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Smoothed particle hydrodynamics(SPH) is a Lagrangian meshfree particle method, and has been widely applied to different areas including incompressible or pseudo-incompressible flows with multiphase interfaces and moving boundaries. In this paper, an instability problem has been identified when the conventional SPH method is applied to modeling the Poiseuille flow problem at long-term simulations. It is found that this instability resulted from the particle inconsistency inherent to the SPH method, which originates from the discrete particle approximation and is a fundamental cause for poor approximation accuracy. A new particle approximation approach has been used to restore the particle consistency. We show that this particle consistency restoring approach can produce stable solutions for both regular and irregular particle distributions even at long-term simulations.
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Keywords:
- smoothed particle hydrodynamics /
- particle approximation /
- consistency /
- stability
[1] ［1］Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[2] ［2］Lucy L B 1977 Astron. J. 82 1013
[3] ［3］Chang J Z, Liu M B, Liu H T 2008 Acta Phys. Sin. 57 3954 (in Chinese) ［常建忠、刘谋斌、刘汉涛 2008 57 3954］
[4] ［4］Zhao Y, Ji Z Z, Feng T 2004 Acta Phys. Sin. 53 671 (in Chinese) ［赵颖、季仲贞、冯涛 2004 53 671］
[5] ［5］Zhang A M 2008 Chin. Phys. B 17 927
[6] ［6］Sun Z H, Han R J 2008 Chin. Phys. B 17 3185
[7] ［7］Zhang A M, Yao X L 2008 Acta Phys. Sin. 57 339 (in Chinese) ［张阿漫、姚熊亮 2008 57 339］
[8] ［8］Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Meshfree Particle Method (Singapore: World Scientific)
[9] ［9］Liu M B, Liu G R, Zong Z 2008 Int. J. Comput. Meth. 5 135
[10] ］Monaghan J J 2005 Rep. Prog. Phys. 68 1703
[11] ］Swegle J W, Attaway S W 1995 J. Comput. Phys. 116 123
[12] ］Fu X J, Qiang H F, Yang Y C 2007 Adv. Mech. 37 375 (in Chinese) ［傅学金、强洪夫、杨月诚 2007 力学进展 37 375］
[13] ］Morris J P 1996 Publ. Astron. Soc. Aust. 13 97
[14] ］Belytschko T, Krongauz Y, Dolbow J, Gerlach C 1998 Int. J. Numer. Meth. Eng. 43 785
[15] ］Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P 1996 Comput. Method. Appl. Math. 139 3
[16] ］Li S, Liu W K 2002 Appl. Mech. Rev. 55 1
[17] ］Liu W K, Chen Y, Jun S, Chen J S, Belytschko T, Pan C, Uras R A, Chang C T 1996 Comput. Meth. Eng. 3 3
[18] ］Liu M B, Liu G R, Lam K Y 2003 J. Comput. Appl. Math. 155 263
[19] ］He X Y, Zhang R Y, Chen S Y, Doolen G D 1999 Phys. Fluids 11 1143
[20] ］Liu M B, Xie W P, Liu G R 2005 Appl. Math. Model. 29 1252
[21] ］Monaghan J J 1992 Annu. Rev. Astron. Astr. 30 543
[22] ］Libersky L D, Petschek A G, Carney T C, Hipp J R, Allahdadi F A 1993 J. Comput. Phys. 109 67
[23] ］Liu M B, Liu G R 2006 Appl. Numer. Math. 56 19
[24] ］Monaghan J J 1982 SIAM J. Sci. Stat. Comp. 3 422
[25] ］Hernquist L 1993 Astrophys. J. 404 717
[26] ］Fulk D A 1994 Ph. D. Dissertation (Wright Patterson: Air Force Institute of Technology)
[27] ］Morris J P 1996 Analysis of Smoothed Particle Hydrodynamics with Applications (Melbourne: Monash University)
[28] ］Morris J P, Fox P J, Zhu Y 1997 J. Comput. Phys. 136 214
[29] ］Tao W Q 1988 Numerical Heat Transfer (Xian: Xian Jiaotong University Press) (in Chinese) ［陶文铨 1988 数值传热学 (西安：西安交通大学出版社)］
[30] ］Monaghan J J 1994 J. Comput. Phys. 110 399
[31] ］Liu M B, Liu G R 2005 Comput. Mech. 35 332
[32] ］MathWorks Inc 2008 Matlab Partial Differential Equation Toolbox 2 (Natick: MathWorks Inc.)
[33] ］Fang J N, Parriaux A 2008 J. Comput. Phys. 227 8894
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[1] ［1］Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[2] ［2］Lucy L B 1977 Astron. J. 82 1013
[3] ［3］Chang J Z, Liu M B, Liu H T 2008 Acta Phys. Sin. 57 3954 (in Chinese) ［常建忠、刘谋斌、刘汉涛 2008 57 3954］
[4] ［4］Zhao Y, Ji Z Z, Feng T 2004 Acta Phys. Sin. 53 671 (in Chinese) ［赵颖、季仲贞、冯涛 2004 53 671］
[5] ［5］Zhang A M 2008 Chin. Phys. B 17 927
[6] ［6］Sun Z H, Han R J 2008 Chin. Phys. B 17 3185
[7] ［7］Zhang A M, Yao X L 2008 Acta Phys. Sin. 57 339 (in Chinese) ［张阿漫、姚熊亮 2008 57 339］
[8] ［8］Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Meshfree Particle Method (Singapore: World Scientific)
[9] ［9］Liu M B, Liu G R, Zong Z 2008 Int. J. Comput. Meth. 5 135
[10] ］Monaghan J J 2005 Rep. Prog. Phys. 68 1703
[11] ］Swegle J W, Attaway S W 1995 J. Comput. Phys. 116 123
[12] ］Fu X J, Qiang H F, Yang Y C 2007 Adv. Mech. 37 375 (in Chinese) ［傅学金、强洪夫、杨月诚 2007 力学进展 37 375］
[13] ］Morris J P 1996 Publ. Astron. Soc. Aust. 13 97
[14] ］Belytschko T, Krongauz Y, Dolbow J, Gerlach C 1998 Int. J. Numer. Meth. Eng. 43 785
[15] ］Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P 1996 Comput. Method. Appl. Math. 139 3
[16] ］Li S, Liu W K 2002 Appl. Mech. Rev. 55 1
[17] ］Liu W K, Chen Y, Jun S, Chen J S, Belytschko T, Pan C, Uras R A, Chang C T 1996 Comput. Meth. Eng. 3 3
[18] ］Liu M B, Liu G R, Lam K Y 2003 J. Comput. Appl. Math. 155 263
[19] ］He X Y, Zhang R Y, Chen S Y, Doolen G D 1999 Phys. Fluids 11 1143
[20] ］Liu M B, Xie W P, Liu G R 2005 Appl. Math. Model. 29 1252
[21] ］Monaghan J J 1992 Annu. Rev. Astron. Astr. 30 543
[22] ］Libersky L D, Petschek A G, Carney T C, Hipp J R, Allahdadi F A 1993 J. Comput. Phys. 109 67
[23] ］Liu M B, Liu G R 2006 Appl. Numer. Math. 56 19
[24] ］Monaghan J J 1982 SIAM J. Sci. Stat. Comp. 3 422
[25] ］Hernquist L 1993 Astrophys. J. 404 717
[26] ］Fulk D A 1994 Ph. D. Dissertation (Wright Patterson: Air Force Institute of Technology)
[27] ］Morris J P 1996 Analysis of Smoothed Particle Hydrodynamics with Applications (Melbourne: Monash University)
[28] ］Morris J P, Fox P J, Zhu Y 1997 J. Comput. Phys. 136 214
[29] ］Tao W Q 1988 Numerical Heat Transfer (Xian: Xian Jiaotong University Press) (in Chinese) ［陶文铨 1988 数值传热学 (西安：西安交通大学出版社)］
[30] ］Monaghan J J 1994 J. Comput. Phys. 110 399
[31] ］Liu M B, Liu G R 2005 Comput. Mech. 35 332
[32] ］MathWorks Inc 2008 Matlab Partial Differential Equation Toolbox 2 (Natick: MathWorks Inc.)
[33] ］Fang J N, Parriaux A 2008 J. Comput. Phys. 227 8894
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