Smoothed Particle Hydrodynamics (SPH_ is a mesh-free Lagrangian method renowned for modeling large deformations and free-surface flows, yet classical formulations remain confined to deterministic systems. We introduce Stochastic SPH (S-SPH), which employs orthogonal Polynomial Chaos expansions to represent uncertainties in system parameters, forcing functions, and boundary or initial conditions, while spatial variation is captured via the SPH kernel. Random fields are discretized through Karhunen-Loève expansions, and a Galerkin projection in the polynomial basis transforms the underlying SPDE into a coupled system of ordinary differential equations governing the time evolution of expansion coefficients. To enforce Dirichlet and Neumann conditions in a mesh-free context, ghost-particle techniques augmented by a gradient-correction matrix are employed, and a predictor-corrector integration scheme ensures numerical stability. We validate S-SPH on benchmark problems, including one-dimensional advection with stochastic advection speed, inviscid Burgers' equations with random initial amplitudes, and two-dimensional Burgers' flows with uncertain Fourier-mode initial fields and viscosity, demonstrating excellent agreement with Monte Carlo simulation statistics of mean and variance. Remarkably, S-SPH achieves up to three orders of magnitude reduction in computational cost relative to direct sampling approaches. The proposed framework thus provides an efficient, accurate, and fully mesh-free methodology for uncertainty quantification in complex mechanics applications.

翻译：摘要：平滑粒子流体动力学（SPH）是一种无网格拉格朗日方法，以模拟大变形和自由表面流动而著称，但经典公式仍局限于确定性系统。我们提出了随机SPH（S-SPH），该方法采用正交多项式混沌展开来表示系统参数、驱动力函数以及边界或初始条件中的不确定性，同时通过SPH核函数捕捉空间变化。随机场通过Karhunen-Loève展开离散化，在多项式基上的伽辽金投影将底层随机偏微分方程转化为一个耦合的常微分方程组，用于控制展开系数的时间演化。为了在无网格背景下施加狄利克雷和诺依曼边界条件，采用了结合梯度修正矩阵的虚粒子技术，并使用预测-校正积分方案确保数值稳定性。我们在基准问题上验证了S-SPH，包括具有随机平流速度的一维对流问题、具有随机初始振幅的无粘伯格斯方程、以及具有不确定傅里叶模态初始场和粘性的二维伯格斯流动，结果与蒙特卡洛模拟的均值和方差统计量高度吻合。值得注意的是，相较于直接采样方法，S-SPH将计算成本降低了多达三个数量级。因此，所提出的框架为复杂力学应用中的不确定性量化提供了一种高效、准确且完全无网格的方法。