Fast Fourier transform (FFT)-based computational homogenization methods [1, 2] offer a robust and efficient framework for solving homogenization problems on regular grids. Despite their favorable numerical properties, conventional FFT solvers often suffer from reduced accuracy when modeling materials with complex microstructures, particularly when material interfaces do not align with the underlying grid. To address this limitation, we introduce an X-FFT solver for three-dimensional mechanical homogenization problems based on previous work that was limited to two-dimensional thermal homogenization [3]. Our X-FFT solver achieves the accuracy of interface-conforming finite elements while maintaining the efficiency of FFT-based solvers. By incorporating the extended finite element method (X-FEM) with a modified absolute enrichment strategy [4], the X-FFT solver accurately captures weak discontinuities across material boundaries through enriched shape functions. To overcome the conditioning problems of standard X-FEM formulations, the X-FFT solver employs a preconditioning technique derived from the strongly stable generalized finite element method (GFEM) [5]. In this talk, we present the underpinnings of the X-FFT solver for three-dimensional small-strain mechanics. We discuss its theoretical foundations, highlight its current range of applications, and present results from benchmark computational experiments demonstrating its accuracy, efficiency, and robustness. REFERENCES [1] H. Moulinec and P. Suquet, A fast numerical method for computing the linear and nonlinear mechanical properties of composites, C. R. Acad. Sci. II, Vol. 318 (11), pp. 1417–1423, 1994. [2] H. Moulinec and P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, CMAME, Vol. 157(1-2), pp. 69–94, 1998. [3] F. Gehrig and M. Schneider An X-FFT Solver for Two-Dimensional Thermal Homogenization Problems, IJNME, Vol. 126 (7), pp. e70022, 2025. [4] N. Moës, M. Cloirec, P. Cartraud, and J. F. Remacle, A computational approach to handle complex microstructure geometries, CMAME, Vol. 192 (28-30), pp. 3163–3177, 2003. [5] I. Babuška, U. Banerjee, and K. Kergrene, Strongly stable generalized finite element method: Application to interface problems, CMAME, Vol. 327, pp. 58–92, 2017.