesh. This strategy allows for the numerical solution of coupled large scale problems based on nonconforming structured and unstructured grids. The framework is validated with the Turek-Hron benchmark and a newly proposed benchmark modelling the flow-induced oscillation of an inert plate. A three-dimensional simulation of an elastic beam in transitional flow is provided to show the solver's capability of coping with anisotropic elastic structures immersed in complex fluid flow.