“Constrained projective dynamics: real-time simulation of deformable objects with energy-momentum conservation” by Kee, Um, Jeong and Han

This paper proposes a novel energy-momentum conserving integration method. Adopting Projective Dynamics, the proposed method extends its unconstrained minimization for time integration into the constrained form with the position-based energy-momentum constraints. This resolves the well-known problem of unwanted dissipation of energy and momenta without compromising the real-time performance and simulation stability. The proposed method also enables users to directly control the energy and momenta so as to easily create the vivid deformable and global motions they want, which is a fascinating feature for many real-time applications such as virtual/augmented reality and games.

1. David Baraff and Andrew Witkin. 1998. Large steps in cloth simulation. In Proceedings of the 25th annual conference on Computer graphics and interactive techniques. 43–54.Google ScholarDigital Library
2. Sofien Bouaziz, Sebastian Martin, Tiantian Liu, Ladislav Kavan, and Mark Pauly. 2014. Projective dynamics: fusing constraint projections for fast simulation. ACM Transactions on Graphics (TOG) 33, 4 (2014), 1–11.Google ScholarDigital Library
3. Christopher Brandt, Elmar Eisemann, and Klaus Hildebrandt. 2018. Hyper-Reduced Projective Dynamics. ACM Trans. Graph. 37, 4, Article 80 (July 2018), 13 pages. Google ScholarDigital Library
4. George E. Brown, Matthew Overby, Zahra Forootaninia, and Rahul Narain. 2018. Accurate Dissipative Forces in Optimization Integrators. ACM Trans. Graph. 37, 6, Article 282 (Dec. 2018), 14 pages. Google ScholarDigital Library
5. Yu Ju Chen, Seung Heon Sheen, Uri M Ascher, and Dinesh K Pai. 2020. SIERE: A Hybrid Semi-Implicit Exponential Integrator for Efficiently Simulating Stiff Deformable Objects. ACM Transactions on Graphics (TOG) 40, 1 (2020), 1–12.Google ScholarDigital Library
6. Kwang-Jin Choi and Hyeong-Seok Ko. 2005. Stable but responsive cloth. In ACM SIGGRAPH 2005 Courses. 1–es.Google ScholarDigital Library
7. Alex Dahl and Adam Bargteil. 2019. Global Momentum Preservation for Position-Based Dynamics. In Motion, Interaction and Games (MIG ’19). Association for Computing Machinery, New York, NY, USA, 1–5. Google ScholarDigital Library
8. Dimitar Dinev, Tiantian Liu, and Ladislav Kavan. 2018a. Stabilizing Integrators for Real-Time Physics. ACM Transactions on Graphics (TOG) 37, 1 (2018), 9.Google ScholarDigital Library
9. Dimitar Dinev, Tiantian Liu, Jing Li, Bernhard Thomaszewski, and Ladislav Kavan. 2018b. FEPR: fast energy projection for real-time simulation of deformable objects. ACM Transactions on Graphics (TOG) 37, 4 (2018), 1–12.Google ScholarDigital Library
10. Theodore F Gast, Craig Schroeder, Alexey Stomakhin, Chenfanfu Jiang, and Joseph M Teran. 2015. Optimization integrator for large time steps. IEEE transactions on visualization and computer graphics 21, 10 (2015), 1103–1115.Google ScholarDigital Library
11. Ernst Hairer. 2006. Long-time energy conservation of numerical integrators. Foundations of computational mathematics, Santander 2005 (2006), 162–180.Google Scholar
12. Thomas J Hughes, TK Caughey, and WK Liu. 1978. Finite-element methods for nonlinear elastodynamics which conserve energy. Journal of Applied Mechanics, Transactions ASME 45, 2 (1978), 366–370.Google ScholarCross Ref
13. Couro Kane, Jerrold E Marsden, Michael Ortiz, and Matthew West. 2000. Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. International Journal for numerical methods in engineering 49, 10 (2000), 1295–1325.Google ScholarCross Ref
14. L. Kharevych, Weiwei Yang, Y. Tong, E. Kanso, J. E. Marsden, P. Schröder, and M. Desbrun. 2006. Geometric, Variational Integrators for Computer Animation. In Proceedings of the 2006 ACM SIGGRAPH/Eurographics Symposium on Computer Animation (Vienna, Austria) (SCA ’06). Eurographics Association, Goslar, DEU, 43–51.Google Scholar
15. Martin Komaritzan and Mario Botsch. 2019. Fast Projective Skinning. In Motion, Interaction and Games (Newcastle upon Tyne, United Kingdom) (MIG ’19). Association for Computing Machinery, New York, NY, USA, Article 22, 10 pages. Google ScholarDigital Library
16. Tassilo Kugelstadt, Dan Koschier, and Jan Bender. 2018. Fast Corotated FEM using Operator Splitting. Computer Graphics Forum (SCA) 37, 8 (2018).Google Scholar
17. Robert A LaBudde and Donald Greenspan. 1975. Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion. Numer. Math. 25, 4 (1975), 323–346.Google ScholarDigital Library
18. M. Levi. 2014. Classical Mechanics with Calculus of Variations, and Optimal Control: An Intuitive Introduction. American Mathematical Society. https://books.google.co.kr/books?id=uVSYswEACAAJGoogle Scholar
19. Jing Li, Tiantian Liu, and Ladislav Kavan. 2018. Laplacian Damping for Projective Dynamics. In VRIPHYS2018: 14th Workshop on Virtual Reality Interaction and Physical Simulation.Google Scholar
20. Jing Li, Tiantian Liu, and Ladislav Kavan. 2019. Fast Simulation of Deformable Characters with Articulated Skeletons in Projective Dynamics. In Proceedings of the 18th Annual ACM SIGGRAPH/Eurographics Symposium on Computer Animation (Los Angeles, California) (SCA ’19). Association for Computing Machinery, New York, NY, USA, Article 1, 10 pages. Google ScholarDigital Library
21. Tiantian Liu, Adam W Bargteil, James F O’Brien, and Ladislav Kavan. 2013. Fast simulation of mass-spring systems. ACM Transactions on Graphics (TOG) 32, 6 (2013), 1–7.Google ScholarDigital Library
22. Tiantian Liu, Sofien Bouaziz, and Ladislav Kavan. 2017. Quasi-newton methods for real-time simulation of hyperelastic materials. ACM Transactions on Graphics (TOG) 36, 3 (2017), 1–16.Google ScholarDigital Library
23. Fabian Löschner, Andreas Longva, Stefan Jeske, Tassilo Kugelstadt, and Jan Bender. 2020. Higher-Order Time Integration for Deformable Solids. Computer Graphics Forum 39, 8 (2020), 157–169. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1111/cgf.14110 Google ScholarDigital Library
24. Miles Macklin, Matthias Müller, and Nuttapong Chentanez. 2016. XPBD: position-based simulation of compliant constrained dynamics. In Proceedings of the 9th International Conference on Motion in Games. 49–54.Google ScholarDigital Library
25. Sebastian Martin, Bernhard Thomaszewski, Eitan Grinspun, and Markus Gross. 2011. Example-Based Elastic Materials. ACM Trans. Graph. 30, 4, Article 72 (July 2011), 8 pages. Google ScholarDigital Library
26. Dominik L. Michels, Vu Thai Luan, and Mayya Tokman. 2017. A Stiffly Accurate Integrator for Elastodynamic Problems. ACM Trans. Graph. 36, 4, Article 116 (July 2017), 14 pages. Google ScholarDigital Library
27. Dominik L. Michels, Gerrit A. Sobottka, and Andreas G. Weber. 2014. Exponential Integrators for Stiff Elastodynamic Problems. ACM Trans. Graph. 33, 1, Article 7 (Feb. 2014), 20 pages. Google ScholarDigital Library
28. Matthias Müller, Bruno Heidelberger, Marcus Hennix, and John Ratcliff. 2007. Position based dynamics. Journal of Visual Communication and Image Representation 18, 2 (2007), 109–118.Google ScholarDigital Library
29. Jorge Nocedal and Stephen Wright. 2006. Numerical optimization. Springer Science & Business Media.Google Scholar
30. Matthew Overby, George E. Brown, Jie Li, and Rahul Narain. 2017. ADMM ⊇ Projective Dynamics: Fast Simulation of Hyperelastic Models with Dynamic Constraints. IEEE Transactions on Visualization and Computer Graphics 23, 10 (Oct 2017), 2222–2234. Google ScholarDigital Library
31. J. Rojas, T. Liu, and L. Kavan. 2019. Average Vector Field Integration for St. Venant-Kirchhoff Deformable Models. IEEE Transactions on Visualization and Computer Graphics 25, 8 (2019), 2529–2539. Google ScholarCross Ref
32. Eftychios Sifakis and Jernej Barbic. 2012. FEM simulation of 3D deformable solids: a practitioner’s guide to theory, discretization and model reduction. In Acm siggraph 2012 courses. 1–50.Google Scholar
33. Juan C Simo, N Tarnow, and KK1187632 Wong. 1992. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer methods in applied mechanics and engineering 100, 1 (1992), 63–116.Google Scholar
34. Ari Stern and Mathieu Desbrun. 2006. Discrete Geometric Mechanics for Variational Time Integrators. In ACM SIGGRAPH 2006 Courses (Boston, Massachusetts) (SIGGRAPH ’06). Association for Computing Machinery, New York, NY, USA, 75–80. Google ScholarDigital Library
35. A. Stuart and A.R. Humphries. 1998. Dynamical Systems and Numerical Analysis. Number vol. 8 in Cambridge Monographs on Applie. Cambridge University Press. https://books.google.fr/books?id=ymoQA8s5pNICGoogle Scholar
36. Jonathan Su, Rahul Sheth, and Ronald Fedkiw. 2012. Energy conservation for the simulation of deformable bodies. IEEE Transactions on Visualization and Computer Graphics 19, 2 (2012), 189–200.Google ScholarDigital Library
37. Demetri Terzopoulos, John Platt, Alan Barr, and Kurt Fleischer. 1987. Elastically deformable models. In Proceedings of the 14th annual conference on Computer graphics and interactive techniques. 205–214.Google ScholarDigital Library
38. Matthias Teschner, Bruno Heidelberger, Matthias Müller, Danat Pomerantes, and Markus H Gross. 2003. Optimized spatial hashing for collision detection of deformable objects.. In Vmv, Vol. 3. 47–54.Google Scholar
39. Huamin Wang. 2015. A Chebyshev Semi-Iterative Approach for Accelerating Projective and Position-Based Dynamics. ACM Trans. Graph. 34, 6, Article 246 (Oct. 2015), 9 pages. Google ScholarDigital Library
40. Marcel Weiler, Dan Koschier, and Jan Bender. 2016. Projective fluids. In Proceedings of the 9th International Conference on Motion in Games. 79–84.Google ScholarDigital Library
41. Matthew West. 2004. Variational integrators. Ph.D. Dissertation. California Institute of Technology.Google Scholar
42. Hongyi Xu and Jernej Barbič. 2017. Example-Based Damping Design. ACM Trans. Graph. 36, 4, Article 53 (July 2017), 14 pages. Google ScholarDigital Library