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Numerical Study of the 3D Variable Coefficient Heat Transfer Problem by Using the Finite Pointset Method

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Abstract

In this paper, a parallel mesh-free finite pointset method (FPM) for the 3D variable coefficient transient heat conduction problem (I-FPM-3D) on regular/irregular region is proposed by coupling several techniques as follows. The partial differential equation with the high-order derivatives is first decomposed into several first-order equations to improve the numerical stability, reduce the computational complexity and easily enforce the Neumann boundary condition. Then, each first-order derivative is solved by the FPM repeatedly. Moreover, the MPI parallel technique is introduced to enhance the computational efficiency, and an appropriate Wendland kernel function is employed which has higher accuracy than the Gaussian function. 3D transient heat conduction problems with different boundary conditions, including the case in a complex cylindrical domain, are investigated and compared with the analytical solutions to illustrate the flexibility and the accuracy of the parallel I-FPM-3D. The convergence and the computational efficiency of the parallel I-FPM-3D are also analyzed. Finally, the temperature field in a 3D functional gradient material is predicted by the parallel I-FPM-3D and compared with the other numerical results. The numerical results indicate that the temperature variation process in the functional gradient materials can be visualized accurately.

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References

  1. Sutradhar, A.; Paulino, G.H.; Gray, L.J.: Transient heat conduction in homogeneous and non-homogeneous materials by the laplace transform galerkin boundary element method. Eng. Anal. Boundary Elem. 26(2), 119–132 (2002)

    Article  Google Scholar 

  2. Ren, J.; Ouyang, J.; Jiang, T.: An improved particle method for simulation of the non-isothermal viscoelastic fluid mold filling process. Int. J. Heat Mass Transf. 85, 543–560 (2015)

    Article  Google Scholar 

  3. Nithiarasu, P.; Lewis, R.W.; Seetharamu, K.N.: Fundamentals of the finite Element Method for Heat and Mass Transfer. Wiley, Hoboken (2016)

    MATH  Google Scholar 

  4. Nazir, U.; Nawaz, M.; Alqarni, M.M.; Saleem, S.: Finite element study of flow of partially ionized fluid containing nanoparticles. Arab. J. Sci. Eng. 44, 10257–10268 (2019)

    Article  Google Scholar 

  5. Liu, W.K.; Jun, S.; Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Methods Fluids 20(8–9), 1081–1106 (1995)

    Article  MathSciNet  Google Scholar 

  6. Dong, L.; Gong, S.; Cheng, P.: Direct numerical simulations of film boiling heat transfer by a phase-change lattice Boltzmann method. Int. Commun. Heat Mass Transfer 91, 109–116 (2018)

    Article  Google Scholar 

  7. Sheikholeslami, M.; Shehzad, S.A.: Magnetohydrodynamic nanofluid convective flow in a porous enclosure by means of LBM. Int. J. Heat Mass Transf. 113, 796-85 (2017)

    Google Scholar 

  8. Jiang, T.; Ouyang, J.; Ren, J.-L.; Yang, B.-X.: A mixed corrected symmetric SPH (MC-SSPH) method for computational dynamic problems. Comput. Phys. Commun. 183(1), 50–62 (2012)

    Article  MathSciNet  Google Scholar 

  9. Shams, S.; Soltani, B.: Buckling of laminated carbon nanotube-reinforced composite plates on elastic foundations using a meshfree method. Arab. J. Sci. Eng. 41, 1981–1993 (2016)

    Article  Google Scholar 

  10. Tiwari, S.; Kuhner, J.: Modeling of two-phase flows with surface tension by finite pointset method (FPM). J. Comput. Appl. Math. 203, 376–386 (2007)

    Article  MathSciNet  Google Scholar 

  11. Tiwari, S.; Kuhnert, J.: A numerical scheme for solving incompressible and low Mach number flows by finite pointset method, Springer lecture notes in computational science and engineering: meshfree methods for partial differential equations II, vol. 43. Springer, Berlin (2005)

    MATH  Google Scholar 

  12. Fang, J.; Parriaux, A.: A regularized Lagrangian finite point method for the simulation of incompressible viscous flows. J. Comput. Phys. 227, 8894–8908 (2008)

    Article  MathSciNet  Google Scholar 

  13. Lucy, L.B.: A numerical approach to the testing of the fission hypothesis. Astronom. J. 82, 1013–1024 (1977)

    Article  Google Scholar 

  14. Farrokhpanah, A.; Bussmann, M.; Mostaghimi, J.: New smoothed particle hydrodynamics (SPH) formulation for modeling heat conduction with solidification and melting. Numer. Heat Transf. Part B: Fund. 71(4), 299–312 (2017)

    Article  Google Scholar 

  15. Li, L.; Shen, L.; Nguyen, G.D.; El-Zein, A.; Maggi, F.: A smoothed particle hydrodynamics framework for modelling multiphase interactions at meso-scale. Comput. Mech. 64, 1–15 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Ren, J.; Jiang, T.; Weigang, L.; Li, G.: An improved parallel SPH approach to solve 3D transient generalized Newtonian free surface flows. Comput. Phys. Commun. 205, 87–105 (2016)

    Article  MathSciNet  Google Scholar 

  17. Jiang, T.; Ren, J.L.; Lu, W.G.; Xu, B.: A corrected particle method with high-order Taylor expansion for solving the viscoelastic fluid flow. Acta. Mech. Sin. 33(1), 20–39 (2017)

    Article  MathSciNet  Google Scholar 

  18. Liu, M.B.; Liu, G.R.: Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch. Comput. Methods Eng. 17, 25–76 (2010)

    Article  MathSciNet  Google Scholar 

  19. Sun, P.N.; Colagrossi, A.; Marrone, S.; Antuono, M.; Zhang, A.M.: Multi-resolution delta-plus-SPH with tensile instability control: towards high Reynolds number flows. Comput. Phys. Commun. 224, 63–80 (2018)

    Article  MathSciNet  Google Scholar 

  20. Zhang, Z.; Wang, J.; Cheng, Y.; Liew, K.M.: The improved element-free galerkin method for three-dimensional transient heat conduction problems. Sci. China Phys. Mech. Astron. 56(8), 1568–1580 (2013)

    Article  Google Scholar 

  21. Drumm, C.; Tiwari, S.; Kuhnert, J.; Bart, H.-J.: Finite pointset method for simulation of the liquid–liquid flow field in an extractor. Comput. Chem. Eng. 32(12), 2946–2957 (2008)

    Article  Google Scholar 

  22. Uhlmann, E.; Gerstenberger, R.; Kuhnert, J.: Cutting simulation with the meshfree finite pointset method. Procedia CIRP 8, 391–396 (2013)

    Article  Google Scholar 

  23. Reséndiz-Flores, E.O.; García-Calvillo, I.D.: Application of the finite pointset method to non-stationary heat conduction problems. Int. J. Heat Mass Transf. 71, 720–723 (2014)

    Article  Google Scholar 

  24. Reséndiz-Flores, E.O.; García-Calvillo, I.D.: Numerical solution of 3D non-stationary heat conduction problems using the finite pointset method. Int. J. Heat Mass Transf. 87, 104–110 (2015)

    Article  Google Scholar 

  25. Saucedo-Zendejo, F.R.; Reséndiz-Flores, E.O.: Transient heat transfer and solidification modeling in direct-chill casting using a generalized finite differences method. J. Min. Metall. Sect. B-Metall. 55, 47–54 (2019)

    Article  Google Scholar 

  26. Tiwari, S.; Kuhner, J.: Finite pointset method based on the projection method for simulations of the incompressible Navier-Stokes equations. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations, pp. 373–387. Springer, Berlin (2003)

    Chapter  Google Scholar 

  27. Reséndiz-Flores, E.O.; Saucedo-Zendejo, F.R.: Meshfree numerical simulation of free surface thermal flows in mould filling processes using the Finite Pointset Method. Int. J. Therm. Sci. 127, 29–40 (2018)

    Article  Google Scholar 

  28. Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics 4, 389–396 (1995)

    Article  MathSciNet  Google Scholar 

  29. Jin-Lian, R.; Ren Heng-Fei, L.; Wei-Gang, J.T.: Simulation of two-dimensional nonlinear problem with solitary wave based on split-step finite pointset method. Acta Physica Sinica 68(14), 140203 (2019)

    Article  Google Scholar 

  30. Fatehi, R.; Manzari, M.T.: A consistent and fast weakly compressible smoothed particle hydrodynamics with a new wall boundary condition. Int. J. Numer. Methods Fluids 68(7), 905–921 (2012)

    Article  MathSciNet  Google Scholar 

  31. Jiang, T.; Chen, Z.-C.; Ren, J.-L.; Li, G.: Simulation of three-dimensional transient heat conduction problem with variable coefficients based on the improved parallel smoothed particle hydrodynamics method. Acta Physica Sinica 66, 130201 (2017)

    Article  Google Scholar 

  32. Jiang, T.; Chen, Z.; Weigang, L.; Yuan, J.; Wang, D.: An efficient split-step and implicit pure mesh-free method for the 2D/3D nonlinear Gross-Pitaevskii equations. Comput. Phys. Commun. 231, 19–30 (2018)

    Article  MathSciNet  Google Scholar 

  33. Barton, M.L.; Withers, G.R.: Computing performance as a function of the speed, quantity, and cost of the processors. In: 1989 ACM/IEEE Conference on Supercomputing, pp. 759–764.

  34. Qu, W.; Fan, C.M.; Zhang, Y.M.: Analysis of three-dimensional heat conduction in functionally graded materials by using a hybrid numerical method. Int. J. Heat Mass Transf. 145, 118771 (2019)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank Prof. George Em Karniadarkis at Brown University and acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 11501495, 51779215, 11672259), the Postdoctoral Science Foundation of China (Grant Nos. 2014M550310, 2015M581869, 2015T80589), the Natural Science Foundation of Jiangsu Province (Grant No. BK20150436), the Jiangsu Government Scholarship for Overseas Studies (Grant No. JS-2017-227), the Top-notch Academic Programs Project of Jiangsu High Education Institutions (Grant No. PPZY2015B109), the Foundation of the Developing Center of INCTMat at Federal University of Parana, Brazil (Grant No. 465591/2014-0), and the Science and Technology Innovation Cultivation Fund of Yangzhou University (Grant No. 2019CXJ003).

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Authors and Affiliations

  1. School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu, China

    Jinlian Ren, Kang Xu, Hengfei Ren & Tao Jiang

  2. Applied Mathematics, Brown University, Providence, USA

    Jinlian Ren

  3. Departmento de Matemática, Universidade Federal do Paraná, Centro Politécnico, Curitiba, Brazil

    Jinyun Yuan

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  1. Jinlian Ren
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  2. Kang Xu
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  3. Hengfei Ren
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  4. Tao Jiang
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Correspondence to Tao Jiang.

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Ren, J., Xu, K., Ren, H. et al. Numerical Study of the 3D Variable Coefficient Heat Transfer Problem by Using the Finite Pointset Method. Arab J Sci Eng 46, 3483–3502 (2021). https://doi.org/10.1007/s13369-020-05139-5

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  • DOI: https://doi.org/10.1007/s13369-020-05139-5

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