A unified second-order accurate in time MPM formulation for simulating viscoelastic liquids with phase change

Abstract

We assume that the viscous forces in any liquid are simultaneously local and non-local, and introduce the extended POM-POM model [McLeish and Larson 1998; Oishi et al. 2012; Verbeeten et al. 2001] to computer graphics to design a unified constitutive model for viscosity that generalizes prior models, such as Oldroyd-B, the Upper-convected Maxwell (UCM) model [Sadeghy et al. 2005], and classical Newtonian viscosity under one umbrella, recovering each of them with different parameter values. Implicit discretization of our model via backward Euler recovers the variational Stokes solver of [Larionov et al. 2017] for Newtonian viscosity. For greater accuracy, however, we introduce the second-order accurate Generalized Single Step Single Solve (GS4) scheme [Tamma et al. 2000; Zhou and Tamma 2004] to computer graphics, which recovers all prior second-order accurate time integration schemes to date. Using GS4 and our generalized constitutive model, we present a Material Point Method (MPM) for simulating various viscoelastic liquid behaviors, such as classical liquid rope coiling, buckling, folding, and shear thinning/thickening. In addition, we show how to couple our viscoelastic liquid simulator with the recently introduced non-Fourier heat diffusion solver [Xue et al. 2020] for simulating problems with phase change, such as melting chocolate and digital fabrication with 3D printing. While the discretization of heat diffusion is slightly different within GS4, we show that it can still be efficiently solved using an assembly-free Multigrid-preconditioned Conjugate Gradients solver. We present end-to-end 3D simulations to demonstrate the versatility of our framework.

References

1. Mridul Aanjaneya, Chengguizi Han, Ryan Goldade, and Christopher Batty. 2019. An Efficient Geometric Multigrid Solver for Viscous Liquids. Proceedings of the ACM on Computer Graphics and Interactive Techniques 2, 2 (2019), 1--21.
2. Stefan Band, Christoph Gissler, Andreas Peer, and Matthias Teschner. 2018. MLS Pressure Extrapolation for the Boundary Handling in Divergence-Free SPH. In Proceedings of the 14th Workshop on Virtual Reality Interactions and Physical Simulations (VRIPHYS '18). Eurographics Association, 55--63.
3. Héctor Barreiro, Ignacio García-Fernández, Iván Alduán, and Miguel A. Otaduy. 2017. Conformation Constraints for Efficient Viscoelastic Fluid Simulation. ACM Trans. Graph. 36, 6, Article 221 (2017), 11 pages.
4. Cx K Batchelor and GK Batchelor. 2000. An introduction to fluid dynamics. Cambridge university press.
5. Christopher Batty and Robert Bridson. 2008. Accurate Viscous Free Surfaces for Buckling, Coiling, and Rotating Liquids.. In Symposium on Computer Animation. 219--228.
6. Christopher Batty and Ben Houston. 2011. A simple finite volume method for adaptive viscous liquids. In Proceedings of the 2011 ACM SIGGRAPH/Eurographics Symposium on Computer Animation. 111--118.
7. Christopher Batty, Andres Uribe, Basile Audoly, and Eitan Grinspun. 2012. Discrete viscous sheets. ACM Transactions on Graphics (TOG) 31, 4 (2012), 1--7.
8. Jan Bender and Dan Koschier. 2017. Divergence-Free SPH for Incompressible and Viscous Fluids. IEEE Transactions on Visualization and Computer Graphics 23, 3 (2017), 1193--1206.
9. J. Bender, Tassilo Kugelstadt, M. Weiler, and Dan Koschier. 2020. Implicit Frictional Boundary Handling for SPH. IEEE Transactions on Visualization and Computer Graphics 26 (2020), 2982--2993.
10. Miklós Bergou, Basile Audoly, Etienne Vouga, Max Wardetzky, and Eitan Grinspun. 2010. Discrete Viscous Threads. ACM Trans. Graph. 29, 4, Article 116 (2010), 10 pages.
11. Volodymyr Valeriyovych Bilovol. 2003. Mould filling simulations during powder injection moulding. (2003).
12. G Bishko, TCB McLeish, OG Harlen, and RG Larson. 1997. Theoretical molecular rheology of branched polymers in simple and complex flows: The pom-pom model. Physical Review Letters 79, 12 (1997), 2352.
13. Javier Bonet and Richard D Wood. 1997. Nonlinear continuum mechanics for finite element analysis. Cambridge university press.
14. EK Borisenkova, VE Dreval, GV Vinogradov, MK Kurbanaliev, VV Moiseyev, and VG Shalganova. 1982. Transition of polymers from the fluid to the forced high-elastic and leathery states at temperatures above the glass transition temperature. Polymer 23, 1 (1982), 91--99.
15. Robert Bridson. 2015. Fluid simulation for computer graphics. CRC Press.
16. George E Brown, Matthew Overby, Zahra Forootaninia, and Rahul Narain. 2018. Accurate dissipative forces in optimization integrators. ACM Transactions on Graphics (TOG) 37, 6 (2018), 1--14.
17. Rachel Caiden, Ronald P Fedkiw, and Chris Anderson. 2001. A numerical method for two-phase flow consisting of separate compressible and incompressible regions. J. Comput. Phys. 166, 1 (2001), 1--27.
18. Mark Carlson, Peter J Mucha, R Brooks Van Horn III, and Greg Turk. 2002. Melting and flowing. In Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation. 167--174.
19. Carlo Cattaneo. 1948. Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3 (1948), 83--101.
20. CI Christov. 2009. On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction. Mechanics Research Communications 36, 4 (2009), 481--486.
21. Pascal Clausen, Martin Wicke, Jonathan R Shewchuk, and James F O'brien. 2013. Simulating liquids and solid-liquid interactions with lagrangian meshes. ACM Transactions on Graphics (TOG) 32, 2 (2013), 1--15.
22. Nicolas Clemeur and Benoit Debbaut. 2007. A pragmatic approach for deriving constitutive equations endowed with pom-pom attributes. Rheologica acta 46, 9 (2007), 1187--1196.
23. Mengyuan Ding, Xuchen Han, Stephanie Wang, Theodore F Gast, and Joseph M Teran. 2019. A thermomechanical material point method for baking and cooking. ACM Transactions on Graphics (TOG) 38, 6 (2019), 192.
24. A Cemal Eringen. 1992. Vistas of nonlocal continuum physics. International journal of engineering science 30, 10 (1992), 1551--1565.
25. A Cemal Eringen and DGB Edelen. 1972. On nonlocal elasticity. International journal of engineering science 10, 3 (1972), 233--248.
26. Yu Fang, Minchen Li, Ming Gao, and Chenfanfu Jiang. 2019. Silly rubber: an implicit material point method for simulating non-equilibrated viscoelastic and elastoplastic solids. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1--13.
27. Adolph Fick. 1855. V. On liquid diffusion. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 10, 63 (1855), 30--39.
28. Wilhelm Flügge. 2013. Viscoelasticity. Springer Science & Business Media.
29. Jean-Baptiste Joseph Fourier. 1878. Théorie analytique de la chaleur. Paris 1822. Engl. Übersetzung:"Theory o: Heat Transfer", Cambridge (1878).
30. Albrecht Fröhlich and R Sack. 1946. Theory of the rheological properties of dispersions. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 185, 1003 (1946), 415--430.
31. Sumit Gajjar, Jaydeep Bhadani, Pramit Dutta, and Naveen Rastogi. 2017. Complete coverage path planning algorithm for known 2d environment. In 2017 2nd IEEE International Conference on Recent Trends in Electronics, Information & Communication Technology (RTEICT). IEEE, 963--967.
32. Ming Gao, Xinlei Wang, Kui Wu, Andre Pradhana, Eftychios Sifakis, Cem Yuksel, and Chenfanfu Jiang. 2018. GPU optimization of material point methods. ACM Transactions on Graphics (TOG) 37, 6 (2018), 1--12.
33. Tolga G Goktekin, Adam W Bargteil, and James F O'Brien. 2004. A method for animating viscoelastic fluids. In ACM SIGGRAPH 2004 Papers. 463--468.
34. Ryan Goldade, Yipeng Wang, Mridul Aanjaneya, and Christopher Batty. 2019. An adaptive variational finite difference framework for efficient symmetric octree viscosity. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1--14.
35. Mehdi Habibi, Maniya Maleki, Ramin Golestanian, Neil M Ribe, and Daniel Bonn. 2006. Dynamics of liquid rope coiling. Physical Review E 74, 6 (2006), 066306.
36. Francis H. Harlow and J. Eddie Welch. 1965. Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface. Physics of Fluids 8, 12 (1965), 2182--2189.
37. Kei Iwasaki, Hideyuki Uchida, Yoshinori Dobashi, and Tomoyuki Nishita. 2010. Fast particle-based visual simulation of ice melting. In Computer graphics forum, Vol. 29. Wiley Online Library, 2215--2223.
38. Daniel D Joseph and Luigi Preziosi. 1989. Heat waves. Reviews of Modern Physics 61, 1 (1989), 41.
39. J Murali Krishnan, Abhijit P Deshpande, and PB Sunil Kumar. 2010. Rheology of complex fluids. Springer.
40. E Kröner. 1967. Elasticity theory of materials with long range cohesive forces. International Journal of Solids and Structures 3, 5 (1967), 731--742.
41. Egor Larionov, Christopher Batty, and Robert Bridson. 2017. Variational stokes: a unified pressure-viscosity solver for accurate viscous liquids. ACM Transactions on Graphics (TOG) 36, 4 (2017), 1--11.
42. Ronald G Larson. 1999. The structure and rheology of complex fluids. Vol. 150. Oxford university press New York.
43. C-Y David Lu, Peter D Olmsted, and RC Ball. 2000. Effects of non-local stress on the determination of shear banding flow. Physical Review Letters 84, 4 (2000), 642.
44. A. McAdams, E. Sifakis, and J. Teran. 2010. A Parallel Multigrid Poisson Solver for Fluids Simulation on Large Grids. In Proceedings of the 2010 ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA '10). 65--74.
45. TCB McLeish and RG Larson. 1998. Molecular constitutive equations for a class of branched polymers: The pom-pom polymer. Journal of Rheology 42, 1 (1998), 81--110.
46. Matthias Müller, Simon Schirm, and Matthias Teschner. 2004. Interactive blood simulation for virtual surgery based on smoothed particle hydrodynamics. Technology and Health Care 12, 1 (2004), 25--31.
47. Meraj Mustafa. 2015. Cattaneo-Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid. Aip Advances 5, 4 (2015), 047109.
48. Kentaro Nagasawa, Takayuki Suzuki, Ryohei Seto, Masato Okada, and Yonghao Yue. 2019. Mixing sauces: A viscosity blending model for shear thinning fluids. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1--17.
49. James F O'brien and Jessica K Hodgins. 1999. Graphical modeling and animation of brittle fracture. In Proceedings of the 26th annual conference on Computer graphics and interactive techniques. 137--146.
50. Raymond W Ogden. 1997. Non-linear elastic deformations. Courier Corporation.
51. CM Oishi, FP Martins, MF Tomé, and MA Alves. 2012. Numerical simulation of drop impact and jet buckling problems using the eXtended Pom-Pom model. Journal of Non-Newtonian Fluid Mechanics 169 (2012), 91--103.
52. CM Oishi, FP Martins, Murilo F Tome, JA Cuminato, and Sean Mckee. 2011. Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows. Journal of Non-Newtonian Fluid Mechanics 166, 3-4 (2011), 165--179.
53. James G Oldroyd. 1950. On the formulation of rheological equations of state. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 200, 1063 (1950), 523--541.
54. Afonso Paiva, Fabiano Petronetto, Thomas Lewiner, and Geovan Tavares. 2009. Particle-based viscoplastic fluid/solid simulation. Computer-Aided Design 41, 4 (2009), 306--314.
55. Peyman Pakdel and Gareth H McKinley. 1996. Elastic instability and curved streamlines. Physical Review Letters 77, 12 (1996), 2459.
56. Andreas Peer, Markus Ihmsen, Jens Cornelis, and Matthias Teschner. 2015. An implicit viscosity formulation for SPH fluids. ACM Transactions on Graphics (TOG) 34, 4 (2015), 1--10.
57. Daniel Ram, Theodore Gast, Chenfanfu Jiang, Craig Schroeder, Alexey Stomakhin, Joseph Teran, and Pirouz Kavehpour. 2015. A material point method for viscoelastic fluids, foams and sponges. In Proceedings of the 14th ACM SIGGRAPH/Eurographics Symposium on Computer Animation. 157--163.
58. Kayvan Sadeghy, Amir-Hosain Najafi, and Meghdad Saffaripour. 2005. Sakiadis flow of an upper-convected Maxwell fluid. International Journal of Non-Linear Mechanics 40, 9 (2005), 1220--1228.
59. Mohamed Shaat, Esmaeal Ghavanloo, and S Ahmad Fazelzadeh. 2020. Review on nonlocal continuum mechanics: physics, material applicability, and mathematics. Mechanics of Materials (2020), 103587.
60. M Shimada and Kumar K Tamma. 2012. Conserving/dissipative algorithms and designs for a system of N particles: Total energy framework and single-field form. Computers & structures 112 (2012), 380--405.
61. SL Sobolev. 2014. Nonlocal diffusion models: Application to rapid solidification of binary mixtures. International Journal of Heat and Mass Transfer 71 (2014), 295--302.
62. Arun R Srinivasa and JN Reddy. 2017. An overview of theories of continuum mechanics with nonlocal elastic response and a general framework for conservative and dissipative systems. Applied Mechanics Reviews 69, 3 (2017).
63. Jos Stam. 1999. Stable Fluids. In Proc. of ACM SIGGRAPH (SIGGRAPH '99). 121--128.
64. Alexey Stomakhin, Craig Schroeder, Lawrence Chai, Joseph Teran, and Andrew Selle. 2013. A material point method for snow simulation. ACM Transactions on Graphics (TOG) 32, 4 (2013), 1--10.
65. Alexey Stomakhin, Craig Schroeder, Chenfanfu Jiang, Lawrence Chai, Joseph Teran, and Andrew Selle. 2014. Augmented MPM for phase-change and varied materials. ACM Transactions on Graphics (TOG) 33, 4 (2014), 1--11.
66. Dan Stora, Pierre-Olivier Agliati, Marie-Paule Cani, Fabrice Neyret, and Jean-Dominique Gascuel. 1999. Animating lava flows.
67. Tetsuya Takahashi and Christopher Batty. 2020. Monolith: A Monolithic Pressure-Viscosity-Contact Solver for Strong Two-Way Rigid-Rigid Rigid-Fluid Coupling. ACM Trans. Graph. 39, 6, Article 182 (2020).
68. Tetsuya Takahashi, Yoshinori Dobashi, Issei Fujishiro, Tomoyuki Nishita, and Ming C Lin. 2015. Implicit formulation for SPH-based viscous fluids. In Computer Graphics Forum, Vol. 34. Wiley Online Library, 493--502.
69. T. Takahashi and M. Lin. 2019. A Geometrically Consistent Viscous Fluid Solver with Two-Way Fluid-Solid Coupling. Computer Graphics Forum 38 (2019).
70. Kumar K Tamma, X Zhou, and D Sha. 2000. The time dimension: a theory towards the evolution, classification, characterization and design of computational algorithms for transient/dynamic applications. Archives of Computational Methods in Engineering 7, 2 (2000), 67--290.
71. RI Tanner. 1970. A theory of die-swell. Journal of Polymer Science Part A-2: Polymer Physics 8, 12 (1970), 2067--2078.
72. Demetri Terzopoulos and Kurt Fleischer. 1988. Modeling inelastic deformation: viscolelasticity, plasticity, fracture. In Proceedings of the 15th annual conference on Computer graphics and interactive techniques. 269--278.
73. Ulrich Trottenberg, Cornelius W. Oosterlee, and Anton Schuller. 2001. Multigrid. Academic Press.
74. BM Tymrak, Megan Kreiger, and Joshua M Pearce. 2014. Mechanical properties of components fabricated with open-source 3-D printers under realistic environmental conditions. Materials & Design 58 (2014), 242--246.
75. Da Yu Tzou. 2014. Macro-to microscale heat transfer: the lagging behavior. John Wiley & Sons.
76. Wilco MH Verbeeten, Gerrit WM Peters, and Frank PT Baaijens. 2001. Differential constitutive equations for polymer melts: The extended Pom-Pom model. Journal of Rheology 45, 4 (2001), 823--843.
77. Vaughan R Voller and CR Swaminathan. 1991. ERAL Source-based method for solidification phase change. Numerical Heat Transfer, Part B Fundamentals 19, 2 (1991), 175--189.
78. Marcel Weiler, Dan Koschier, Magnus Brand, and Jan Bender. 2018. A Physically Consistent Implicit Viscosity Solver for SPH Fluids. Computer Graphics Forum (Eurographics) 37, 2 (2018).
79. Chris Wojtan and Greg Turk. 2008. Fast viscoelastic behavior with thin features. In ACM SIGGRAPH 2008 papers. 1--8.
80. Tao Xue, Haozhe Su, Chengguizi Han, Chenfanfu Jiang, and Mridul Aanjaneya. 2020. A novel discretization and numerical solver for non-fourier diffusion. ACM Transactions on Graphics (TOG) 39, 6 (2020), 1--14.
81. Tao Xue, Kumar K Tamma, and Xiaobing Zhang. 2016. A consistent moving particle system simulation method: applications to parabolic/hyperbolic heat conduction type problems. International Journal of Heat and Mass Transfer 101 (2016), 365--372.
82. Tao Xue, Xiaobing Zhang, and Kumar K Tamma. 2018. Generalized heat conduction model involving imperfect thermal contact surface: application of the GSSSS-1 differential-algebraic equation time integration. International Journal of Heat and Mass Transfer 116 (2018), 889--896.
83. Yonghao Yue, Breannan Smith, Christopher Batty, Changxi Zheng, and Eitan Grinspun. 2015. Continuum foam: A material point method for shear-dependent flows. ACM Transactions on Graphics (TOG) 34, 5 (2015), 1--20.
84. X Zhou and Kumar K Tamma. 2004. Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics. Internat. J. Numer. Methods Engrg. 59, 5 (2004), 597--668.
85. Bo Zhu, Minjae Lee, Ed Quigley, and Ronald Fedkiw. 2015. Codimensional non-Newtonian fluids. ACM Transactions on Graphics (TOG) 34, 4 (2015), 1--9.

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation

      • Physical simulation