We present a hybrid particle/grid approach for simulating incompressible fluids on collocated velocity grids. Our approach supports both particle‐based Lagrangian advection in very detailed regions of the flow and efficient Eulerian grid‐based advection in other regions of the flow. A novel Backward Semi‐Lagrangian method is derived to improve accuracy of grid based advection. Our approach utilizes the implicit formula associated with solutions of the inviscid Burgers’ equation. We solve this equation using Newton's method enabled by C1 continuous grid interpolation. We enforce incompressibility over collocated, rather than staggered grids. Our projection technique is variational and designed for B‐spline interpolation over regular grids where multiquadratic interpolation is used for velocity and multilinear interpolation for pressure. Despite our use of regular grids, we extend the variational technique to allow for cut‐cell definition of irregular flow domains for both Dirichlet and free surface boundary conditions.

中文翻译：

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一种用于流体模拟的混合拉格朗日/欧拉并置速度平流和投影方法

我们提出了一种混合粒子/网格方法，用于在并置速度网格上模拟不可压缩流体。我们的方法既支持流的非常详细区域中基于粒子的拉格朗日平流，也支持流的其他区域中基于欧拉网格的高效平流。推导了一种新的后向半拉格朗日方法，以提高基于网格的对流精度。我们的方法利用与无粘性 Burgers 方程的解相关的隐式公式。我们使用由 C1 连续网格插值启用的牛顿法求解该方程。我们在并置的网格上强制不可压缩，而不是交错的网格。我们的投影技术是变分的，设计用于规则网格上的 B 样条插值，其中多二次插值用于速度，多线性插值用于压力。