适用于自由表面流动的平衡型弱可压缩光滑粒子流体动力学（WCSPH）格式及其GPU实现
A well-balanced weakly compressible SPH formulation for free-surface flows and its GPU implementation
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中文总结 AI 辅助
本研究提出一种平衡型WCSPH格式，通过引入辅助势变量和黎曼基梯度近似解决离散层面的流体静平衡问题，在GPU上实现后将伪速度误差降至10^-13量级，可高效模拟千万粒子规模的自由表面流动。
中文摘要 AI 辅助
本研究提出一种适用于自由表面流动的平衡型弱可压缩光滑粒子流体动力学（WCSPH）格式，该格式在离散层面精确保持流体静平衡，这一特性对可靠的长期模拟至关重要。尽管平衡型格式在基于网格的方法中已得到充分发展，但WCSPH中该特性仍未得到解决，其压力梯度的粒子近似无法精确平衡重力，这种不平衡源于两个难点：一是压力梯度与密度项的非线性，二是通过粒子求和计算梯度的近似误差。第一个难点通过引入辅助势变量解决，该变量将非线性项重构为单个标量的梯度，在流体静条件下简化为位置的线性函数；第二个难点通过结合核校正的黎曼基梯度近似解决，该近似具有一阶一致性，可精确恢复线性场。这两个要素确保离散势梯度能精确平衡重力，δ-SPH、粒子移位、拉伸不稳定性控制等广泛使用的技术可轻松集成。该格式进一步扩展至三维，并通过适配架构的优化在GPU上实现。针对矩形、三角形和高斯底部地形的流体静测试表明，所提格式达到了机器精度的平衡特性，将传统SPH的伪速度误差从10^-3降至10^-13量级；更复杂的基准测试证实了其鲁棒性、准确性和低压振荡特性，在单个消费级GPU上完成了多达1753万个粒子的模拟。
英文摘要
This study proposes a well-balanced formulation of weakly compressible smoothed particle hydrodynamics (WCSPH) for free-surface flows, which preserves hydrostatic equilibrium exactly at the discrete level--a property essential for reliable long-term simulations. Although well-balanced schemes are well established for mesh-based methods, the property remains largely unaddressed in WCSPH, where the particle approximation of the pressure gradient fails to balance the gravitational force exactly. The imbalance stems from two difficulties: the nonlinearity of the pressure-gradient-over-density term, and the approximation error of gradients evaluated by particle summation. The first is resolved by introducing an auxiliary potential variable that recasts the nonlinear term as the gradient of a single scalar, which reduces to a linear function of position under hydrostatic conditions. The second is resolved by a Riemann-based gradient approximation with kernel correction, which is first-order consistent and recovers linear fields exactly. These two ingredients ensure that the discrete potential gradient balances gravitational force exactly. Widely used techniques, including $δ-$SPH, particle shifting and tensile instability control, are readily incorporated. The formulation is further extended to three dimensions and implemented on GPU with architecture-tailored optimizations. Hydrostatic tests with rectangular, triangular and Gaussian bottom topographies show that the proposed formulation attains the well-balanced property to machine precision, reducing the spurious velocity error of conventional SPH from $10^{-3}$ to the order of $10^{-13}$. More complex benchmarks confirm its robustness, accuracy and low pressure oscillation, with simulations of up to 17.53 million particles performed on a single consumer-grade GPU.
