AI 中文总结
该研究针对不规则几何中二维斯托克斯流，提出非拟合边界代数方程方法。通过构建特定格林函数，经局部插值等操作求解。利用离散卡尔德隆预处理器，实现二阶收敛，降低条件数，在多种场景验证有效性，解决了相关数值模拟问题。
AI 中文摘要
我们提出了一种用于交错 MAC 网格上二维外部/内部斯托克斯方程的非拟合边界代数方程方法。通过从自由空间拉普拉斯晶格格林函数构造显式的自由空间速度和压力晶格格林函数对，我们使用仅支撑在薄交错边界层上的源来表示均匀场。该公式通过局部插值在切割点施加物理狄利克雷数据，同时采样法线秩更新消除与单个或多个障碍物相关的静水零模式。工作流程与经典边界积分公式平行，外部流动不需要人工边界条件，但遵循离散然后表示的路线，不需要奇异/近奇异求积。通过 GMRES 求解由此产生的密集边界系统，利用由标量拉普拉斯核和填充 FFT 构建的逐分量离散卡尔德隆预处理器进行快速体积卷积。广泛的数值验证，包括多连通域、窄间隙和莫法特涡旋，证实了求解器精度的离散不可压缩性并恢复了预期的莫法特涡旋缩放。我们实现了二阶速度和压力收敛，并在数值精度内限制了最大离散散度。离散卡尔德隆预处理器将条件数降低了几个数量级，并在外部配置中产生了几乎与网格无关的条件，而对于窄间隙和细网格内部问题仍然有效——尽管要求更高。
英文摘要
We present an unfitted boundary algebraic equation method for the two-dimensional exterior/interior Stokes equations on a staggered MAC grid. By constructing an explicit free-space pair of velocity and pressure lattice Green's functions (LGFs) from free-space Laplace LGFs, we represent homogeneous fields using sources supported exclusively on thin staggered boundary layers. This formulation imposes physical Dirichlet data at cut points via local interpolation, while sampled-normal rank updates remove hydrostatic null modes associated with single or multiple obstacles. The workflow parallels that of classical boundary integral formulations and requires no artificial boundary conditions for exterior flows, but follows a discretize-then-represent route and does not require singular/near-singular quadrature. The resulting dense boundary system is solved via GMRES, utilizing a componentwise discrete Calderón preconditioner built from the scalar Laplace kernel and padded FFTs for fast volume convolutions. Extensive numerical validation, including multiply connected domains, narrow gaps, and Moffatt eddies, confirms discrete incompressibility to solver accuracy and recovers the expected Moffatt eddy scaling. We achieve second-order velocity and pressure convergence and bound maximum discrete divergence within numerical accuracy. The discrete Calderón preconditioner reduces the condition number by orders of magnitude and yields nearly mesh-independent conditioning in exterior configurations, while remaining effective---though more demanding---for narrow-gap and fine-grid interior problems.
