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
DOI: 10.12677/mos.2024.135499, PDF,    国家自然科学基金支持
作者: 陈肖潇, 赵嘉毅：上海理工大学能源与动力工程学院，上海
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；Meshless Method； Convergence； Stability； Taylor Expansion； Weight； Operator Matrix； SPH
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。
Abstract: 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, traditional 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.
文章引用：陈肖潇, 赵嘉毅. 基于局部泰勒展开的高精度无网格算法[J]. 建模与仿真, 2024, 13(5): 5513-5526. https://doi.org/10.12677/mos.2024.135499

参考文献

[1] Lele, S.K. (1992) Compact Finite Difference Schemes with Spectral-Like Resolution. Journal of Computational Physics, 103, 16-42. [Google Scholar] [CrossRef] 
[2] Gottlieb, D. and Orszag, S.A. (1977) Numerical Analysis of Spectral Methods. Society for Industrial and Applied Mathematics. [Google Scholar] [CrossRef] 
[3] Gingold, R.A. and Monaghan, J.J. (1977) Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars. Monthly Notices of the Royal Astronomical Society, 181, 375-389. [Google Scholar] [CrossRef] 
[4] Lucy, L.B. (1977) A Numerical Approach to the Testing of the Fission Hypothesis. The Astronomical Journal, 82, 1013-1024. [Google Scholar] [CrossRef] 
[5] Monaghan, J.J. (2012) Smoothed Particle Hydrodynamics and Its Diverse Applications. Annual Review of Fluid Mechanics, 44, 323-346. [Google Scholar] [CrossRef] 
[6] Monaghan, J.J. (2005) Smoothed Particle Hydrodynamics. Reports on Progress in Physics, 68, 1703-1759. [Google Scholar] [CrossRef] 
[7] Quinlan, N.J., Basa, M. and Lastiwka, M. (2006) Truncation Error in Mesh‐Free Particle Methods. International Journal for Numerical Methods in Engineering, 66, 2064-2085. [Google Scholar] [CrossRef] 
[8] Lind, S.J., Xu, R., Stansby, P.K. and Rogers, B.D. (2012) Incompressible Smoothed Particle Hydrodynamics for Free-Surface Flows: A Generalised Diffusion-Based Algorithm for Stability and Validations for Impulsive Flows and Propagating Waves. Journal of Computational Physics, 231, 1499-1523. [Google Scholar] [CrossRef] 
[9] Dehnen, W. and Aly, H. (2012) Improving Convergence in Smoothed Particle Hydrodynamics Simulations without Pairing Instability. Monthly Notices of the Royal Astronomical Society, 425, 1068-1082. [Google Scholar] [CrossRef] 
[10] Bonet, J. and Lok, T.-L. (1999) Variational and Momentum Preservation Aspects of Smooth Particle Hydrodynamic Formulations. Computer Methods in Applied Mechanics and Engineering, 180, 97-115. [Google Scholar] [CrossRef] 
[11] Chen, J.K., Beraun, J.E. and Carney, T.C. (1999) A Corrective Smoothed Particle Method for Boundary Value Problems in Heat Conduction. International Journal for Numerical Methods in Engineering, 46, 231-252. [Google Scholar] [CrossRef] 
[12] Chen, J.K. and Beraun, J.E. (2000) A Generalized Smoothed Particle Hydrodynamics Method for Nonlinear Dynamic Problems. Computer Methods in Applied Mechanics and Engineering, 190, 225-239. [Google Scholar] [CrossRef] 
[13] Zhang, G.M. and Batra, R.C. (2004) Modified Smoothed Particle Hydrodynamics Method and Its Application to Transient Problems. Computational Mechanics, 34, 137-146. [Google Scholar] [CrossRef] 
[14] Liu, M.B., Xie, W.P. and Liu, G.R. (2005) Modeling Incompressible Flows Using a Finite Particle Method. Applied Mathematical Modelling, 29, 1252-1270. [Google Scholar] [CrossRef] 
[15] Liu, M.B. and Liu, G.R. (2006) Restoring Particle Consistency in Smoothed Particle Hydrodynamics. Applied Numerical Mathematics, 56, 19-36. [Google Scholar] [CrossRef] 
[16] Asprone, D., Auricchio, F., Manfredi, G., Prota, A., Reali, A. and Sangalli, G. (2010) Particle Methods for a 1D Elastic Model Problem: Error Analysis and Development of a Second-Order Accurate Formulation. Computer Modeling in Engineering & Sciences, 62, 1-21, [Google Scholar] [CrossRef] 
[17] Asprone, D., Auricchio, F. and Reali, A. (2011) Novel Finite Particle Formulations Based on Projection Methodologies. International Journal for Numerical Methods in Fluids, 65, 1376-1388. [Google Scholar] [CrossRef] 
[18] Lind, S.J. and Stansby, P.K. (2016) High-Order Eulerian Incompressible Smoothed Particle Hydrodynamics with Transition to Lagrangian Free-Surface Motion. Journal of Computational Physics, 326, 290-311. [Google Scholar] [CrossRef] 
[19] Fourtakas, G., Stansby, P.K., Rogers, B.D. and Lind, S.J. (2018) An Eulerian-Lagrangian Incompressible SPH Formulation (ELI-SPH) Connected with a Sharp Interface. Computer Methods in Applied Mechanics and Engineering, 329, 532-552. [Google Scholar] [CrossRef] 
[20] King, J.R.C., Lind, S.J. and Nasar, A.M.A. (2020) High Order Difference Schemes Using the Local Anisotropic Basis Function Method. Journal of Computational Physics, 415, Article No. 109549. [Google Scholar] [CrossRef] 

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