An FFT-based method is proposed to solve dynamic problems in heterogeneous d-dimensional rectangular domains [1]. Time discretization is performed using an unconditionally stable beta-Newmark approach. The method allows one to prescribe the displacement as a function of time in a subregion of the domain, emulating the application of Dirichlet boundary conditions on an outer face. To this aim, a body force field is imposed on the boundary subregion, and its value is obtained at each time step to fulfill the prescribed conditions. The displacement is solved transforming the equilibrium equations into Fourier space and using Krylov solvers. The elasto-dynamic algorithm is first used to simulate the propagation of an elastic wave through the RVE in the time domain, and study wave attenuation. Then, the framework proposed is used to simulate a Resonant Ultrasound Spectroscopy (RUS) experiment. Finally, the method is extended for non-linear materials solving for the body source together with the unknown displacement field using a minimization procedure. [1] R. Sancho, V. Rey-De-Pedraza, P. Lafourcade, R.A. Lebensohn, J. Segurado, An implicit FFT-based method for wave propagation in elastic heterogeneous media, Computer Methods Applied Mechanics and Engineering 404 115772, 2023