“Efficient and Accurate Collision Response for Elastically Deformable Models” by Verschoor and Jalba

  • ©Mickeal Verschoor and Andrei Jalba




    Efficient and Accurate Collision Response for Elastically Deformable Models

Session/Category Title: MPM and Collision



    Simulating (elastically) deformable models that can collide with each other and with the environment remains a challenging task. The resulting contact problems can be elegantly approached using Lagrange multipliers to represent the unknown magnitude of the response forces. Typical methods construct and solve a Linear Complementarity Problem (LCP) to obtain the response forces. This requires the inverse of the generalized mass matrix, which is generally hard to obtain for deformable-body problems. In this article, we tackle such contact problems by directly solving the Mixed Linear Complementarity Problem (MLCP) and omitting the construction of an LCP matrix. Since a convex quadratic program with linear constraints is equivalent to an MLCP, we propose to use a Conjugate Residual (CR) solver as the backbone of our collision response system. By dynamically updating the set of active constraints, the MLCP with inequality constraints can be solved efficiently. We also propose a simple yet efficient preconditioner that ensures faster convergence. Finally, our approach is faster than existing methods (at the same accuracy), and it allows accurate treatment of friction.


    1. Jérémie Allard, François Faure, Hadrien Courtecuisse, Florent Falipou, Christian Duriez, and Paul G. Kry. 2010. Volume contact constraints at arbitrary resolution. ACM Trans. Graph. 29, 4 (2010), 82:1–82:10.
    2. Mihai Anitescu and Gary D. Hart. 2004. A fixed-point iteration approach for multibody dynamics with contact and small friction. Math. Program. 101, 1 (2004), 3–32.
    3. David Baraff. 1989. Analytical methods for dynamic simulation of non-penetrating rigid bodies. In Proc. of the 16th Annual Conf. on Computer Graphics and Interactive Techniques (SIGGRAPH’89). ACM, New York, NY, 223–232.
    4. David Baraff. 1994. Fast contact force computation for nonpenetrating rigid bodies. In Proc. of the 21st Annual Conf. on Computer Graphics and Interactive Techniques (SIGGRAPH’94). ACM, New York, NY, 23–34.
    5. David Baraff. 1996. Linear-time dynamics using Lagrange multipliers. In Proc. of the 23rd Annual Conf. on Computer Graphics and Interactive Techniques (SIGGRAPH’96). ACM, New York, NY, 137–146.
    6. David Baraff and Andrew Witkin. 1992. Dynamic simulation of non-penetrating flexible bodies. In Proc. of the 19th Annual Conf. on Computer Graphics and Interactive Techniques (SIGGRAPH’92). ACM, New York, NY, 303–308.
    7. David Baraff and Andrew Witkin. 1998. Large steps in cloth simulation. In Proc. of the 25th Annual Conf. on Computer Graphics and Interactive Techniques (SIGGRAPH’98). ACM, New York, NY, 43–54.
    8. Jernej Barbič and Doug James. 2007. Time-critical distributed contact for 6-DoF haptic rendering of adaptively sampled reduced deformable models. In Proc. of the 2007 ACM SIGGRAPH/Eurographics Symp. on Computer Animation (SCA’07). 171–180.
    9. Jan Bender, Kenny Erleben, and Jeff Trinkle. 2014. Interactive simulation of rigid body dynamics in computer graphics. Comput. Graph. Forum 33, 1 (2014), 246–270.
    10. Florence Bertails-Descoubes, Florent Cadoux, Gilles Daviet, and Vincent Acary. 2011. A nonsmooth Newton solver for capturing exact Coulomb friction in fiber assemblies. ACM Trans. Graph. 30, 1 (2011), 6:1–6:14.
    11. Stephen Boyd and Lieven Vandenberghe. 2004. Convex Optimization. Cambridge University Press.
    12. Robert Bridson, Ronald Fedkiw, and John Anderson. 2002. Robust treatment of collisions, contact and friction for cloth animation. ACM Trans. Graph. 21, 3 (2002), 594–603.
    13. Erin Catto. 2005. Iterative dynamics with temporal coherence. In Proc. of the 2005 Game Developer Conf.
    14. Richard Cottle, Jong-Shi Pang, and Richard E. Stone. 1992. The Linear Complementarity Problem. Academic Press.
    15. Gilles Daviet, Florence Bertails-Descoubes, and Laurence Boissieux. 2011. A hybrid iterative solver for robustly capturing Coulomb friction in hair dynamics. ACM Trans. Graph. 30, 6 (2011), 139:1–139:12.
    16. Christian Duriez, Frederic Dubois, Abderrahmane Kheddar, and Claude Andriot. 2006. Realistic haptic rendering of interacting deformable objects in virtual environments. IEEE Trans. Vis. Comput. Graph. 12, 1 (2006), 36–47.
    17. Stanley C. Eisenstat, Howard C. Elman, and Martin H. Schultz. 1983. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20, 2 (1983), 345–357.
    18. Kenny Erleben. 2007. Velocity-based shock propagation for multibody dynamics animation. ACM Trans. Graph. 26, 2 (2007), Article 12.
    19. Kenny Erleben. 2013. Numerical methods for linear complementarity problems in physics-based animation. In ACM SIGGRAPH 2013 Courses (SIGGRAPH’13). ACM, New York, NY, 8:1–8:42.
    20. David C. Fong and Michael Saunders. 2011. CG Versus MINRES: An Empirical Comparison. Technical Report. Stanford University, Stanford, CA.
    21. Nico Galoppo, Miguel A. Otaduy, Paul Mecklenburg, Markus Gross, and Ming C. Lin. 2006. Fast simulation of deformable models in contact using dynamic deformation textures. In Proc. of the 2006 ACM SIGGRAPH/Eurographics Symp. on Computer Animation (SCA’06). 73–82.
    22. Gene H. Golub and Charles F. Van Loan. 1996. Matrix Computations (3rd ed.). Johns Hopkins University Press, Baltimore, MD.
    23. Eran Guendelman, Robert Bridson, and Ronald Fedkiw. 2003. Nonconvex rigid bodies with stacking. ACM Trans. Graph. 22, 3 (2003), 871–878.
    24. Ken Hayami. 2002. On the behaviour of the conjugate residual method for singular systems. In Proc. of the 5th China-Japan Seminar on Numerical Mathematics. 117–126.
    25. T. Heyn, M. Anitescu, A. Tasora, and D. Negrut. 2012. Using Krylov subspace and spectral methods for solving complementarity problems in many-body contact dynamics simulation. Int. J. Numer. Meth. Eng. 95, 7 (2012), 541–561.
    26. Geoffrey Irving, Craig Schroeder, and Ronald Fedkiw. 2007. Volume conserving finite element simulations of deformable models. ACM Trans. Graph. 26, 3 (2007), Article 13.
    27. G. Irving, J. Teran, and R. Fedkiw. 2004. Invertible finite elements for robust simulation of large deformation. In Proc. of the 2004 ACM SIGGRAPH/Eurographics Symp. on Computer Animation (SCA’04). 131–140.
    28. J. Judice and F. Pires. 1994. A block principal pivoting algorithm for large-scale strictly monotone linear complementarity problems. Comput. Oper. Res. 21, 5 (1994), 587–596.
    29. Danny M. Kaufman, Shinjiro Sueda, Doug L. James, and Dinesh K. Pai. 2008. Staggered projections for frictional contact in multibody systems. ACM Trans. Graph. 27, 5 (2008), 164:1–164:11.
    30. H. W. Kuhn and A. W. Tucker. 1950. Nonlinear programming. In Proc. of the 2nd Berkeley Symp. on Mathematical Statistics and Probability. 481–492.
    31. Siwang Li, Zherong Pan, Jin Huang, Hujun Bao, and Xiaogang Jin. 2015. Deformable objects collision handling with fast convergence. Comput. Graph. Forum 34, 7 (2015), 269–278.
    32. D. Lobo, M. Saraç, M. Verschoor, M. Solazzi, A. Frisoli, and M. A. Otaduy. 2017. Proxy-based haptic rendering for underactuated haptic devices. In Proc. of the 2017 IEEE World Haptics Conf. (WHC’17). 48–53.
    33. David G. Luenberger. 1970. The conjugate residual method for constrained minimization problems. SIAM J. Numer. Anal. 7, 3 (1970), 390–398.
    34. David G. Luenberger. 1973. An approach to nonlinear programming. J. Optimiz. Theory App. 11, 3 (1973), 219–227.
    35. H. Mazhar, T. Heyn, D. Negrut, and A. Tasora. 2015. Using Nesterov’s method to accelerate multibody dynamics with friction and contact. ACM Trans. Graph. 34, 3 (2015), 32:1–32:14.
    36. Eder Miguel and Miguel A. Otaduy. 2011. Efficient simulation of contact between rigid and deformable objects. In ECCOMAS—Multibody Dynamics 2011.
    37. Matthew Moore and Jane Wilhelms. 1988. Collision detection and response for computer animation. In Proc. of the 15th Annual Conf. on Computer Graphics and Interactive Techniques (SIGGRAPH’88). ACM, New York, NY, 289–298.
    38. Matthias Müller and Markus Gross. 2004. Interactive virtual materials. In Proc. of Graphics Interface 2004 (GI’04). 239–246.
    39. Matthias Müller, Bruno Heidelberger, Marcus Hennix, and John Ratcliff. 2007. Position based dynamics. J. Vis. Comun. Image Represent. 18, 2 (2007), 109–118.
    40. Miguel A. Otaduy, Rasmus Tamstorf, Denis Steinemann, and Markus Gross. 2009. Implicit contact handling for deformable objects. Comput. Graph. Forum 28, 2 (2009), 559–568.
    41. C. C. Paige and M. A. Saunders. 1975. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 4 (1975), 617–629.
    42. Mark Pauly, Dinesh K. Pai, and Leonidas J. Guibas. 2004. Quasi-rigid objects in contact. In Proc. of the 2004 ACM SIGGRAPH/Eurographics Symp. on Computer Animation (SCA’04). 109–119.
    43. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. 1992. Numerical Recipes: The Art of Scientific Computing (2nd ed.). Cambridge University Press.
    44. Laks Raghupathi and François Faure. 2006. QP-collide: A new approach to collision treatment. In journées du groupe de travail Animation et Simulation (GTAS) (Annual French Working Group on Animation and Simulation). Institut de Recherche en Informatique de Toulouse, 91–101.
    45. Alison Ramage and Andrew J. Wathen. 1994. Iterative solution techniques for the stokes and Navier-Stokes equations. Int. J. Numer. Methods Fluids 19 (1994), 67–83.
    46. Stéphane Redon, Abderrahmane Kheddar, and Sabine Coquillart. 2002. Gauss’ least constraints principle and rigid body simulations. In Proc. of the IEEE Int. Conf. on Robotics and Automation. 11–15.
    47. Mathieu Renouf and Pierre Alart. 2005. Conjugate gradient type algorithms for frictional multi-contact problems: Applications to granular materials. Comput. Methods Appl. Mech. Eng. 194, 18–20 (2005), 2019–2041.
    48. Youcef Saad and Martin H. Schultz. 1986. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 3 (1986), 856–869.
    49. Tamar Shinar, Craig Schroeder, and Ronald Fedkiw. 2008. Two-way coupling of rigid and deformable bodies. In Proc. of the 2008 ACM SIGGRAPH/Eurographics Symp. on Computer Animation (SCA’08). 95–103.
    50. Morten Silcowitz-Hansen, Sarah Maria Niebe Abel, and Kenny Erleben. 2010a. Projected Gauss-Seidel subspace minimization method for interactive rigid body dynamics: Improving animation quality using a projected Gauss–Seidel subspace minimization method. In Proc. of the Int. Conf. on Computer Graphics Theory and Applications (GRAPP’10), Vol. 1. 38–45.
    51. Morten Silcowitz-Hansen, Sarah Niebe, and Kenny Erleben. 2010b. A nonsmooth nonlinear conjugate gradient method for interactive contact force problems. Vis. Comput. 26, 6–8 (2010), 893–901.
    52. Funshing Sin, Yufeng Zhu, Yongqiang Li, and Daniel Schroeder. 2011. Invertible isotropic hyperelasticity using SVD gradients. In Proc. of the ACM SIGGRAPH/Eurographics Symp. on Computer Animation (Posters).
    53. J. Spillmann, M. Becker, and M. Teschner. 2007. Non-iterative computation of contact forces for deformable objects. J. WSCG 14, (2007), 1–3.
    54. D. Stewart and J. C. Trinkle. 1996. An implicit time-stepping scheme for rigid body dynamics with coulomb friction. Int. J. Numer. Meth. Eng. 39 (1996), 2673–2691.
    55. Alexey Stomakhin, Russell Howes, Craig Schroeder, and Joseph M. Teran. 2012. Energetically consistent invertible elasticity. In Proc. of the ACM SIGGRAPH/Eurographics Symp. on Computer Animation (SCA’12). 25–32.
    56. Min Tang, Dinesh Manocha, Miguel A. Otaduy, and Ruofeng Tong. 2012. Continuous penalty forces. ACM Trans. Graph. 31, 4 (2012), 107:1–107:9.
    57. M. Teschner, S. Kimmerle, B. Heidelberger, G. Zachmann, L. Raghupathi, A. Fuhrmann, et al. 2004. Collision detection for deformable objects. Comput. Graph. Forum 24, 1 (2004), 61–81.
    58. Richard Tonge, Feodor Benevolenski, and Andrey Voroshilov. 2012. Mass splitting for jitter-free parallel rigid body simulation. ACM Trans. Graph. 31, 4 (2012), 105:1–105:8.
    59. Mickeal Verschoor and Andrei C. Jalba. 2012. Analysis and performance estimation of the conjugate gradient method on multiple GPUs. Parallel Comput. 38, 10–11 (2012), 552–575.
    60. Pascal Volino and Nadia Magnenat Thalmann. 1994. Efficient self-collision detection on smoothly discretized surface animations using geometrical shape regularity. Comput. Graph. Forum 13, 3 (1994), 155–166.
    61. P. Wriggers. 2002. Computational Contact Mechanics. Wiley.
    62. H. Xu, Y. Zhao, and J. Barbič. 2014. Implicit multibody penalty-based distributed contact. IEEE Trans. Vis. Comput. Graph. 20, 9 (2014), 1266–1279.

ACM Digital Library Publication:

Overview Page: