With rheology applications in mind, we present a fast solver for the time-dependent effective viscosity of an infinite lattice containing one or more neutrally buoyant smooth rigid particles per unit cell, in a two-dimensional Stokes fluid with given shear rate. At each time, the mobility problem is reformulated as a 2nd-kind boundary integral equation, then discretized to spectral accuracy by the Nyström method and solved iteratively, giving typically 10 digits of accuracy. Its solution controls the evolution of particle locations and angles in a first-order system of ordinary differential equations. The formulation is placed on a rigorous footing by defining a generalized periodic Green's function for the skew lattice. Numerically, the periodized integral operator is split into a near image sum—applied in linear time via the fast multipole method—plus a correction field solved cheaply via proxy Stokeslets. We use barycentric quadratures to evaluate particle interactions and velocity fields accurately, even at distances much closer than the node spacing. Using first-order time-stepping we simulate, for example, 25 ellipses per unit cell to 3-digit accuracy on a desktop in 1 hour per shear time. Our examples show equilibration at long times, force chains, and two types of blow-ups (jamming) whose power laws match lubrication theory asymptotics.

考虑到流变学的应用，我们提出了一种快速求解器，用于求解二维Stokes流体中具有给定剪切速率的无限晶格随时间变化的有效粘度，该无限晶格每单元包含一个或多个中性浮力的光滑刚性颗粒。每次将迁移率问题重新表述为第二类边界积分方程，然后通过Nyström方法离散化为光谱精度，然后迭代求解，通常给出10位精度。它的解决方案控制一阶常微分方程组中粒子位置和角度的演变。通过定义偏斜晶格的广义周期格林函数，可将公式置于严格的基础上。在数值上 周期积分算子被分成一个近似的图像总和（通过快速多极方法在线性时间中应用）以及一个通过代理Stokeslets廉价解决的校正场。我们使用重心正交函数来精确评估粒子相互作用和速度场，即使距离远小于节点间距。使用一阶时间步长，我们可以在每个剪切时间1小时内，例如在台式机上模拟每单位像元25个椭圆至3位精度。我们的示例显示了长时间的平衡，力链和两种类型的爆燃（卡塞），它们的功率定律与润滑理论渐近一致。使用一阶时间步长，我们在每个剪切时间1小时内，例如在台式机上模拟每单位像元25个椭圆至3位精度。我们的示例显示了长时间的平衡，力链以及两种类型的爆炸（卡塞），其功率定律与润滑理论的渐近性一致。使用一阶时间步长，我们可以在每个剪切时间1小时内，例如在台式机上模拟每单位像元25个椭圆至3位精度。我们的示例显示了长时间的平衡，力链以及两种类型的爆炸（卡塞），其功率定律与润滑理论的渐近性一致。