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In this work, an improved parallel SPH method is proposed to accurately solve the three-dimensional (3D) transient heat conduction equation with variable coefficients. The improvements are described as follows. Firstly, the first-order symmetric smoothed particle hydrodynamics (SPH) method is extended to the simulating of the 3D problem based on Taylor expansion. Secondly, the concept of stabilized up-wind technique is introduced into the convection term. Thirdly, the MPI parallel technique based on the neighboring particle mark method is introduced into the above improved SPH method, and named the corrected parallel SPH method for 3D problems (CPSPH-3D). Subsequently, the accuracy, convergence and the computational efficiency of the proposed CPSPH-3D method are tested by simulating the 3D transient heat conduction problems with constant/variable coefficient, and compared with the analytical solution. Meanwhile, the capacity of the proposed CPSPH-3D for solving the 3D heat conduction problems with the Dirichlet and Newmann boundaries is illustrated, in which the change of temperature with time under the complex cylindrical area is also considered. The numerical results show that:1) the proposed CPSPH-3D method has the better stability, higher accuracy and computational efficiency than the conventional SPH method no matter whether the particle distribution is uniform; 2) the calculating time can be well reduced by increasing the number of CPUs when the particle number is refined in the simulations of CPSPH-3D. Finally, the temperature variation in the 3D functionally gradient material is predicted by the corrected parallel SPH method, and compared with the other numerical results. The process of temperature variation in the functionally gradient material can be shown accurately.
[1] Zhang Z, Wang J F, Cheng Y M, Liew K M 2013 Sci. China 56 1568
[2] Sutradhar A, Paulino G H, Gray J J 2002 Engin. Anal. Bound. Elements 26 119
[3] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[4] Zhang J Q, Niea L, Zhang X Y, Chen R Y 2014 Eur. Phys. J. B 87 285
[5] Nie L R, Yu L L, Zheng Z G, Shu C Z 2013 Phys. Rev. E 87 062142
[6] Lewis R W, Nithiarasu P, Seetharamu K N 2004 Fundamentals of the Finite Element Method for Heat and Fluid Flow (Chichester:John Wiley)
[7] Akil J H 2008 J. Computat. Appl. Math. 220 335
[8] Zhong C W, Xie J F, Zhuo C S, Xiong S W, Yin D C 2009 Chin. Phys. B 18 4083
[9] Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[10] Liu M B, Liu G R 2010 Arch. Computat. Methods Engin. 17 25
[11] Jeong J H, Jhon M S, Halow J S, Osdol J V 2003 Comput. Phys. Commun. 153 71
[12] Chen J K, Beraun J E, Carney T C 1999 Int. J. Num. Meth. Eng. 46 231
[13] Zhang G M, Batra R C 2004 Comp. Mech. 34 137
[14] Bonet J, Lok T S L 1999 Comput. Meth. Appl. Mech. Eng. 180 97
[15] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 3654 (in Chinese)[刘谋斌, 常建忠 2010 59 3654]
[16] Liu M B, Liu G R 2006 Appl. Num. Math. 56 19
[17] Jiang T, Ouyang J, Li X J, Zhang L, Ren J L 2011 Acta Phys. Sin. 60 090206 (in Chinese)[蒋涛, 欧阳洁, 栗雪娟, 张林, 任金莲 2011 60 090206]
[18] Holman J P 2002 Heat Transfer (9th Ed.) (Singapore:McGraw-Hill)
[19] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics:A Mesh-free Particle Method (Singapore:World Scientific)
[20] Young D L, Tsai C C, Murugesan K, Fan C M, Chen C 2004 Engin. Anal. Bound. Elements 28 1463
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[1] Zhang Z, Wang J F, Cheng Y M, Liew K M 2013 Sci. China 56 1568
[2] Sutradhar A, Paulino G H, Gray J J 2002 Engin. Anal. Bound. Elements 26 119
[3] Wang J F, Sun F X, Cheng R J 2010 Chin. Phys. B 19 060201
[4] Zhang J Q, Niea L, Zhang X Y, Chen R Y 2014 Eur. Phys. J. B 87 285
[5] Nie L R, Yu L L, Zheng Z G, Shu C Z 2013 Phys. Rev. E 87 062142
[6] Lewis R W, Nithiarasu P, Seetharamu K N 2004 Fundamentals of the Finite Element Method for Heat and Fluid Flow (Chichester:John Wiley)
[7] Akil J H 2008 J. Computat. Appl. Math. 220 335
[8] Zhong C W, Xie J F, Zhuo C S, Xiong S W, Yin D C 2009 Chin. Phys. B 18 4083
[9] Gingold R A, Monaghan J J 1977 Mon. Not. R. Astron. Soc. 181 375
[10] Liu M B, Liu G R 2010 Arch. Computat. Methods Engin. 17 25
[11] Jeong J H, Jhon M S, Halow J S, Osdol J V 2003 Comput. Phys. Commun. 153 71
[12] Chen J K, Beraun J E, Carney T C 1999 Int. J. Num. Meth. Eng. 46 231
[13] Zhang G M, Batra R C 2004 Comp. Mech. 34 137
[14] Bonet J, Lok T S L 1999 Comput. Meth. Appl. Mech. Eng. 180 97
[15] Liu M B, Chang J Z 2010 Acta Phys. Sin. 59 3654 (in Chinese)[刘谋斌, 常建忠 2010 59 3654]
[16] Liu M B, Liu G R 2006 Appl. Num. Math. 56 19
[17] Jiang T, Ouyang J, Li X J, Zhang L, Ren J L 2011 Acta Phys. Sin. 60 090206 (in Chinese)[蒋涛, 欧阳洁, 栗雪娟, 张林, 任金莲 2011 60 090206]
[18] Holman J P 2002 Heat Transfer (9th Ed.) (Singapore:McGraw-Hill)
[19] Liu G R, Liu M B 2003 Smoothed Particle Hydrodynamics:A Mesh-free Particle Method (Singapore:World Scientific)
[20] Young D L, Tsai C C, Murugesan K, Fan C M, Chen C 2004 Engin. Anal. Bound. Elements 28 1463
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