“CD-MPM: continuum damage material point methods for dynamic fracture animation” by Wolper, Fang, Li, Lu, Gao, et al. …
Conference:
Type:
Title:
- CD-MPM: continuum damage material point methods for dynamic fracture animation
Session/Category Title: MPM and Collision
Presenter(s)/Author(s):
Abstract:
We present two new approaches for animating dynamic fracture involving large elastoplastic deformation. In contrast to traditional mesh-based techniques, where sharp discontinuity is introduced to split the continuum at crack surfaces, our methods are based on Continuum Damage Mechanics (CDM) with a variational energy-based formulation for crack evolution. Our first approach formulates the resulting dynamic material damage evolution with a Ginzburg-Landau type phase-field equation and discretizes it with the Material Point Method (MPM), resulting in a coupled momentum/damage solver rooted in phase field fracture: PFF-MPM. Although our PFF-MPM approach achieves convincing fracture with or without plasticity, we also introduce a return mapping algorithm that can be analytically solved for a wide range of general non-associated plasticity models, achieving more than two times speedup over traditional iterative approaches. To demonstrate the efficacy of the algorithm, we also develop a Non-Associated Cam-Clay (NACC) plasticity model with a novel fracture-friendly hardening scheme. Our NACC plasticity paired with traditional MPM composes a second approach to dynamic fracture, as it produces a breadth of organic, brittle material fracture effects on its own. Though NACC and PFF can be combined, we focus on exploring their material effects separately. Both methods can be easily integrated into any existing MPM solver, enabling the simulation of various fracturing materials with extremely high visual fidelity while requiring little additional computational overhead.
References:
1. S. M. Allen and J. W. Cahn. 1972. Ground state structures in ordered binary alloys with second neighbor interactions. Acta Metallurgica 20, 3 (1972), 423–433.Google ScholarCross Ref
2. M. Ambati, R. Kruse, and L. De Lorenzis. 2016. A phase-field model for ductile fracture at finite strains and its experimental verification. Comp. Mech. 57, 1 (2016), 149–167. Google ScholarDigital Library
3. H. Amor, J.-J. Marigo, and C. Maurini. 2009. Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments. J. of the Mech. and Phys. of Solids 57, 8 (2009), 1209–1229.Google ScholarCross Ref
4. K. Aoki, N. H. Dong, T. Kaneko, and S. Kuriyama. 2004. Physically Based Simulation of Cracks on Drying 3D Solids. In Proc. of the Comp. Graph. Int. 357–364. Google ScholarDigital Library
5. I. S. Aranson, V. A. Kalatsky, and V. M. Vinokur. 2000. Continuum field description of crack propagation. Physical Review Letters 85, 1 (2000), 118–121.Google ScholarCross Ref
6. Z. Bao, J. M. Hong, J. Teran, and R. Fedkiw. 2007. Fracturing Rigid Materials. IEEE Trans. on Vis. and Comp. Graph. 13, 2 (2007), 370–378. Google ScholarDigital Library
7. G. I. Barenblatt. 1962. The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. Advances in Applied Mechanics, Vol. 7. 55 — 129.Google ScholarCross Ref
8. T. Belytschko and T. Black. 1999. Elastic crack growth in finite elements with minimal remeshing. Int. J. for Num. Meth. in Eng. 45, 5 (1999), 601–620.Google ScholarCross Ref
9. T. Belytschko, D. Organ, and Y. Krongauz. 1995. A coupled finite element-element-free Galerkin method. Comp. Mech. 17, 3 (1995), 186–195.Google ScholarCross Ref
10. H. Bhatacharya, Y. Gao, and A. Bargteil. 2011. A Level-set Method for Skinning Animated Particle Data. In Symp. Comp. Anim. 17–24. Google ScholarDigital Library
11. M. J. Borden, T. J.R. Hughes, C. M. Landis, A. Anvari, and I. J. Lee. 2016. A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Comp. Meth. in Applied Mech. and Eng. 312 (2016), 130–166.Google ScholarCross Ref
12. M. J. Borden, C. V. Verhoosel, M. A. Scott, T. JR. Hughes, and C. M. Landis. 2012. A phase-field description of dynamic brittle fracture. Comp. Meth. in Applied Mech. and Eng. 217 (2012), 77–95.Google ScholarCross Ref
13. S. Bouaziz, S. Martin, T. Liu, L. Kavan, and M. Pauly. 2014. Projective Dynamics: Fusing Constraint Projections for Fast Simulation. ACM Trans. Graph. 33, 4 (2014), 154:1–154:11. Google ScholarDigital Library
14. B. Bourdin, G. A. Francfort, and J-J. Marigo. 2000. Numerical experiments in revisited brittle fracture. J. of the Mech. and Phys. of Solids 48, 4 (2000), 797 — 826.Google ScholarCross Ref
15. B. Bourdin, G. A. Francfort, and J.-J. Marigo. 2008. The variational approach to fracture. Journal of elasticity 91, 1–3 (2008), 5–148.Google ScholarCross Ref
16. J. Brackbill and H. Ruppel. 1986. FLIP: A method for adaptively zoned, Particle-In-Cell calculations of fluid flows in two dimensions. J Comp Phys 65 (1986), 314–343. Google ScholarDigital Library
17. O. Busaryev, T. K. Dey, and H. Wang. 2013. Adaptive Fracture Simulation of Multi-layered Thin Plates. ACM Trans. Graph. 32, 4, Article 52 (2013), 6 pages. Google ScholarDigital Library
18. J. W. Cahn and J. E. Hilliard. 1958. Free energy of a nonuniform system. I. interfacial free energy. The Journal of Chemical Physics 28, 2 (1958), 258–267.Google ScholarCross Ref
19. M. Cervera and M. Chiumenti. 2006. Mesh objective tensile cracking via a local continuum damage model and a crack tracking technique. Comp. Meth. in Applied Mech. and Eng. 196, 1–3 (2006), 304–320.Google ScholarCross Ref
20. F. Chen, C. Wang, B. Xie, and H. Qin. 2013. Flexible and rapid animation of brittle fracture using the smoothed particle hydrodynamics formulation. Computer Animation and Virtual Worlds 24, 3–4 (2013), 215–224.Google ScholarCross Ref
21. Z. Chen, M. Yao, R. Feng, and H. Wang. 2014. Physics-inspired Adaptive Fracture Refinement. ACM Trans. Graph. 33, 4, Article 113 (2014), 7 pages. Google ScholarDigital Library
22. J. Choo and W. C. Sun. 2018. Coupled phase-field and plasticity modeling of geological materials: From brittle fracture to ductile flow. Comp. Meth. in Applied Mech. and Eng. 330 (2018), 1–32.Google ScholarCross Ref
23. F. Da, C. Batty, and E. Grinspun. 2014. Multimaterial Mesh-based Surface Tracking. ACM Trans. Graph. 33, 4, Article 112 (2014), 11 pages. Google ScholarDigital Library
24. F. Da, D. Hahn, C. Batty, C. Wojtan, and E. Grinspun. 2016. Surface-only Liquids. ACM Trans. Graph. 35, 4, Article 78 (2016), 12 pages. Google ScholarDigital Library
25. G. Daviet and F. Bertails-Descoubes. 2016. A semi-implicit material point method for the continuum simulation of granular materials. ACM Trans. Graph. 35, 4 (2016). Google ScholarDigital Library
26. Y. (R.) Fei, C. Batty, E. Grinspun, and C. Zheng. 2018. A Multi-scale Model for Simulating Liquid-fabric Interactions. ACM Trans. Graph. 37, 4 (2018), 51:1–51:16. Google ScholarDigital Library
27. G. A. Francfort and J.-J. Marigo. 1998. Revisiting brittle fracture as an energy minimization problem. J. of the Mech. and Phys. of Solids 46, 8 (1998), 1319–1342.Google ScholarCross Ref
28. M. Gao, A. Pradhana, X. Han, Q. Guo, G. Kot, E. Sifakis, and C. Jiang. 2018a. Animating fluid sediment mixture in particle-laden flows. ACM Trans. Graph. 37, 4 (2018), 149. Google ScholarDigital Library
29. M. Gao, X. Wang, K. Wu, A. Pradhana, E. Sifakis, C. Yuksel, and C. Jiang. 2018b. GPU Optimization of Material Point Methods. ACM Trans. Graph. 37, 6, Article 254 (2018), 12 pages. Google ScholarDigital Library
30. J. Gaume, T. Gast, J. Teran, A. van Herwijnen, and C. Jiang. 2018. Dynamic anticrack propagation in snow. Nature communications 9, 1 (2018), 3047.Google Scholar
31. L. Glondu, M. Marchal, and G. Dumont. 2013. Real-Time Simulation of Brittle Fracture Using Modal Analysis. IEEE Trans. on Vis. and Comp. Graph. 19, 2 (2013), 201–209. Google ScholarDigital Library
32. L. Glondu, S. C. Schvartzman, M. Marchal, G. Dumont, and M. A. Otaduy. 2014. Fast Collision Detection for Fracturing Rigid Bodies. IEEE Trans. on Vis. and Comp. Graph. 20, 1 (2014), 30–41. Google ScholarDigital Library
33. P. Grassl and M. Jirásek. 2004. On mesh bias of local damage models for concrete. (2004).Google Scholar
34. A. A. Griffith and M. Eng. 1921. VI. The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. A 221, 582–593 (1921), 163–198.Google ScholarCross Ref
35. Q. Guo, X. Han, C. Fu, T. Gast, R. Tamstorf, and J. Teran. 2018. A material point method for thin shells with frictional contact. ACM Trans. Graph. 37, 4 (2018), 147. Google ScholarDigital Library
36. D. Hahn and C. Wojtan. 2015. High-resolution brittle fracture simulation with boundary elements. ACM Trans. Graph. 34, 4, Article 151 (2015), 12 pages. Google ScholarDigital Library
37. D. Hahn and C. Wojtan. 2016. Fast approximations for boundary element based brittle fracture simulation. ACM Trans. Graph. 35, 4, Article 104 (2016), 11 pages. Google ScholarDigital Library
38. X. He, H. Wang, and E. Wu. 2018. Projective peridynamics for modeling versatile elastoplastic materials. IEEE Trans. on Vis. and Comp. Graph. 24, 9 (2018), 2589–2599.Google ScholarCross Ref
39. X. He, H. Wang, F. Zhang, H. Wang, G. Wang, K. Zhou, and E. Wu. 2015. Simulation of fluid mixing with interface control. In Symp. Comp. Anim. 129–135. Google ScholarDigital Library
40. J. Hegemann, C. Jiang, C. Schroeder, and J. Teran. 2013. A level set method for ductile fracture. In Proc ACM SIGGRAPH/Eurograp Symp Comp Anim. 193–201. Google ScholarDigital Library
41. K. Hirota, Y. Tanoue, and T. Kaneko. 1998. Generation of crack patterns with a physical model. The Visual Computer 14, 3 (1998), 126–137.Google ScholarCross Ref
42. K. Hirota, Y. Tanoue, and T. Kaneko. 2000. Simulation of three-dimensional cracks. The Visual Computer 16, 7 (2000), 371–378.Google ScholarCross Ref
43. M. A. Homel and E. B. Herbold. 2017. Field-gradient partitioning for fracture and frictional contact in the material point method. Int. J. for Num. Meth. in Eng. 109, 7 (2017), 1013–1044.Google ScholarCross Ref
44. 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. Graph. 37, 4 (2018), 150. Google ScholarDigital Library
45. G. R. Irwin. 1957. Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate. J. Appl. Mech. (1957).Google Scholar
46. D. L. James and D. K. Pai. 1999. ArtDefo: Accurate real time deformable objects. In Proc. of the 26th Ann. Conf. on Comp. Graph. and Inter. Tech. 65–72. Google ScholarDigital Library
47. C. Jiang, T. Gast, and J. Teran. 2017. Anisotropic elastoplasticity for cloth, knit and hair frictional contact. ACM Trans. Graph. 36, 4 (2017). Google ScholarDigital Library
48. C. Jiang, C. Schroeder, A. Selle, J. Teran, and A. Stomakhin. 2015. The affine particle-in-cell method. ACM Trans. Graph. 34, 4 (2015), 51:1–51:10. Google ScholarDigital Library
49. C. Jiang, C. Schroeder, J. Teran, A. Stomakhin, and A. Selle. 2016. The material point method for simulating continuum materials. In SIGGRAPH Course. 24:1–24:52. Google ScholarDigital Library
50. B. Jones, A. Martin, J. A. Levine, T. Shinar, and A. W. Bargteil. 2016. Ductile Fracture for Clustered Shape Matching. Proc. of the ACM SIGGRAPH symp. on Int. 3D graph. and games (2016). Google ScholarDigital Library
51. L. M. Kachanov. 1999. Rupture Time Under Creep Conditions. Int. J. of Fracture 97, 1 (1999), 11–18.Google ScholarCross Ref
52. P. Kaufmann, S. Martin, M. Botsch, and M. Gross. 2008. Flexible simulation of deformable models using discontinuous Galerkin FEM. In Symp. Comp. Anim. 105–116. Google ScholarDigital Library
53. T. Kim, M. Henson, and M. C. Lin. 2004. A hybrid algorithm for modeling ice formation. In Symp. Comp. Anim. 305–314. Google ScholarDigital Library
54. G. Klár, T. Gast, A. Pradhana, C. Fu, C. Schroeder, C. Jiang, and J. Teran. 2016. Drucker-prager elastoplasticity for sand animation. ACM Trans. Graph. 35, 4 (2016), 103:1–103:12. Google ScholarDigital Library
55. D. Koschier, J. Bender, and N. Thuerey. 2017. Robust eXtended finite elements for complex cutting of deformables. ACM Trans. Graph. 36, 4, Article 55 (2017), 13 pages. Google ScholarDigital Library
56. L. D. Landau and E. M. Lifshitz. 1971. The classical theory of fields. (1971).Google Scholar
57. J. A. Levine, A. W. Bargteil, C. Corsi, J. Tessendorf, and R. Geist. 2014. A peridynamic perspective on spring-mass fracture. In Symp. Comp. Anim. 47–55. Google ScholarDigital Library
58. X. Li, S. Andrews, B. Jones, and A. Bargteil. 2018. Energized rigid body fracture. Proc. ACM Comput. Graph. Interact. Tech. 1, 1, Article 9 (2018), 9 pages. Google ScholarDigital Library
59. N. Liu, X. He, S. Li, and G. Wang. 2011. Meshless Simulation of Brittle Fracture. Comput. Animat. Virtual Worlds 22, 2–3 (2011), 115–124. Google ScholarDigital Library
60. T. Liu, A. Bargteil, J. O’Brien, and L. Kavan. 2013. Fast Simulation of Mass-Spring Systems. ACM Trans. Graph. 32, 6 (2013), 209:1–7. Google ScholarDigital Library
61. T. Liu, S. Bouaziz, and L. Kavan. 2017. Quasi-Newton Methods for Real-Time Simulation of Hyperelastic Materials. ACM Trans. Graph. 36, 3, Article 116a (2017). Google ScholarDigital Library
62. Y. Y. Lu, T. Belytschko, and M. Tabbara. 1995. Element-free Galerkin method for wave propagation and dynamic fracture. Comp. Meth. in Applied Mech. and Eng. 126, 1 (1995), 131 — 153.Google ScholarCross Ref
63. C. Miehe, M. Hofacker, and F. Welschinger. 2010a. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Comp. Meth. in Applied Mech. and Eng. 199, 45–48 (2010), 2765–2778.Google ScholarCross Ref
64. C. Miehe, L. M. Schänzel, and H. Ulmer. 2015. Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comp. Meth. in Applied Mech. and Eng. 294 (2015), 449 — 485.Google ScholarCross Ref
65. C. Miehe, F. Welschinger, and M. Hofacker. 2010b. Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations. Int. J. for Num. Meth. in Eng. 83, 10 (2010), 1273–1311.Google ScholarCross Ref
66. N. Mitchell, M. Aanjaneya, R. Setaluri, and E. Sifakis. 2015. Non-manifold level sets: A multivalued implicit surface representation with applications to self-collision processing. ACM Trans. Graph. 34, 6 (2015), 247. Google ScholarDigital Library
67. N. Moës, J. Dolbow, and T. Belytschko. 1999. A finite element method for crack growth without remeshing. Int. J. for Num. Meth. in Eng. 46, 1 (1999), 131–150.Google ScholarCross Ref
68. N. Molino, Z. Bao, and R. Fedkiw. 2005. A Virtual Node Algorithm for Changing Mesh Topology During Simulation. In ACM SIGGRAPH 2005 Courses. Article 4. Google ScholarDigital Library
69. D. Mould. 2005. Image-guided Fracture. In Proc. of Graphics Interface 2005. 219–226. Google ScholarDigital Library
70. M. Müller and M. Gross. 2004. Interactive Virtual Materials. In Proc. of Graphics Interface 2004. 239–246. Google ScholarDigital Library
71. M. Müller, R. Keiser, A. Nealen, M. Pauly, M. Gross, and M. Alexa. 2004. Point Based Animation of Elastic, Plastic and Melting Objects. In Symp. Comp. Anim. 141–151. Google ScholarDigital Library
72. Ken Museth, Jeff Lait, John Johanson, Jeff Budsberg, Ron Henderson, Mihai Alden, Peter Cucka, David Hill, and Andrew Pearce. 2013. OpenVDB: an open-source data structure and toolkit for high-resolution volumes. In Acm siggraph 2013 courses. 19. Google ScholarDigital Library
73. J. A. Nairn. 2003. Material point method calculations with explicit cracks. Comp. Mod. in Eng. And Sci. 4 (2003), 649–663.Google Scholar
74. R. Narain, M. Overby, and G. E. Brown. 2016. ADMM ⊇ Projective Dynamics: Fast Simulation of General Constitutive Models. In Symp Comp Anim. 21–28. Google ScholarDigital Library
75. M. Neff and E. Fiume. 1999. A Visual Model for Blast Waves and Fracture. In Proc. of the 1999 Conference on Graphics Interface ’99. 193–202. Google ScholarDigital Library
76. D. Ngo and A. C. Scordelis. 1967. Finite Element Analysis of Reinforced Concrete Beams. Journal Proceedings 64. Issue 3.Google Scholar
77. J. Ning, H. Xu, B. Wu, L. Zeng, S. Li, and Y. Xiong. 2013. Modeling and animation of fracture of heterogeneous materials based on CUDA. The Visual Computer 29, 4 (2013), 265–275. Google ScholarDigital Library
78. A. Norton, G. Turk, B. Bacon, J. Gerth, and P. Sweeney. 1991. Animation of fracture by physical modeling. The Visual Computer 7, 4 (1991), 210–219. Google ScholarDigital Library
79. J. F. O’Brien, A. W. Bargteil, and J. K. Hodgins. 2002. Graphical modeling and animation of ductile fracture. ACM Trans. Graph. 21, 3 (2002), 291–294. Google ScholarDigital Library
80. J. F. O’Brien and J. K. Hodgins. 1999. Graphical Modeling and Animation of Brittle Fracture. In Proc. of the 26th Ann. Conf. on Comp. Graph. and Inter. Tech. 137–146. Google ScholarDigital Library
81. M. Pauly, R. Keiser, B. Adams, P. Dutré, M. Gross, and L. J Guibas. 2005. Meshless animation of fracturing solids. ACM Trans. Graph. 24, 3 (2005), 957–964. Google ScholarDigital Library
82. T. Pfaff, R. Narain, J. M. de Joya, and J. F. O’Brien. 2014. Adaptive Tearing and Cracking of Thin Sheets. ACM Trans. Graph. 33, 4, Article 110 (2014), 9 pages. Google ScholarDigital Library
83. A. Pradhana, 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. Graph. 36, 4 (2017). Google ScholarDigital Library
84. R. Radovitzky and M. Ortiz. 1999. Error estimation and adaptive meshing in strongly nonlinear dynamic problems. Comp. Meth. in Applied Mech. and Eng. 172, 1–4 (1999), 203–240.Google ScholarCross Ref
85. D. Ram, T. Gast, C. Jiang, C. Schroeder, A. Stomakhin, J. Teran, and P. Kavehpour. 2015. A material point method for viscoelastic fluids, foams and sponges. In Symp. Comp. Anim. 157–163. Google ScholarDigital Library
86. Y. R. Rashid. 1968. Ultimate strength analysis of prestressed concrete pressure vessels. Nuclear Engineering and Design 7, 4 (1968), 334 — 344.Google ScholarCross Ref
87. K. Roscoe and J. Burland. 1968. On the generalised stress-strain behaviour of wet clay. Eng Plast (1968), 535–609.Google Scholar
88. P. Roy, S. P. Deepu, A. Pathrikar, D. Roy, and J. N. Reddy. 2017. Phase field based peridynamics damage model for delamination of composite structures. Composite Structures 180 (2017), 972 — 993.Google ScholarCross Ref
89. E. Sifakis, K. G. Der, and R. Fedkiw. 2007. Arbitrary cutting of deformable tetrahedralized objects. In Symp. Comp. Anim. 73–80. Google ScholarDigital Library
90. S. A. Silling. 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. of the Mech. and Phys. of Solids 48, 1 (2000), 175 — 209.Google ScholarCross Ref
91. S. A. Silling and E. Askari. 2005. A meshfree method based on the peridynamic model of solid mechanics. Computers /& Structures 83, 17 (2005), 1526 — 1535. Advances in Meshfree Methods. Google ScholarDigital Library
92. J. C. Simo. 1988. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation. Comp. Meth. in Applied Mech. and Eng. 66, 2 (1988), 199–219. Google ScholarDigital Library
93. B. Smith, F. de Goes, and T. Kim. 2018. Stable Neo-Hookean Flesh Simulation. ACM Trans. Graph. 37, 2 (2018), 12. Google ScholarDigital Library
94. A. Stomakhin, R. Howes, C. Schroeder, and J. M. Teran. 2012. Energetically consistent invertible elasticity. In Symp. Comp. Anim. 25–32. Google ScholarDigital Library
95. A. Stomakhin, C. Schroeder, L. Chai, J. Teran, and A. Selle. 2013. A material point method for snow simulation. ACM Trans. Graph. 32, 4 (2013), 102:1–102:10. Google ScholarDigital Library
96. A. Stomakhin, C. Schroeder, C. Jiang, L. Chai, J. Teran, and A. Selle. 2014. Augmented MPM for phase-change and varied materials. ACM Trans. Graph. 33, 4 (2014), 138:1–138:11. Google ScholarDigital Library
97. N. Sukumar, N. Moës, B. Moran, and T. Belytschko. 2000. Extended finite element method for three-dimensional crack modelling. Int. J. for Num. Meth. in Eng. 48, 11 (2000), 1549–1570.Google ScholarCross Ref
98. N. Sukumar, B. Moran, T. Black, and T. Belytschko. 1997. An element-free Galerkin method for three-dimensional fracture mechanics. Comp. Mech. 20, 1 (1997), 170–175. Google ScholarDigital Library
99. D. Sulsky, Z. Chen, and H. L. Schreyer. 1994. A particle method for history-dependent materials. Comp. Meth. in Applied Mech. and Eng. 118, 1–2 (1994), 179–196.Google ScholarCross Ref
100. D. Sulsky, S. Zhou, and H. Schreyer. 1995. Application of a particle-in-cell method to solid mechanics. Comp Phys Comm 87, 1 (1995), 236–252.Google ScholarCross Ref
101. D. Terzopoulos and K. Fleischer. 1988. Modeling inelastic deformation: viscolelasticity, plasticity, fracture. In ACM Siggraph Computer Graphics, Vol. 22. 269–278. Google ScholarDigital Library
102. H. Wang and Y. Yang. 2016. Descent Methods for Elastic Body Simulation on the GPU. ACM Trans. Graph. 35, 6, Article 212 (2016), 10 pages. Google ScholarDigital Library
103. Y. Wang, C. Jiang, C. Schroeder, and J. Teran. 2014. An adaptive virtual node algorithm with robust mesh cutting. In Symp. Comp. Anim. 77–85. Google ScholarDigital Library
104. M. Wicke, D. Ritchie, B. M. Klingner, S. Burke, J. R. Shewchuk, and J. F. O’Brien. 2010. Dynamic local remeshing for elastoplastic simulation. ACM Transactions on Graphics 29, 4 (2010), 49:1–49:11. Google ScholarDigital Library
105. J. Wolper, Y. Fang, M. Li, J. Lu, M. Gao, and C. Jiang. 2019. CD-MPM: Continuum damage material point methods for dynamic fracture animation: Supplemental document. ACM Trans. Graph. (2019). Google ScholarDigital Library
106. J. Wretborn, R. Armiento, and K. Museth. 2017. Animation of crack propagation by means of an extended multi-body solver for the material point method. Computers & Graphics 69 (2017), 131 — 139. Google ScholarDigital Library
107. J. Y. Wu. 2017. A unified phase-field theory for the mechanics of damage and quasi-brittle failure. J. of the Mech. and Phys. of Solids 103 (2017), 72 — 99.Google ScholarCross Ref
108. J. Y. Wu. 2018. A geometrically regularized gradient-damage model with energetic equivalence. Comp. Meth. in Applied Mech. and Eng. 328 (2018), 612 — 637.Google ScholarCross Ref
109. T. Yang, J. Chang, M. C. Lin, R. R. Martin, J. J. Zhang, and S. Hu. 2017. A Unified particle system framework for multi-phase, multi-material visual simulations. ACM Trans. Graph. 36, 6, Article 224 (2017), 13 pages. Google ScholarDigital Library
110. T. Yang, J. Chang, B. Ren, M. C. Lin, J. J. Zhang, and S. Hu. 2015. Fast multiple-fluid simulation using Helmholtz free energy. ACM Trans. Graph. 34, 6, Article 201 (2015), 11 pages. Google ScholarDigital Library
111. J. Yu, C. Wojtan, G. Turk, and C. Yap. 2012. Explicit Mesh Surfaces for Particle Based Fluids. Comput. Graph. Forum 31, 2pt4 (2012), 815–824. Google ScholarDigital Library
112. Y. Yue, B. Smith, C. Batty, C. Zheng, and E. Grinspun. 2015. Continuum foam: a material point method for shear-dependent flows. ACM Trans. Graph. 34, 5 (2015), 160:1–160:20. Google ScholarDigital Library
113. 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. ACM Trans. Graph. 37, 6, Article 283 (2018), 19 pages. Google ScholarDigital Library
114. Y. Zhu and R. Bridson. 2005. Animating sand as a fluid. ACM Trans. Graph. 24, 3 (2005), 965–972. Google ScholarDigital Library
115. Y. Zhu, R. Bridson, and C. Greif. 2015. Simulating rigid body fracture with surface meshes. ACM Trans. Graph. 34, 4, Article 150 (2015), 11 pages. Google ScholarDigital Library
