We propose to augment standard grid-based fluid solvers with pointwise divergence-free velocity interpolation, thereby ensuring exact incompressibility down to the sub-cell level. Our method takes as input a discretely divergence-free velocity field generated by a staggered grid pressure projection, and first recovers a corresponding discrete vector potential. Instead of solving a costly vector Poisson problem for the potential, we develop a fast parallel sweeping strategy to find a candidate potential and apply a gauge transformation to enforce the Coulomb gauge condition and thereby make it numerically smooth. Interpolating this discrete potential generates a pointwise vector potential whose analytical curl is a pointwise incompressible velocity field. Our method further supports irregular solid geometry through the use of level set-based cut-cells and a novel Curl-Noise-inspired potential ramping procedure that simultaneously offers strictly non-penetrating velocities and incompressibility. Experimental comparisons demonstrate that the vector potential reconstruction procedure at the heart of our approach is consistently faster than prior such reconstruction schemes, especially those that solve vector Poisson problems. Moreover, in exchange for its modest extra cost, our overall Curl-Flow framework produces significantly improved particle trajectories that closely respect irregular obstacles, do not suffer from spurious sources or sinks, and yield superior particle distributions over time.

翻译：我们提出对标准网格基流体求解器进行增强，通过引入逐点散度为零的速度插值，从而在子网格尺度上确保精确不可压缩性。本方法以交错网格压力投影生成的离散无散速度场为输入，首先重构对应的离散矢量势。为避免求解计算代价高昂的矢量泊松问题，我们开发了一种快速并行扫描策略来获得候选势，并施加规范变换以强制满足库仑规范条件，从而保证数值光滑性。对该离散势进行插值可生成逐点矢量势，其解析旋度即为逐点不可压缩速度场。本方法通过使用基于水平集的切割网格和新型旋度噪声启发势斜坡过程，进一步支持不规则固体几何体，该过程可同时实现严格无穿透速度与不可压缩性。实验对比表明，本方法核心的矢量势重构过程始终优于现有重构方案（尤其是求解矢量泊松问题的方案）。此外，仅需付出适度的额外计算成本，我们提出的完整Curl-Flow框架即可显著改善粒子轨迹——这些轨迹能紧密贴合不规则障碍物、不受伪源汇影响，并在时间演化中保持更优的粒子分布。