The fast Fourier transform (FFT) based homogenization method of Moulinec and Suquet has been established as a fast, accurate, and robust tool for periodic homogenization in solid mechanics. In a finite element context, Pahr and Zysset have introduced nonperiodic boundary conditions (PMUBC) for homogenization problems. We show how to implement PMUBC efficiently in an FFT-based code using discrete sine and cosine transforms. Compared with the domain mirroring approach, we reduce the runtime by a factor of 2 to 3, and the memory requirements by a factor of 8. We show that the use of periodic boundary conditions for nonperiodic geometries yields vastly different results than with PMUBC. Furthermore, we examine the influence of the discretization method by comparing the staggered grid discretization with a finite element discretization.

中文翻译：

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具有混合均匀边界条件的基于 FFT 的均匀化

Moulinec 和 Suquet 基于快速傅里叶变换 (FFT) 的均质化方法已被确立为固体力学中周期性均质化的快速、准确和稳健的工具。在有限元环境中，Pahr 和 Zysset 为均匀化问题引入了非周期边界条件 (PMUBC)。我们展示了如何使用离散正弦和余弦变换在基于 FFT 的代码中有效地实现 PMUBC。与域镜像方法相比，我们将运行时间减少了 2 到 3 倍，内存需求减少了 8 倍。我们表明，对非周期性几何使用周期性边界条件会产生与 PMUBC 截然不同的结果。此外，我们通过比较交错网格离散化和有限元离散化来检验离散化方法的影响。