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法（FFT-PIM），这是一种灵活且计算高效的方法，用于计算由无限周期阵列中一个晶格内的叠加和给出的标量势。在同一框架下，FFT-PIM 能够计算一维、二维和三维周期性问题（包括动态和静态问题）中的势，这些问题可能包含或不包含周期性相位偏移。对于 N 个空间位置重合的源点和观测点，FFT-PIM 的计算复杂度为 $O(N \log N)$。FFT-PIM 使用作为叠加和核函数的格林函数的快速收敛级数表示。基于这些表示，FFT-PIM 将势分解为近区分量和远区分量：近区分量包含感兴趣晶格周围少量镜像源的影响，远区分量则包含其余无限多个镜像源的影响。远区分量的计算通过将非均匀分布的源投影到稀疏的均匀网格上，在该稀疏网格上执行叠加和，然后将势从均匀网格插值到非均匀观测点来完成。近区分量则采用一种基于 FFT 的方法进行评估，该方法经过调整，能高效处理周期晶格内非均匀的源-观测点分布。FFT-PIM 可广泛应用于各类问题，例如计算电磁学和声学中的积分方程周期问题、微磁求解器以及密度泛函理论求解器等。