Fluid simulations based on memory-efficient spatial representations like implicit neural spatial representations (INSRs) and Gaussian spatial representation (GSR), where the velocity fields are parameterized by neural networks or weighted Gaussian functions, has been an emerging research area. Though advantages over traditional discretizations like spatial adaptivity and continuous differentiability of these spatial representations are leveraged by fluid solvers, solving the time-dependent PDEs that governs the fluid dynamics remain challenging, especially in incompressible fluids where the divergence-free constraint is enforced. In this paper, we propose a grid-free solver Dynamic Divergence-Free Kernels (DDFKs) for incompressible flows based on divergence-free kernels (DFKs). Each DFK is incorporated with a matrix-valued radial basis function and a vector-valued weight, yielding a divergence-free vector field. We model the continuous flow velocity as the sum of multiple DFKs, thus enforcing incompressibility while being able to preserve different level of details. Quantitative and qualitative results show that our method achieves comparable accuracy, robustness, ability to preserve vortices, time and memory efficiency and generality across diverse phenomena to state-of-the-art methods using memory-efficient spatial representations, while excels at maintaining incompressibility. Though our first-order solver are slower than fluid solvers with traditional discretizations, our approach exhibits significantly lower numerical dissipation due to reduced discretization error. We demonstrate our method on diverse incompressible flow examples with rich vortices and various solid boundary conditions.

翻译：基于内存高效空间表示（如隐式神经空间表示（INSRs）与高斯空间表示（GSR））的流体模拟已成为新兴研究领域，其中速度场通过神经网络或加权高斯函数进行参数化。尽管流体求解器利用了这些空间表示相较于传统离散化方法（如空间网格）的优势——包括空间自适应性及连续可微性，但求解控制流体动力学的时变偏微分方程仍然具有挑战性，特别是在需要强制满足无散约束的不可压缩流体中。本文提出一种基于无散核（DFKs）的、适用于不可压缩流动的无网格求解器——动态无散核（DDFKs）。每个DFK由一个矩阵值径向基函数与一个向量值权重构成，从而生成一个无散向量场。我们将连续流动速度建模为多个DFK的叠加，从而在保持不同层次细节的同时强制满足不可压缩性。定量与定性结果表明，本方法在精度、鲁棒性、涡旋保持能力、时间与内存效率以及跨多种现象的泛化性方面，与采用内存高效空间表示的先进方法相当，同时在保持不可压缩性方面表现更优。尽管我们的一阶求解器速度低于采用传统离散化的流体求解器，但由于离散误差减小，本方法展现出显著更低的数值耗散。我们在具有丰富涡旋及多种固体边界条件的不可压缩流动算例中验证了本方法的有效性。