Energetically consistent inelasticity for optimization time integration

Abstract

In this paper, we propose Energetically Consistent Inelasticity (ECI), a new formulation for modeling and discretizing finite strain elastoplasticity/viscoelasticity in a way that is compatible with optimization-based time integrators. We provide an in-depth analysis for allowing plasticity to be implicitly integrated through an augmented strain energy density function. We develop ECI on the associative von-Mises J2 plasticity, the non-associative Drucker-Prager plasticity, and the finite strain viscoelasticity. We demonstrate the resulting scheme on both the Finite Element Method (FEM) and the Material Point Method (MPM). Combined with a custom Newton-type optimization integration scheme, our method enables simulating stiff and large-deformation inelastic dynamics of metal, sand, snow, and foam with larger time steps, improved stability, higher efficiency, and better accuracy than existing approaches.

References

1. Iván Alduán and Miguel A Otaduy. 2011. SPH granular flow with friction and cohesion. In Proceedings of the 2011 Symp. Computer animation. 25--32.
2. A. W. Bargteil, J. K. Hodgins C. Wojtan, and G. Turk. 2007. A finite element method for animating large viscoplastic flow. ACM Trans. on Graph. 26, 3 (2007).
3. Christopher Batty, Florence Bertails, and Robert Bridson. 2007. A fast variational framework for accurate solid-fluid coupling. ACM Trans. on Graph. 26, 3 (2007).
4. Jan Bender, Matthias Müller, and Miles Macklin. 2017. A survey on position based dynamics, 2017. In European Association for Computer Graphics: Tutorials. 1--31.
5. Javier Bonet and Richard D Wood. 1997. Nonlinear continuum mechanics for finite element analysis. Cambridge university press.
6. S. Bouaziz, S. Martin, T. Liu, L. Kavan, and M. Pauly. 2014. Projective dynamics: Fusing constraint projections for fast simulation. ACM Trans. on Graph. 33, 4 (2014).
7. J. U. Brackbill and H. M. Ruppel. 1986. FLIP: A method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions. J. Comput. Phys. 65, 2 (1986).
8. G.E. Brown, M. Overby, Z. Forootaninia, and R. Narain. 2018. Accurate dissipative forces in optimization integrators. ACM Trans. on Graph. 37, 6 (2018), 1--14.
9. George E Brown and Rahul Narain. 2021. WRAPD: weighted rotation-aware ADMM for parameterization and deformation. ACM Trans. on Graph. 40, 4 (2021), 1--14.
10. Jagabandhu Chakrabarty and WJ Drugan. 1988. Theory of plasticity. (1988).
11. Wei Chen, Fei Zhu, Jing Zhao, Sheng Li, and Guoping Wang. 2018. Peridynamics-Based Fracture Animation for Elastoplastic Solids. In Computer Graphics Forum, Vol. 37.
12. Simon Clavet, Philippe Beaudoin, and Pierre Poulin. 2005. Particle-based viscoelastic fluid simulation. In Proceedings of the 2005 Symp. Computer animation. 219--228.
13. G. Daviet and F. Bertails-Descoubes. 2016. A semi-implicit material point method for the continuum simulation of granular materials. ACM Trans. on Graph. 35, 4 (2016).
14. Dimitar Dinev, Tiantian Liu, and Ladislav Kavan. 2018. Stabilizing integrators for real-time physics. ACM Trans. on Graph. 37, 1 (2018), 1--19.
15. Michael Falkenstein, Ben Jones, Joshua A Levine, Tamar Shinar, and Adam W Bargteil. 2017. Reclustering for large plasticity in clustered shape matching. In Proceedings of the Tenth International Conference on Motion in Games. 1--6.
16. Y. Fang, M. Li, M. Gao, and C. Jiang. 2019. Silly rubber: an implicit material point method for simulating non-equilibrated viscoelastic and elastoplastic solids. ACM Trans. on Graph. 38, 4 (2019), 1--13.
17. Y. Fei, C. Batty, E. Grinspun, and C. Zheng. 2019. A multi-scale model for coupling strands with shear-dependent liquid. ACM Trans. on Graph. 38, 6 (2019), 1--20.
18. Zachary Ferguson, Minchen Li, Teseo Schneider, Francisca Gil-Ureta, Timothy Langlois, Chenfanfu Jiang, Denis Zorin, Danny M Kaufman, and Daniele Panozzo. 2021. Intersection-free rigid body dynamics. ACM Transactions on Graphics 40, 4 (2021).
19. M. Gao, X. Wang, K. Wu, A. Pradhana, E. Sifakis, C. Yuksel, and C. Jiang. 2018. GPU optimization of material point methods. ACM Trans. on Graph. 37, 6 (2018).
20. T. F. Gast, C. Schroeder, A. Stomakhin, C. Jiang, and J.M. Teran. 2015. Optimization integrator for large time steps. trans. on vis. and comp. graph. 21, 10 (2015).
21. J. Gaume, T. Gast, J. Teran, A. van Herwijnen, and C. Jiang. 2018. Dynamic anticrack propagation in snow. Nature Communications 9, 1 (2018), 3047.
22. Dan Gerszewski, Haimasree Bhattacharya, and Adam W Bargteil. 2009. A point-based method for animating elastoplastic solids. In Symp. Computer animation. 133--138.
23. C. Gissler, A. Henne, S. Band, A. Peer, and M. Teschner. 2020. An implicit compressible SPH solver for snow simulation. ACM Trans. on Graph. 39, 4 (2020).
24. Francis H Harlow. 1964. The particle-in-cell computing method for fluid dynamics. Methods Comput. Phys. 3 (1964), 319--343.
25. Xiaowei He, Huamin Wang, and Enhua Wu. 2017. Projective peridynamics for modeling versatile elastoplastic materials. trans. on vis. and comp. graph. 24, 9 (2017).
26. Jan Hegemann, Chenfanfu Jiang, Craig Schroeder, and Joseph M Teran. 2013. A level set method for ductile fracture. In Symp. Computer animation. 193--201.
27. Y. Hu, Y. Fang, Z. Ge, Z. Qu, Y. Zhu, A. Pradhana, and C. Jiang. 2018. A moving least squares material point method with displacement discontinuity and two-way rigid body coupling. ACM Trans. on Graph. 37, 4 (2018), 150.
28. Tiffany Inglis, M-L Eckert, James Gregson, and Nils Thuerey. 2017. Primal-dual optimization for fluids. In Computer Graphics Forum, Vol. 36.
29. Geoffrey Irving, Joseph Teran, and Ronald Fedkiw. 2004. Invertible finite elements for robust simulation of large deformation. In Symp. Computer animation. 131--140.
30. Geoffrey Irving, Joseph Teran, and Ronald Fedkiw. 2006. Tetrahedral and hexahedral invertible finite elements. Graphical Models 68, 2 (2006), 66--89.
31. Chenfanfu Jiang, Craig Schroeder, Andrew Selle, Joseph Teran, and Alexey Stomakhin. 2015. The Affine Particle-in-Cell Method. ACM Trans. Graph. 34, 4 (2015).
32. Chenfanfu Jiang, Craig Schroeder, Joseph Teran, Alexey Stomakhin, and Andrew Selle. 2016. The material point method for simulating continuum materials. In ACM SIGGRAPH 2016 Courses. 1--52.
33. Ben Jones, April Martin, Joshua A Levine, Tamar Shinar, and Adam W Bargteil. 2016a. Ductile fracture for clustered shape matching. In Proceedings of the 20th ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games. 65--70.
34. Ben Jones, Nils Thuerey, Tamar Shinar, and Adam W Bargteil. 2016b. Example-based plastic deformation of rigid bodies. ACM Trans. on Graph. 35, 4 (2016).
35. Ben Jones, Stephen Ward, Ashok Jallepalli, Joseph Perenia, and Adam W Bargteil. 2014. Deformation embedding for point-based elastoplastic simulation. ACM Trans. on Graph. 33, 2 (2014), 1--9.
36. I. Karamouzas, R. Sohre, N.and Narain, and S.J. Guy. 2017. Implicit crowds: Optimization integrator for robust crowd simulation. ACM Trans. on Graph. 36, 4 (2017).
37. G. Klár, T. Gast, A. Pradhana, C. Fu, C. Schroeder, C. Jiang, and J. Teran. 2016. Drucker-prager elastoplasticity for sand animation. ACM Trans. on Graph. 35, 4 (2016).
38. Lei Lan, Yin Yang, Danny Kaufman, Junfeng Yao, Minchen Li, and Chenfanfu Jiang. 2021. Medial IPC: accelerated incremental potential contact with medial elastics. ACM Trans. on Graph. 40, 4 (2021), 1--16.
39. Jing Li, Tiantian Liu, and Ladislav Kavan. 2018. Laplacian damping for projective dynamics. In Proceedings of the 14th Workshop on Virtual Reality Interactions and Physical Simulations. 29--36.
40. M. Li, Z. Ferguson, T. Schneider, T. Langlois, D. Zorin, D. Panozzo, C. Jiang, and D.M. Kaufman. 2020. Incremental potential contact: Intersection-and inversion-free, large-deformation dynamics. ACM transactions on graphics (2020).
41. M. Li, M. Gao, T. Langlois, C. Jiang, and D. M. Kaufman. 2019. Decomposed optimization time integrator for large-step elastodynamics. ACM Trans. on Graph. 38, 4 (2019).
42. Minchen Li, Danny M. Kaufman, and Chenfanfu Jiang. 2021b. Codimensional Incremental Potential Contact. ACM Transactions on Graphics 40, 4 (2021).
43. X. Li, Y. Fang, M. Li, and C. Jiang. 2021a. BFEMP: Interpenetration-free MPM-FEM coupling with barrier contact. Comp. meth. applied mech. eng. (2021).
44. Yue Li, Xuan Li, Minchen Li, Yixin Zhu, Bo Zhu, and Chenfanfu Jiang. 2021c. Lagrangian-Eulerian multidensity topology optimization with the material point method. Internat. J. Numer. Methods Engrg. 122, 14 (2021), 3400--3424.
45. Tiantian Liu, Sofien Bouaziz, and Ladislav Kavan. 2017. Quasi-newton methods for real-time simulation of hyperelastic materials. ACM Trans. on Graph. 36, 3 (2017).
46. Neil Molino, Zhaosheng Bao, and Ron Fedkiw. 2004. A virtual node algorithm for changing mesh topology during simulation. ACM Trans. on Graph. 23, 3 (2004).
47. Matthias Müller, Bruno Heidelberger, Marcus Hennix, and John Ratcliff. 2007. Position based dynamics. J. Visual Communication and Image Repres. 18, 2 (2007), 109--118.
48. M. Müller, B. Heidelberger, M. Teschner, and M. Gross. 2005. Meshless deformations based on shape matching. ACM Trans. on Graph. 24, 3 (2005).
49. Matthias Müller, Richard Keiser, Andrew Nealen, Mark Pauly, Markus Gross, and Marc Alexa. 2004. Point based animation of elastic, plastic and melting objects. In Proceedings of the 2004 Symp. Computer animation. 141--151.
50. Rahul Narain, Abhinav Golas, and Ming C Lin. 2010. Free-flowing granular materials with two-way solid coupling. In ACM SIGGRAPH Asia 2010 papers. 1--10.
51. R. Narain, M. Overby, and G.E. Brown. 2016. ADMM ⊇ projective dynamics: fast simulation of general constitutive models.. In Symp. Comput. Animat., Vol. 1. 2016.
52. Rahul Narain, Tobias Pfaff, and James F O'Brien. 2013. Folding and crumpling adaptive sheets. ACM Trans. on Graph. 32, 4 (2013), 1--8.
53. Rahul Narain, Armin Samii, and James F O'Brien. 2012. Adaptive anisotropic remeshing for cloth simulation. ACM Trans. on Graph. 31, 6 (2012), 1--10.
54. James F O'Brien, Adam W Bargteil, and Jessica K Hodgins. 2002. Graphical modeling and animation of ductile fracture. In Proceedings of the 29th annual conference on Computer graphics and interactive techniques. 291--294.
55. Michael Ortiz and Laurent Stainier. 1999. The variational formulation of viscoplastic constitutive updates. Comp. meth. applied mech. eng. 171, 3--4 (1999), 419--444.
56. R. Radovitzky and M. Ortiz. 1999. Error estimation and adaptive meshing in strongly nonlinear dynamic problems. Comp. meth. applied mech. eng. 172, 1--4 (1999).
57. Stewart A Silling. 2000. Reformulation of elasticity theory for discontinuities and longrange forces. Journal of the Mechanics and Physics of Solids 48, 1 (2000), 175--209.
58. Juan C Simo. 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comp. meth. applied mech. eng. 99, 1 (1992), 61--112.
59. Juan C Simo and Thomas JR Hughes. 1998. Computational inelasticity. Springer-Verlag.
60. Jason Smith and Scott Schaefer. 2015. Bijective parameterization with free boundaries. ACM Trans. on Graph. 34, 4 (2015), 1--9.
61. Alexey Stomakhin, Russell Howes, Craig Schroeder, and Joseph M Teran. 2012. Energetically consistent invertible elasticity. In Proceedings of the 11th ACM SIGGRAPH/Eurographics conference on Computer Animation. 25--32.
62. A. Stomakhin, C. Schroeder, L. Chai, J. Teran, and A. Selle. 2013. A material point method for snow simulation. ACM Trans. on Graph. 32, 4 (2013), 102.
63. D. Sulsky, Z. Chen, and H. L Schreyer. 1994. A particle method for history-dependent materials. Comp. meth. applied mech. eng. 118, 1--2 (1994).
64. Yunxin Sun, Tamar Shinar, and Craig Schroeder. 2020. Effective time step restrictions for explicit MPM simulation. In Computer Graphics Forum, Vol. 39.
65. A. P. Tampubolon, T. Gast, G. Klár, C. Fu, J. Teran, C. Jiang, and K. Museth. 2017. Multi-species simulation of porous sand and water mixtures. ACM Trans. on Graph. 36, 4 (2017), 105.
66. Joseph Teran, Eftychios Sifakis, Geoffrey Irving, and Ronald Fedkiw. 2005. Robust quasistatic finite elements and flesh simulation. In Proceedings of the 2005 Symp. Computer animation. 181--190.
67. 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.
68. Huamin Wang and Yin Yang. 2016. Descent methods for elastic body simulation on the GPU. ACM Trans. on Graph. 35, 6 (2016), 1--10.
69. Xinlei Wang, Minchen Li, Yu Fang, Xinxin Zhang, Ming Gao, Min Tang, Danny M Kaufman, and Chenfanfu Jiang. 2020. Hierarchical optimization time integration for cfl-rate mpm stepping. ACM Trans. on Graph. 39, 3 (2020), 1--16.
70. Chris Wojtan, Nils Thürey, Markus Gross, and Greg Turk. 2009. Deforming meshes that split and merge. In ACM SIGGRAPH 2009 papers.
71. Chris Wojtan and Greg Turk. 2008. Fast viscoelastic behavior with thin features. In ACM SIGGRAPH 2008 papers. 1--8.
72. Tao Yang, Jian Chang, Ming C Lin, Ralph R Martin, Jian J Zhang, and Shi-Min Hu. 2017. A unified particle system framework for multi-phase, multi-material visual simulations. ACM Trans. on Graph. 36, 6 (2017), 1--13.
73. Youtube. 2018. Crushing Long Steel Pipes with Hydraulic Press. https://www.youtube.com/watch?v=TM5dyY8zfxs&t=215s
74. Youtube. 2021. Satisfying steel pipe crush video. https://www.youtube.com/watch?v=1s53ejidJgY&t=4s
75. Y. Yue, B. Smith, C. Batty, C. Zheng, and E. Grinspun. 2015. Continuum foam: A material point method for shear-dependent flows. ACM Trans. on Graph. 34, 5 (2015), 160.
76. Y. Yue, B. Smith, P. Y. Chen, M. Chantharayukhonthorn, K. Kamrin, and E. Grinspun. 2018. Hybrid grains: Adaptive coupling of discrete and continuum simulations of granular media. In SIGGRAPH Asia 2018 Technical Papers. ACM, 283.

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation

      • Physical simulation