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In order to improve the accuracy and the stability of the conventional smoothed particle hydrodynamics (SPH) method for simulating the transient heat conduction problems, a first order symmetric smoothed particle hydrodynamics (FO-SPH) method is proposed.In order to solve the heat conduction problem with second derivative, the proposed FO-SSPH method is first to decompose the problem into two first order partial differential equations (PDEs), and then the first order kernel gradient is corrected based on the discretization of gradient and the concept of Taylor series. Finally, the obtained local matrix is locally symmetrized. All the numerical results demonstrate that the FO-SSPH possesses a higher accuracy and better stability than the SPH method, that the mixed boundary conditions can be well imposed using FO-SSPH method, and that the reliability and the flexibility of the FO-SSPH method can also be observed for PDEs with multi-boundary conditions. Finally, the one-dimensional nonlinear heat conduction problem is investigated by the FO-SSPH method, and the phenomena of concave and bulge are observed when the temperature achieves the stable state, in which the influence of the coefficients for heat flux is discussed.
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Keywords:
- transient heat conduction /
- SPH /
- non-linear
[1] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[2] Lopez H, Sigalotti L D G 2006 Phys. Rev. E 73 051201-1
[3] Lewis R W, Nithiarasu P, Seetharamu K N 2004 Fundamentals of the Finite Element Method for Heat and Fluid Flow (Chichester: John Wiley)
[4] Zhong C W, Xie J F, Zhuo C S, Xiong S W, and Yin D C 2009 Chin. Phys. B 18 4083
[5] Zhang D H, Liu F G, Zhang J C, Rui X F 2010 Chin. J. Comp. Phys. 27 699 (in Chinese) [张东辉、刘方贵、张金存、芮孝芳 2010 计算物理 27 699]
[6] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[7] Cheng R J, Cheng Y M, Ge H X 2009 Chin. Phys. B 18 4059
[8] Gingold R A, Monaghan J J 1977 Mon, Not. R. Astron. Soc. 181 375
[9] Lucy L B 1977 Astron. J. 82 1013
[10] Liu M B, Liu G R 2010 Archives of Computational Methods in Engineering 17 25
[11] Jeong J H, Jhon M S, Halow J S, Osdol J V 2003 Computer Phys. Comm. 153 71
[12] Cheng R J, Ge H X 2010 Chin. Phys. B 19 090201
[13] Chen J K, Beraun J E, Carney T C 1999 Int. J. Num. Meth. Eng. 46 231
[14] Zhang G M, Batra R C 2004 Comp. Mech. 34 137
[15] Liu M B, Xie W P, Liu G R 2005 Appl. Math. Model. 29 1252
[16] Bonet J, Lok T S L 1999 Comput. Meth. Appl. Mech. Eng. 180 97
[17] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 3654 (in Chinese) [刘谋斌、常建忠 2010 59 3654]
[18] Liu M B, Liu G R 2006 Appl. Num. Math. 56 19
[19] Zhang X H, Ouyang J, Zhang L 2010 Int. J. Heat Mas Transfer 52 2161
[20] Holman J P 2002 Heat Transfer, ninth ed. (Singapore: McGraw-Hill)
[21] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Mesh-free Particle Method (Singapore: World Scientific)
[22] Jiang T, Ouyang J, Zhao X K, Ren J L 2011 Acta Phys. Sin. 60 ( in Chinese) [蒋涛、欧阳洁、赵晓凯、任金莲 2011 60]
[23] Lewis R W, Nithiarasu P, Seetharamu K N 2004 Fundamentals of the Finite Element Method for Heat and Fluid Flow (Chichester: Wiley)
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[1] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[2] Lopez H, Sigalotti L D G 2006 Phys. Rev. E 73 051201-1
[3] Lewis R W, Nithiarasu P, Seetharamu K N 2004 Fundamentals of the Finite Element Method for Heat and Fluid Flow (Chichester: John Wiley)
[4] Zhong C W, Xie J F, Zhuo C S, Xiong S W, and Yin D C 2009 Chin. Phys. B 18 4083
[5] Zhang D H, Liu F G, Zhang J C, Rui X F 2010 Chin. J. Comp. Phys. 27 699 (in Chinese) [张东辉、刘方贵、张金存、芮孝芳 2010 计算物理 27 699]
[6] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[7] Cheng R J, Cheng Y M, Ge H X 2009 Chin. Phys. B 18 4059
[8] Gingold R A, Monaghan J J 1977 Mon, Not. R. Astron. Soc. 181 375
[9] Lucy L B 1977 Astron. J. 82 1013
[10] Liu M B, Liu G R 2010 Archives of Computational Methods in Engineering 17 25
[11] Jeong J H, Jhon M S, Halow J S, Osdol J V 2003 Computer Phys. Comm. 153 71
[12] Cheng R J, Ge H X 2010 Chin. Phys. B 19 090201
[13] Chen J K, Beraun J E, Carney T C 1999 Int. J. Num. Meth. Eng. 46 231
[14] Zhang G M, Batra R C 2004 Comp. Mech. 34 137
[15] Liu M B, Xie W P, Liu G R 2005 Appl. Math. Model. 29 1252
[16] Bonet J, Lok T S L 1999 Comput. Meth. Appl. Mech. Eng. 180 97
[17] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 3654 (in Chinese) [刘谋斌、常建忠 2010 59 3654]
[18] Liu M B, Liu G R 2006 Appl. Num. Math. 56 19
[19] Zhang X H, Ouyang J, Zhang L 2010 Int. J. Heat Mas Transfer 52 2161
[20] Holman J P 2002 Heat Transfer, ninth ed. (Singapore: McGraw-Hill)
[21] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics: A Mesh-free Particle Method (Singapore: World Scientific)
[22] Jiang T, Ouyang J, Zhao X K, Ren J L 2011 Acta Phys. Sin. 60 ( in Chinese) [蒋涛、欧阳洁、赵晓凯、任金莲 2011 60]
[23] Lewis R W, Nithiarasu P, Seetharamu K N 2004 Fundamentals of the Finite Element Method for Heat and Fluid Flow (Chichester: Wiley)
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