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A technique to remove the tensile instability in weakly compressible SPH

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Abstract

When smoothed particle hydrodynamics (SPH) is directly applied for the numerical simulations of transient viscoelastic free surface flows, a numerical problem called tensile instability arises. In this paper, we develop an optimized particle shifting technique to remove the tensile instability in SPH. The basic equations governing free surface flow of an Oldroyd-B fluid are considered, and approximated by an improved SPH scheme. This includes the implementations of the correction of kernel gradient and the introduction of Rusanov flux into the continuity equation. To verify the effectiveness of the optimized particle shifting technique in removing the tensile instability, the impacting drop, the injection molding of a C-shaped cavity, and the extrudate swell, are conducted. The numerical results obtained are compared with those simulated by other numerical methods. A comparison among different numerical techniques (e.g., the artificial stress) to remove the tensile instability is further performed. All numerical results agree well with the available data.

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References

  1. Hirt CW, Nicholls BD (1981) Volume of fluid (VOF) method for dynamics of free boundaries. J Comput Phys 39:201–221

    Article  Google Scholar 

  2. McKee S, Tomé MF, Ferreira VG, Cuminato JA, Castelo A, Sousa FS, Mangiavacchi N (2008) The MAC method. Comput Fluids 37:907–930

    Article  MathSciNet  Google Scholar 

  3. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton Jacobi formulations. J Comput Phys 79:12–49

    Article  MathSciNet  Google Scholar 

  4. Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55:1–34

    Article  Google Scholar 

  5. Li S, Liu WK (2004) Meshfree particle methods. Springer, Berlin

    MATH  Google Scholar 

  6. Liu MB, Liu GR (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch Comput Methods Eng 17:25–76

    Article  MathSciNet  Google Scholar 

  7. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–389

    Article  Google Scholar 

  8. Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 83:1013–1024

    Article  Google Scholar 

  9. Shadloo MS, Zainali A, Sadek SH, Yildiz M (2011) Improved incompressible smoothed particle hydrodynamics method for simulating flow around bluff bodies. Comput Methods Appl Mech Eng 200:1008–1020

    Article  MathSciNet  Google Scholar 

  10. Shadloo MS, Zainali A, Yildiz M, Suleman A (2012) A robust weakly compressible SPH method and its comparison with an incompressible SPH. Int J Numer Methods Eng 89:939–956

    Article  MathSciNet  Google Scholar 

  11. Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110:399–406

    Article  Google Scholar 

  12. Ozbulut M, Yildiz M, Goren O (2014) A numerical investigation into the correction algorithms for SPH method in modeling violent free surface flows. Int J Mech Sci 79:56–65

    Article  Google Scholar 

  13. Xu X, Ouyang J, Jiang T, Li Q (2014) Numerical analysis of the impact of two droplets with a liquid film using an incompressible SPH method. J Eng Math 85:35–53

    Article  MathSciNet  Google Scholar 

  14. Xu X, Deng XL (2016) An improved weakly compressible SPH method for simulating free surface flows of viscous and viscoelastic fluids. Comput Phys Commun 201:43–62

    Article  MathSciNet  Google Scholar 

  15. Monaghan JJ, Kocharyan A (1995) SPH simulation of multi-phase flow. Comput Phys Commun 87:225–235

    Article  Google Scholar 

  16. Zainali A, Tofighi N, Shadloo MS, Yildiz M (2013) Numerical investigation of Newtonian and non-Newtonian multiphase flows using ISPH method. Comput Methods Appl Mech Eng 254:99–113

    Article  MathSciNet  Google Scholar 

  17. Lind SJ, Stansby PK, Rogers BD (2016) Incompressible-compressible flows with a transient discontinuous interface using smoothed particle hydrodynamics (SPH). J Comput Phys 309:129–147

    Article  MathSciNet  Google Scholar 

  18. Tartakovskya AM, Panchenko A (2016) Pairwise force smoothed particle hydrodynamics model for multiphase flow: surface tension and contact line dynamics. J Comput Phys 305:1119–1146

    Article  MathSciNet  Google Scholar 

  19. Ellero M, Kroger M, Hess S (2002) Viscoelastic flows studied by smoothed particle dynamics. J Non-Newton Fluid Mech 105:35–51

    Article  Google Scholar 

  20. Ellero M, Tanner RI (2005) SPH simulations of transient viscoelastic flows at low Reynolds number. J Non-Newton Fluid Mech 132:61–72

    Article  Google Scholar 

  21. Xu X, Ouyang J, Jiang T, Li Q (2012) Numerical simulation of 3D-unsteady viscoelastic free surface flows by improved smoothed particle hydrodynamics method. J Non-Newton Fluid Mech 177:109–120

    Article  Google Scholar 

  22. Xu X, Ouyang J (2013) A SPH-based particle method for simulating 3D transient free surface flows of branched polymer melts. J Non-Newton Fluid Mech 202:54–71

    Article  Google Scholar 

  23. Xu X, Ouyang J, Liu Q, Li W (2014) SPH simulations of 2D transient viscoelastic flows using Brownian configuration fields. J Non-Newton Fluid Mech 208:59–71

    Article  Google Scholar 

  24. Xu X, Yu P (2016) A multiscale SPH method for simulating transient viscoelastic flows using bead-spring chain model. J Non-Newton Fluid Mech 229:27–42

    Article  MathSciNet  Google Scholar 

  25. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluid 20:1081–1106

    Article  MathSciNet  Google Scholar 

  26. Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity, part I—formulation and theory. Int J Numer Methods Eng 45:251–288

    Article  Google Scholar 

  27. Li S, Liu WK (1999) Reproducing kernel hierarchical partition of unity, part II—applications. Int J Numer Methods Eng 45:289–317

    Article  Google Scholar 

  28. Chen JK, Beraun JE, Jih CJ (1999) An improvement for tensile instability in smoothed particle hydrodynamics. Comput Mech 23:279–287

    Article  Google Scholar 

  29. Chen JK, Beraun JE (2000) A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems. Comput Methods Appl Mech Eng 190:225–239

    Article  Google Scholar 

  30. Bonet J, Lok T-SL (1999) Variational and momentum preservation aspects of smooth particle hydrodynamic formulations. Comput Methods Appl Mech Eng 180:97–115

    Article  MathSciNet  Google Scholar 

  31. Liu MB, Xie WP, Liu GR (2005) Modeling incompressible flows using a finite particle method. Appl Math Model 29:1252–1270

    Article  Google Scholar 

  32. Liu MB, Liu GR (2006) Restoring particle consistency in smoothed particle hydrodynamics. Appl Numer Math 56:19–36

    Article  MathSciNet  Google Scholar 

  33. Zhang GM, Batra RC (2004) Modified smoothed particle hydrodynamics method and its applications to transient problems. Comput Mech 34:137–146

    MATH  Google Scholar 

  34. Zhang GM, Batra RC (2007) Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (MSPH) method. J Comput Phys 222:374–390

    Article  MathSciNet  Google Scholar 

  35. Batra RC, Zhang GM (2007) Search algorithm and simulation of elastodynamic crack propagation by modified smoothed particle hydrodynamics (MSPH) method. Comput Mech 40:531–546

    Article  Google Scholar 

  36. Fang J, Parriaux A, Rentschler M, Ancey C (2009) Improved SPH methods for simulating free surface flows of viscous fluids. Appl Numer Math 59(2):251–271

    Article  MathSciNet  Google Scholar 

  37. Xu R, Stansby PK, Laurence D (2009) Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach. J Comput Phys 228(18):6703–6725

    Article  MathSciNet  Google Scholar 

  38. Lind SJ, Xu R, Stansby PK, Rogers BD (2012) Incompressible smoothed particle hydrodynamics for free-surface flows: a generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves. J Comput Phys 231(4):1499–1523

    Article  MathSciNet  Google Scholar 

  39. Skillen A, Lind S, Stansby PK, Rogers BD (2013) Incompressible smoothed particle hydrodynamics (SPH) with reduced temporal noise and generalised Fickian smoothing applied to body-water slam and efficient wave-body interaction. Comput Methods Appl Mech Eng 265:163–173

    Article  MathSciNet  Google Scholar 

  40. Xenakis AM, Lind SJ, Stansby PK, Rogers BD (2015) An incompressible SPH scheme with improved pressure predictions for free-surface generalised Newtonian flows. J Non-Newton Fluid Mech 218:1–15

    Article  MathSciNet  Google Scholar 

  41. Vila JP (1999) On particle weighted methods and smooth particle hydrodynamics. Math Model Methods Appl Sci 9:161–209

    Article  MathSciNet  Google Scholar 

  42. Ferrari A, Dumbser M, Toro EF, Armanini A (2009) A new 3D parallel SPH scheme for free-surface flows. Comput Fluids 38:1203–1217

    Article  MathSciNet  Google Scholar 

  43. Cherfils JM, Pinon G, Rivoalen E (2012) JOSEPHINE: a parallel SPH code for free surface flows. Comput Phys Commun 183:1468–1480

    Article  MathSciNet  Google Scholar 

  44. Cha SH, Inutsuka S, Nayakshin S (2010) Kelvin–Helmholtz instabilities with Godunov smoothed particle hydrodynamics. Mon Not R Astron Soc 403:1165–1174

    Article  Google Scholar 

  45. Sugiura K, Inutsuka S (2016) An extension of Godunov SPH: application to negative pressure media. J Comput Phys 308:171–197

    Article  MathSciNet  Google Scholar 

  46. Sugiura K, Inutsuka S (2017) An extension of Godunov SPH II: application to elastic dynamics. J Comput Phys 333:78–103

    Article  MathSciNet  Google Scholar 

  47. Molteni D, Colagrossi A (2009) A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH. Comput Phys Commun 180:861–872

    Article  MathSciNet  Google Scholar 

  48. Antuono M, Colagrossi A, Marrone S, Molteni D (2010) Free-surface flows solved by means of SPH schemes with numerical diffusive terms. Comput Phys Commun 181:532–549

    Article  MathSciNet  Google Scholar 

  49. Marrone S, Antuono M, Colagrossi A, Colicchio G, Le Touzé D, Graziani G (2011) \(\delta \)-SPH model for simulating violent impact flows. Comput Methods Appl Mech Eng 200:1526–1542

    Article  MathSciNet  Google Scholar 

  50. Antuono M, Marrone S, Colagrossi A, Bouscasse B (2015) Energy balance in the \(\delta \)-SPH scheme. Comput Methods Appl Mech Eng 289:209–226

    Article  MathSciNet  Google Scholar 

  51. Sun PN, Colagrossi A, Marrone S et al (2017) The \(\delta \)plus-SPH model: simple procedures for a further improvement of the SPH scheme. Comput Methods Appl Mech Eng 315:25–49

    Article  MathSciNet  Google Scholar 

  52. Schuessler I, Schmitt D (1981) Comments on smoothed particle hydrodynamics. Astron Astrophys 97:373–379

    Google Scholar 

  53. Swegle JW, Hicks DL, Attaway SW (1995) Smoothed particle hydrodynamics stability analysis. J Comput Phys 116:123–134

    Article  MathSciNet  Google Scholar 

  54. Dyka CT, Ingel RP (1995) An approach for tension instability in smoothed particle hydrodynamics (SPH). Comput Struct 57:573–580

    Article  Google Scholar 

  55. Dyka CT, Randles PW, Ingel RP (1997) Stress points for tension instability in SPH. Int J Numer Methods Eng 40:2325–2341

    Article  Google Scholar 

  56. Hicks DL, Liebrock LM (1999) SPH hydrocodes can be stabilized with shape-shifting. Comput Math Appl 38:1–16

    Article  MathSciNet  Google Scholar 

  57. Monaghan JJ (2000) SPH without a tensile instability. J Comput Phys 159:290–311

    Article  Google Scholar 

  58. Gray JP, Monaghan JJ, Swift RP (2001) SPH elastic dynamics. Comput Methods Appl Mech Eng 190:6641–6662

    Article  Google Scholar 

  59. Fang J, Owens RG, Tacher L, Parriaux A (2006) A numerical study of the SPH method for simulating transient viscoelastic free surface flows. J Non-Newton Fluid Mech 139:68–84

    Article  Google Scholar 

  60. Nestor RM, Basa M, Lastiwka M et al (2009) Extension of the finite volume particle method to viscous flow. J Comput Phys 228(5):1733–1749

    Article  MathSciNet  Google Scholar 

  61. Xu X, Ouyang J, Yang B, Liu Z (2013) SPH simulations of three-dimensional non-Newtonian free surface flows. Comput Methods Appl Mech Eng 256:101–116

    Article  MathSciNet  Google Scholar 

  62. Khayyer A, Gotoh H, Shimizu Y, Gotoh K (2016) Comparative study on accuracy and conservation properties of particle regularization schemes and proposal of an improved particle shifting scheme. In: Proceedings of the 11th international SPHERIC workshop, pp 416–423

  63. Marrone S, Colagrossi A, Le Touz’e D, Graziani G (2010) Fast free-surface detection and level-set function definition in SPH solvers. J Comput Phys 229:3652–3663

    Article  Google Scholar 

  64. Randles PW, Libersky LD (1996) Smoothed particle hydrodynamics: some recent improvements and applications. Comput Methods Appl Mech Eng 139:375–408

    Article  MathSciNet  Google Scholar 

  65. Oishi CM, Martins FP, Tomé MF, Alves MA (2012) Numerical simulation of drop impact and jet buckling problems using the eXtended Pom–Pom model. J Non-Newton Fluid Mech 169:91–103

    Article  Google Scholar 

  66. Keunings R (1986) On the high Weissenberg number problem. J Non-Newton Fluid Mech 20:209–226

    Article  Google Scholar 

  67. Tanner RI (1970) A theory of die swell. J Polym Sci 8:2067–2078

    Google Scholar 

  68. Crochet MJ, Keunings R (1982) Finite element analysis of die swell of a highly elastic fluid. J Non-Newton Fluid Mech 10:339–356

    Article  Google Scholar 

  69. Tomé MF, Mangiavacchi N, Cuminato JA, Castelo A, McKee S (2002) A finite difference technique for simulating unsteady viscoelastic free surface flows. J Non-Newton Fluid Mech 106:61–106

    Article  Google Scholar 

  70. Sadek SH, Yildiz M (2013) Modeling die swell of second-order fluids using smoothed particle hydrodynamics. J Fluids Eng 135:051103

    Article  Google Scholar 

  71. Han C (ed) (2012) Multiphase flow in polymer processing. Elsevier, Amsterdam

    Google Scholar 

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Acknowledgements

This work is financially supported by National Natural Science Foundation of China (No. 11502132), Young Talent Fund of University Association for Science and Technology in Shaanxi, China (No. 20160127), and Scientific Research Program Funded by Shaanxi Provincial Education Department (No. 17JK0160).

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong, 723000, China

    Xiaoyang Xu

  2. Department of Mechanics and Aerospace Engineering, South University of Science and Technology of China, Shenzhen, 518055, China

    Peng Yu

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  1. Xiaoyang Xu
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  2. Peng Yu
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Correspondence to Xiaoyang Xu.

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Xu, X., Yu, P. A technique to remove the tensile instability in weakly compressible SPH. Comput Mech 62, 963–990 (2018). https://doi.org/10.1007/s00466-018-1542-4

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