Evaluation of the influence tensor of two disks in a periodic setting through the Poisson summation formula

The goal of the calculations presented here is to show numerically that in a periodic setting, the influence tensor of two disks can be expressed as a sum in the real space of the influence tensors of the first disk with all periodic images of the second disk (these interactions being evaluated with boundary conditions at infinity). This result was first published heuristically by Zecevic et Lebenson (2021, DOI:10.1002/nme.6841). However, we show that the published formula is not correct and propose a (heuristic) correction that was introduced in (Brisard, Bertin and Legoll, 2023). The corrected formula is then asserted numerically. See 00-README.html for more details.

本文所呈现的计算工作，旨在通过数值手段证明：在周期性域设置下，两圆盘的影响张量（influence tensor）可表示为第一圆盘的影响张量与第二圆盘全部周期镜像（periodic images）的影响张量在实空间（real space）中的求和形式（此类相互作用通过无穷远边界条件（boundary conditions at infinity）进行计算）。该结论最初由Zecevic与Lebenson于2021年以启发式方式发表（DOI: 10.1002/nme.6841）。不过本文研究表明，该已发表的公式存在错误，因此我们提出了一种启发式修正方案——该方案最早由Brisard、Bertin与Legoll在2023年的研究中提出。随后我们通过数值方式对修正后的公式进行了验证。更多细节请参阅00-README.html。