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Smoothed particle hydrodynamics (SPH) is a Lagrangian meshfree particle method. It has special advantages in modeling large deformation and free surface flow, and has been widely applied to different problems in engineering and science. However, the classical SPH suffers from stress instability which resticts its further development and applications. The fundamental reason of stress instability is that the stress state and the kernel do not match each other. For frequently used bell-shaped kernel function, in tensile state the attraction between particles increases as particle spacing decreases, thereby leading to tensile instability. In a compressible state, the repulsive force between particles increases, and then decreases as particle spacing decreases, thereby leading to compressible instability. In this paper is presented an approach to removing stress instability in SPH by proposing a new kernel function and a modified SPH discrete form. In the modified SPH, the force between particles is always repulsive and it increases as particle spacing decreases. Two numerical examples are given to test the proposed approachs, and the obtained numerical results clearly demonstrate that the new approach can eliminate stress instability effectively.
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Keywords:
- smoothed particle hydrodynamics /
- stability /
- kernel /
- discrete form
[1] Lucy L B 1977 Astron. J. 82 1013
[2] Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[3] Liu G R, Liu M B 2003 Smoothed particle hydrodynamics: a meshfree particle method (Singapore: World Scientific) p36
[4] Liu M B, Liu G R, Zong Z 2008 Int. J. Comput. Methods 5 135
[5] Monaghan J J 2005 Rep. Prog. Phys. 68 1703
[6] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 7556 (in Chinese) [刘谋斌, 常建忠 2010 物理学报 59 7556]
[7] Swegle J W, Hicks D L, Attaway S W 1995 J. Comput. Phys. 116 123
[8] Dyka C T, Ingel R P 1995 Comput. Struct. 57 573
[9] Zhang J Z, Zheng J, Yu K P, Wei Y J 2010 Eng. Mech. 27 65 (in Chinese) [张嘉钟, 郑俊, 于开平, 魏英杰 2010 工程力学 27 65]
[10] Zheng J 2010 Ph. D. Dissertation (Harbin: Harbin Institute of Technoloty) (in Chinese) [郑俊 2010 博士学位论文 (哈尔滨: 哈尔滨工业大学)]
[11] Morris J P 1995 Pulb. Astron. Soc. Aust. 13 97
[12] Dyka C T, Randles P W, Ingel R P 1997 Int. J. Num. Methods Eng. 40 2325
[13] Randles P W, Libersky L D 2000 Int. J. Num. Methods Eng. 48 1445
[14] Morris J P 1996 Ph.D. Dissertation (Melbourne: Monash University)
[15] Johnson G R, Stryk R A, Beissel S R 1996 Comput. Methods Appl. Mech. Eng. 139 347
[16] Johnson G R, Beissel S R 1996 Int. J. Num. Methods Eng. 39 2725
[17] Hicks D L, Liebrock L M 1999 Comput. Math. Appl. 38 1
[18] Belytschko T, Guo Y, Liu W K, Xiao S P 2000 Int. J. Num. Methods Eng. 48 1359
[19] Zhang G M, Wang X J, Hu X Z, Zhou Z 2003 Explosion and Shock Waves 23 219 (in Chinese) [张刚明, 王肖钧, 胡秀章, 周钟 2003 爆炸与冲击 23 219]
[20] Hicks D L, Swegle J W, Attaway S W 1997 Appl. Math. Comput. 85 209
[21] Wen Y, Hicks D L, Swegle J W 1994 Sandia Report SAND94-1932
[22] Chen J K, Beraun J E, Jih C J 1999 Comput. Mech. 23 279
[23] Monaghan J J 2000 J. Comput. Phys. 159 290
[24] Gray J P, Monaghan J J, Swift R P 2001 Comput. Methods Appl. Mech. Eng. 190 6641
[25] Jia B, Ma Z T, Zhang W, Pang B J 2010 J. of Harbin Institute of Technology 42 1369 (in Chinese) [贾斌, 马志涛, 张伟, 庞宝君 2010 哈尔滨工业大学学报 42 1369]
[26] Liu W K, Jun S, Zhang Y F 1995 Int. J. Num. Methods in Fluids 20 1081
[27] Dilts G A 1999 Int. J. Num. Methods Eng. 44 1115
[28] Fu X J, Qiang H F, Yang Y C 2007 Advances in Mech. 37 375 (in Chinese) [傅学金, 强洪夫, 杨月诚 2007 力学进展 37 375]
[29] Morris J P, Fox P J, Zhu Y 1997 J. Comput. Phys. 136 214
[30] Fulk D A, Quinn D W 1996 J. Comput. Phys. 126 165
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[1] Lucy L B 1977 Astron. J. 82 1013
[2] Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[3] Liu G R, Liu M B 2003 Smoothed particle hydrodynamics: a meshfree particle method (Singapore: World Scientific) p36
[4] Liu M B, Liu G R, Zong Z 2008 Int. J. Comput. Methods 5 135
[5] Monaghan J J 2005 Rep. Prog. Phys. 68 1703
[6] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 7556 (in Chinese) [刘谋斌, 常建忠 2010 物理学报 59 7556]
[7] Swegle J W, Hicks D L, Attaway S W 1995 J. Comput. Phys. 116 123
[8] Dyka C T, Ingel R P 1995 Comput. Struct. 57 573
[9] Zhang J Z, Zheng J, Yu K P, Wei Y J 2010 Eng. Mech. 27 65 (in Chinese) [张嘉钟, 郑俊, 于开平, 魏英杰 2010 工程力学 27 65]
[10] Zheng J 2010 Ph. D. Dissertation (Harbin: Harbin Institute of Technoloty) (in Chinese) [郑俊 2010 博士学位论文 (哈尔滨: 哈尔滨工业大学)]
[11] Morris J P 1995 Pulb. Astron. Soc. Aust. 13 97
[12] Dyka C T, Randles P W, Ingel R P 1997 Int. J. Num. Methods Eng. 40 2325
[13] Randles P W, Libersky L D 2000 Int. J. Num. Methods Eng. 48 1445
[14] Morris J P 1996 Ph.D. Dissertation (Melbourne: Monash University)
[15] Johnson G R, Stryk R A, Beissel S R 1996 Comput. Methods Appl. Mech. Eng. 139 347
[16] Johnson G R, Beissel S R 1996 Int. J. Num. Methods Eng. 39 2725
[17] Hicks D L, Liebrock L M 1999 Comput. Math. Appl. 38 1
[18] Belytschko T, Guo Y, Liu W K, Xiao S P 2000 Int. J. Num. Methods Eng. 48 1359
[19] Zhang G M, Wang X J, Hu X Z, Zhou Z 2003 Explosion and Shock Waves 23 219 (in Chinese) [张刚明, 王肖钧, 胡秀章, 周钟 2003 爆炸与冲击 23 219]
[20] Hicks D L, Swegle J W, Attaway S W 1997 Appl. Math. Comput. 85 209
[21] Wen Y, Hicks D L, Swegle J W 1994 Sandia Report SAND94-1932
[22] Chen J K, Beraun J E, Jih C J 1999 Comput. Mech. 23 279
[23] Monaghan J J 2000 J. Comput. Phys. 159 290
[24] Gray J P, Monaghan J J, Swift R P 2001 Comput. Methods Appl. Mech. Eng. 190 6641
[25] Jia B, Ma Z T, Zhang W, Pang B J 2010 J. of Harbin Institute of Technology 42 1369 (in Chinese) [贾斌, 马志涛, 张伟, 庞宝君 2010 哈尔滨工业大学学报 42 1369]
[26] Liu W K, Jun S, Zhang Y F 1995 Int. J. Num. Methods in Fluids 20 1081
[27] Dilts G A 1999 Int. J. Num. Methods Eng. 44 1115
[28] Fu X J, Qiang H F, Yang Y C 2007 Advances in Mech. 37 375 (in Chinese) [傅学金, 强洪夫, 杨月诚 2007 力学进展 37 375]
[29] Morris J P, Fox P J, Zhu Y 1997 J. Comput. Phys. 136 214
[30] Fulk D A, Quinn D W 1996 J. Comput. Phys. 126 165
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