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未与网格对齐的材料界面，导致精度欠佳。为解决此问题，本研究引入了一种新颖的基于FFT的求解器，该求解器在三维力学问题中实现了界面保形精度。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不连续性的能力。我们采用改进的绝对值（abs）富集方法，并基于强稳定广义有限元（GFEM）的概念开发了预处理器，从而缓解了传统X-FEM实现中观察到的条件数问题。我们的计算研究表明，所开发的X-FFT求解器在求解具有光滑材料界面的三维线性弹性均匀化问题时，能够实现界面保形精度、数值效率及稳定性。