计算函数导数的FODF-SPH方法
|
摘要
在光滑粒子流体动力学(Smooth Particle Hydrodynamics:SPH)核近似方法原理的基础上,通过泰勒级数展开提出了计算函数导数的新FODF-SPH(Frist Order Derivative Free:FODF)方法,并分别推导一维、二维及三维情况下,计算函数的导数核估计的离散形式.用不同的粒子间距和不同的光滑长度计算一维和二维函数导数,与传统SPH方法进行误差对比分析.结果表明,与传统方法对比提出的计算方法的误差小、收敛速度快且计算过程避免核函数导数计算等优越性,因此在工程应用和数值计算中具有较强的适用范围.
Based on the basic principle of the smooth particle hydrodynamics(SPH)method's kernel approximation,New FODF-SPH(Frist Order Derivative Free Smoothed Particle Hydrodynamics) method to compute first order derivatives are constructed through Taylor series expansion.The Discrete forms of kernel estimations are deduced in the one,two and three dimensional space.Then selecting different particle distance and setting different smooth length computed one and two dimensional function's derivatives,compared and analyzed errors of traditional SPH method and FODF-SPH methods.The results show that error of new methods is relatively smaller and convergence speed is faster than traditional SPH method;also in calculation process can avoid calculation of kernel function's derivative kernel approximation,so this method have a wider range of application in engineering an numerical calculation.
Based on the basic principle of the smooth particle hydrodynamics(SPH)method's kernel approximation,New FODF-SPH(Frist Order Derivative Free Smoothed Particle Hydrodynamics) method to compute first order derivatives are constructed through Taylor series expansion.The Discrete forms of kernel estimations are deduced in the one,two and three dimensional space.Then selecting different particle distance and setting different smooth length computed one and two dimensional function's derivatives,compared and analyzed errors of traditional SPH method and FODF-SPH methods.The results show that error of new methods is relatively smaller and convergence speed is faster than traditional SPH method;also in calculation process can avoid calculation of kernel function's derivative kernel approximation,so this method have a wider range of application in engineering an numerical calculation.
引文
[1]Lucy L.A numerical approach to the testing of fission hypothesis[J].Astronomical Journal,1977,82(12):1013-1020.
[2]Gingold R,Monaghan J.Smoothed particle hydrodynamics:Theory and application to nonspherical stars[J].Monthly Notices of the Royal Astronomical Society,1977,181:375-389.
[3]Gingold R A,Monaghan J J.Kernel estimates as a basis for general particle methods in hydrodynamics[J].Journal of Computational Physics,1982,46(3):429-453.
[4]Monaghan J.Simulating free surface flows with SPH[J].Journal of Computational Physics,1994,110(2):399-406.
[5]Larry D,Libersky A G.Petschek.Smoothed particle hydrodynamics with Strength of materials[J].Lecture Notes in Physics,1991,395:248-257.
[6]DavidStowe,Ryan Kupchella,HuaPan,JohnCogar.Investigation of S-SPH for Hypervelocity impact calculations[J].Procedia Engineering,2015,103:585-592.
[7]Prapanch Nair,Gaurav Toma.An improved free surface modeling for incompressible SPH[J].Computers and Fluids,2014,102:304-314.
[8]Chen Z,Zong Z,Liu M B.etc.An SPH model for multiphase flows with complex interfaces and large density differences[J],Journal of Computational Physics,2015,283:169-188.
[9]Monaghan J.Smoothed particle hydrodynamics[J].Reports on Progress in Physics,2005,68:1703-1759.
[10]Liu W K,Jun S,Zhang Y.Reproducing kernel particle methods[J].International Journal for Numerical Methods in Fluids,1995,20(8-9):1081-1106.
[11]Johson G R,Beissel S R.Normalized smoothed functions for SPH impact computations[J].International Journal for Numerical Methods in Engineering,1996,39(16):2725-2741.
[12]Chen J K,Beraun J E,Carney T C.A corrective smoothed particle method for boundary value problems in heat conduction[J].International Journal for Numerical Methods in Engineering,1999,46(2):231-252
[13]Zhang G M,Betra R C.Modified smoothed particle hydrodynamics method and its application to transient problems[J].Computational Mechanics,2004,34(2):137-146.
[14]李建成,袁德奎,陶建华.二阶核近似有限粒子方法的改进[J].水动力学研究与进展,2013,28(3):160—173.
[15]郑兴,段文洋.光滑粒子流体动力学二阶算法精度研究[J].水科学进展,2008,19(6):821—827.
[16]热合买提江·依明,买买提明·艾尼.对几种SPH方法的对比研究[J].佳木斯大学,2014,32(4):593—596.
[1 7]热合买提江·依明,买买提明·艾尼.对称SPH核近似方法的研究[J].水动力学与进展,2014,29(4):479—486.
[18]Liu G R,Liu M B.Smoothed Particle Hydrodynamics:A Meshless Particle Method[M].Singapore,World Scientific Publishing,2003:150-156.
[2]Gingold R,Monaghan J.Smoothed particle hydrodynamics:Theory and application to nonspherical stars[J].Monthly Notices of the Royal Astronomical Society,1977,181:375-389.
[3]Gingold R A,Monaghan J J.Kernel estimates as a basis for general particle methods in hydrodynamics[J].Journal of Computational Physics,1982,46(3):429-453.
[4]Monaghan J.Simulating free surface flows with SPH[J].Journal of Computational Physics,1994,110(2):399-406.
[5]Larry D,Libersky A G.Petschek.Smoothed particle hydrodynamics with Strength of materials[J].Lecture Notes in Physics,1991,395:248-257.
[6]DavidStowe,Ryan Kupchella,HuaPan,JohnCogar.Investigation of S-SPH for Hypervelocity impact calculations[J].Procedia Engineering,2015,103:585-592.
[7]Prapanch Nair,Gaurav Toma.An improved free surface modeling for incompressible SPH[J].Computers and Fluids,2014,102:304-314.
[8]Chen Z,Zong Z,Liu M B.etc.An SPH model for multiphase flows with complex interfaces and large density differences[J],Journal of Computational Physics,2015,283:169-188.
[9]Monaghan J.Smoothed particle hydrodynamics[J].Reports on Progress in Physics,2005,68:1703-1759.
[10]Liu W K,Jun S,Zhang Y.Reproducing kernel particle methods[J].International Journal for Numerical Methods in Fluids,1995,20(8-9):1081-1106.
[11]Johson G R,Beissel S R.Normalized smoothed functions for SPH impact computations[J].International Journal for Numerical Methods in Engineering,1996,39(16):2725-2741.
[12]Chen J K,Beraun J E,Carney T C.A corrective smoothed particle method for boundary value problems in heat conduction[J].International Journal for Numerical Methods in Engineering,1999,46(2):231-252
[13]Zhang G M,Betra R C.Modified smoothed particle hydrodynamics method and its application to transient problems[J].Computational Mechanics,2004,34(2):137-146.
[14]李建成,袁德奎,陶建华.二阶核近似有限粒子方法的改进[J].水动力学研究与进展,2013,28(3):160—173.
[15]郑兴,段文洋.光滑粒子流体动力学二阶算法精度研究[J].水科学进展,2008,19(6):821—827.
[16]热合买提江·依明,买买提明·艾尼.对几种SPH方法的对比研究[J].佳木斯大学,2014,32(4):593—596.
[1 7]热合买提江·依明,买买提明·艾尼.对称SPH核近似方法的研究[J].水动力学与进展,2014,29(4):479—486.
[18]Liu G R,Liu M B.Smoothed Particle Hydrodynamics:A Meshless Particle Method[M].Singapore,World Scientific Publishing,2003:150-156.