This paper introduces a novel Lagrangian fluid solver based on covector flow maps. We aim to address the challenges of establishing a robust flow-map solver for incompressible fluids under complex boundary conditions. Our key idea is to use particle trajectories to establish precise flow maps and tailor path integrals of physical quantities along these trajectories to reformulate the Poisson problem during the projection step. We devise a decoupling mechanism based on path-integral identities from flow-map theory. This mechanism integrates long-range flow maps for the main fluid body into a short-range projection framework, ensuring a robust treatment of free boundaries. We show that our method can effectively transform a long-range projection problem with integral boundaries into a Poisson problem with standard boundary conditions -- specifically, zero Dirichlet on the free surface and zero Neumann on solid boundaries. This transformation significantly enhances robustness and accuracy, extending the applicability of flow-map methods to complex free-surface problems.

翻译：本文提出了一种基于协矢量流映射的新型拉格朗日流体求解器。我们旨在解决复杂边界条件下不可压缩流体鲁棒流映射求解器建立的挑战。核心思想是利用粒子轨迹建立精确的流映射，并沿这些轨迹定制物理量的路径积分，以重构投影步骤中的泊松问题。基于流映射理论中的路径积分恒等式，我们设计了一种解耦机制。该机制将主流体区域的长程流映射整合到短程投影框架中，确保对自由边界的鲁棒处理。我们证明，该方法能够有效将具有积分边界条件的长程投影问题转化为具有标准边界条件的泊松问题——具体而言，自由表面为零狄利克雷边界条件，固体边界为零诺伊曼边界条件。这一转化显著增强了鲁棒性与精度，将流映射方法的适用性扩展至复杂的自由表面问题。