Smoothed particle hydrodynamics (SPH) has attracted significant attention in recent decades, and exhibits special advantages in modeling complex flows with multiphysics processes and complex phenomena. Its accuracy depends heavily on the distribution of particles, and will generally be lower if the particles are distributed non-uniformly. A high-order SPH scheme is proposed in the present work for simulating both compressible and incompressible flows. It uses a Gaussian quadrature rule to perform the particle approximation of SPH by introducing Gaussian nodes. Unfortunately, the Gaussian nodes hardly overlap with SPH particles due to the Lagrangian feature, and thus we use a high-order interpolation method to obtain the corresponding physical quantities at the Gaussian nodes. The accuracy and robustness of the proposed Gaussian SPH are demonstrated by several numerical tests, including the Sod problem, Poiseuille flow, Couette flow, cavity flow, Taylor–Green vortex and dam break flow, and a convergence analysis is also conducted to evaluate the effects of particle resolution and distribution for reconstructing a given function. The simulation results for each test case are in good agreements with the available analytical, experimental or numerical results, showing that the proposed Gaussian SPH method is accurate and reliable but expensive for simulating compressible and incompressible flow problems.

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高斯平滑粒子流体动力学：一种高阶无网格粒子方法


近几十年来，平滑粒子流体动力学（SPH）引起了人们的广泛关注，并且在模拟具有多物理过程和复杂现象的复杂流动方面表现出特殊的优势。其精度在很大程度上取决于颗粒的分布，如果颗粒分布不均匀，其精度通常会较低。目前的工作中提出了一种高阶 SPH 方案来模拟可压缩和不可压缩流。它利用高斯求积法则，通过引入高斯节点来进行SPH的粒子逼近。不幸的是，由于拉格朗日特征，高斯节点很难与SPH粒子重叠，因此我们使用高阶插值方法来获得高斯节点处相应的物理量。所提出的高斯 SPH 的准确性和鲁棒性通过多项数值测试得到了证明，包括 Sod 问题、Poiseuille 流、Couette 流、空腔流、Taylor-Green 涡流和溃坝流，并且还进行了收敛分析来评估效果用于重建给定函数的粒子分辨率和分布。每个测试案例的模拟结果与现有的分析、实验或数值结果非常吻合，表明所提出的高斯 SPH 方法准确可靠，但对于模拟可压缩和不可压缩流动问题来说成本高昂。