基于局部泰勒展开的高精度无网格算法  

High-Precision Meshless Algorithm Based on Local Taylor Expansion

作　　者：陈肖潇 赵嘉毅 Xiaoxiao Chen;Jiayi Zhao(School of Energy and Power Engineering,University of Shanghai for Science and Technology,Shanghai)

机构地区：[1]上海理工大学能源与动力工程学院,上海

出　　处：《建模与仿真》2024年第5期5513-5526,共14页Modeling and Simulation

基　　金：国家自然科学基金(12002212)。

摘　　要：光滑粒子动力学方法(Smoothed Particle Hydrodynamics,SPH)所具备的无网格特性使其摆脱了网格划分过程,在处理复杂几何边界上具有潜在优势。然而,传统SPH受限于空间微分高阶一致性缺失,无法实现高精度求解。本文基于SPH粒子的局部泰勒展开,通过邻域粒子空间矩阵求解,实现了满足高阶一致性的SPH微分算子算法。通过一、二维函数证明了该算法在任意粒子分布中微分2、3、4阶一致性。随后,基于热传导方程、Burgers方程、纳维–斯托克斯方程等偏微分方程证明了该算法求解空间算子的精确性与稳定性。基于局部泰勒展开的SPH算法为无网格方法在复杂几何域的高精度数值求解提供了一种可行方法。The meshless nature of Smooth Particle Dynamics(SPH)method eliminates the need for mesh partitioning and has potential advantages in handling complex geometric boundaries.However,tradi-tional SPH is limited by the lack of high-order consistency in spatial differentiation and cannot achieve high-precision solutions.This article is based on the local Taylor expansion of SPH particles,and solves the SPH differential operator algorithm that satisfies high-order consistency through the neighborhood particle space matrix.The algorithm has been proven to have 2nd,3rd,and 4th order consistency in differentiation in any particle distribution through one-dimensional and two-dimensional functions.Subsequently,the accuracy and stability of the algorithm for solving spatial operators were demonstrated based on partial differential equations such as the heat conduction equation,Burgers equation,and Navier Stokes equation.The SPH algorithm based on local Taylor expansion provides a feasible method for high-precision numerical solutions in complex geometric domains.

关 键 词：无网格方法 收敛性 稳定性 泰勒展开 权重 算子矩阵 SPH 

分 类 号：O17[理学—数学]