We propose a Fast Fourier Transform based Periodic Interpolation Method (FFT-PIM), a flexible and computationally efficient approach for computing the scalar potential given by a superposition sum in a unit cell of an infinitely periodic array. Under the same umbrella, FFT-PIM allows computing the potential for 1D, 2D, and 3D periodicities for dynamic and static problems, including problems with and without a periodic phase shift. The computational complexity of the FFT-PIM is of $O(N \log N)$ for $N$ spatially coinciding sources and observer points. The FFT-PIM uses rapidly converging series representations of the Green's function serving as a kernel in the superposition sum. Based on these representations, the FFT-PIM splits the potential into its near-zone component, which includes a small number of images surrounding the unit cell of interest, and far-zone component, which includes the rest of an infinite number of images. The far-zone component is evaluated by projecting the non-uniform sources onto a sparse uniform grid, performing superposition sums on this sparse grid, and interpolating the potential from the uniform grid to the non-uniform observation points. The near-zone component is evaluated using an FFT-based method, which is adapted to efficiently handle non-uniform source-observer distributions within the periodic unit cell. The FFT-PIM can be used for a broad range of applications, such as periodic problems involving integral equations in computational electromagnetic and acoustic, micromagnetic solvers, and density functional theory solvers.

翻译：我们提出了一种基于快速傅里叶变换的周期插值方法（FFT-PIM），这是一种灵活且高效的计算方法，用于求解无限周期阵列中单个晶胞内叠加求和所给出的标量势。FFT-PIM在统一框架下，能够处理一维、二维和三维周期性下的动态与静态问题，包括有无周期相位偏移的情形。该方法的计算复杂度为 O(N log N)，其中 N 为空间重合的源点与观测点数量。FFT-PIM采用格林函数（作为叠加求和中的核函数）的快速收敛级数表示。基于这些表示，FFT-PIM将势能分解为近区分量（包含目标晶胞周围少量镜像）和远区分量（包含其余无限数量镜像）。远区分量通过将非均匀源投影到稀疏均匀网格上，在稀疏网格上执行叠加求和，再将势能从均匀网格插值到非均匀观测点来计算。近区分量则采用基于FFT的方法进行求解，该方法经适应性改进以高效处理晶胞内非均匀源-观测分布。FFT-PIM可广泛应用于涉及积分方程的周期性问题（如计算电磁学与声学）、微磁求解器以及密度泛函理论求解器等场景。