Although FFT-based methods are renowned for their numerical efficiency and stability, traditional discretizations fail to capture material interfaces that are not aligned with the grid, resulting in suboptimal accuracy. To address this issue, the work at hand introduces a novel FFT-based solver that achieves interface-conforming accuracy for three-dimensional mechanical problems. More precisely, we integrate the extended finite element (X-FEM) discretization into the FFT-based framework, leveraging its ability to resolve discontinuities via additional shape functions. We employ the modified abs(olute) enrichment and develop a preconditioner based on the concept of strongly stable GFEM, which mitigates the conditioning issues observed in traditional X-FEM implementations. Our computational studies demonstrate that the developed X-FFT solver achieves interface-conforming accuracy, numerical efficiency, and stability when solving three-dimensional linear elastic homogenization problems with smooth material interfaces.

翻译：尽管基于FFT的方法以其数值效率和稳定性著称，但传统离散化方法无法捕捉与网格不对齐的材料界面，导致精度欠佳。为解决此问题，本文提出一种新型基于FFT的求解器，可在三维力学问题中实现与界面一致的精度。具体而言，我们将扩展有限元（X-FEM）离散化集成到基于FFT的框架中，利用其通过附加形函数解析间断面的能力。我们采用修正的绝对值富集技术，并基于强稳定GFEM概念开发预处理器，以缓解传统X-FEM实现中存在的条件数问题。计算研究表明，所开发的X-FFT求解器在求解具有光滑材料界面的三维线弹性均匀化问题时，能够实现与界面一致的精度、数值效率与稳定性。