A new scheme is presented for imposing periodic boundary conditions on unit cells with arbitrary source distributions. We restrict our attention here to the Poisson, modified Helmholtz, Stokes and modified Stokes equations. The approach extends to the oscillatory equations of mathematical physics, including the Helmholtz and Maxwell equations, but we will address these in a companion paper, since the nature of the problem is somewhat different and includes the consideration of quasiperiodic boundary conditions and resonances. Unlike lattice sum-based methods, the scheme is insensitive to the unit cell's aspect ratio and is easily coupled to adaptive fast multipole methods (FMMs). Our analysis relies on classical "plane-wave" representations of the fundamental solution, and yields an explicit low-rank representation of the field due to all image sources beyond the first layer of neighboring unit cells. When the aspect ratio of the unit cell is large, our scheme can be coupled with the nonuniform fast Fourier transform (NUFFT) to accelerate the evaluation of the induced field. Its performance is illustrated with several numerial examples.
翻译:我们在此只关注Poisson、经修改的Helmholtz、Stokes和经修改的Stokes等方程式。我们的分析延伸至数学物理的血管方程式,包括Helmholtz和Maxwell等方程式,但我们将在一份配套文件中讨论这些问题,因为问题的性质有些不同,包括考虑半周期的边界条件和共鸣。与基于 lattice和基于总和的方法不同,这个方法对单元细胞的侧面比率不敏感,容易与适应性快速多极方法(FMMs)相结合。我们的分析依赖于基本解决方案的经典“平流”表达方式,并产生明显低层次的字段代表度,由于离相邻单元细胞第一层以外的所有图像源。当该单元的方位比例较大时,我们的计划可以与非统一快速Fourier变换(UFFT)相结合,以加速对引出场的评价。我们的分析用几个数字示例说明其表现。