To study the mechanical behavior of a periodic microstructure, the fast Fourier transform (FFT)-based simulation [1] can be used alternatively to the classical finite element (FE)-based simulation. Besides several advantages of this FFT-based method over the FE method, one of its drawbacks is related to the appearance of artificial oscillations occurring within the microstructural fields around sharp contrasts in the local properties. These oscillations, known as Gibbs oscillations, result from considering a truncated set of Fourier modes to approximate the microstructural fields. To smooth the artificial oscillations in the context of an FFT-based simulation, different methods can be used, such as methods based on a finite difference approximation of all occurring spatial derivatives, which is related to the Lanczos σ-approximation. In this work, we use a new approach based on local subvoxel-shifts [2] to reduce the Gibbs ringing within the computed microstructural fields in a post-processing step. This method is based on the idea that the local fields are computed on a discretized grid, while the magnitude of the ringing artifacts depends on how this grid is located with respect to the sharp contrast in the local properties and the oscillation behavior of the microstructural fields. To demonstrate the applicability of the proposed method in the context of an FFT-based microstructure simulation, we will investigate several numerical examples. [1] Moulinec, H., Suquet, P. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Computer methods in applied mechanics and engineering, Vol. 157 (1-2), pp. 69-94, 1998. [2] Kellner, E., Dhital, B., Kiselev, V. G., Reisert, M. Gibbs‐ringing artifact removal based on local subvoxel‐shifts. Magnetic resonance in medicine, Vol. 76 (5), pp. 1574-1581, 2016.