“Volume conserving finite element simulations of deformable models” by Irving, Schroeder and Fedkiw

  • ©Geoffrey Irving, Craig Schroeder, and Ronald Fedkiw




    Volume conserving finite element simulations of deformable models



    We propose a numerical method for modeling highly deformable nonlinear incompressible solids that conserves the volume locally near each node in a finite element mesh. Our method works with arbitrary constitutive models, is applicable to both passive and active materials (e.g. muscles), and works with simple tetrahedra without the need for multiple quadrature points or stabilization techniques. Although simple linear tetrahedra typically suffer from locking when modeling incompressible materials, our method enforces incompressibility per node (in a one-ring), and we demonstrate that it is free from locking. We correct errors in volume without introducing oscillations by treating position and velocity in separate implicit solves. Finally, we propose a novel method for treating both object contact and self-contact as linear constraints during the incompressible solve, alleviating issues in enforcing multiple possibly conflicting constraints.


    1. Angelidis, A., Cani, M., Wyvill, G., and King, S. 2006. Swirling-sweepers: constant volume modeling. Graphical Models 68, 4, 324–32. Google ScholarDigital Library
    2. Bijelonja, I., Demirdžić, I., and Muzaferija, S. 2005. A finite volume method for large strain analysis of incompressible hyperelastic materials. Int. J. Numer. Meth. Eng. 64, 1594–1609.Google ScholarCross Ref
    3. Bijelonja, I., Demirdžić, I., and Muzaferija, S. 2006. A finite volume method for incompressible linear elasticity. Comput. Methods Appl. Meth. Eng. 195, 6378–90.Google ScholarCross Ref
    4. Bonet, J., and Burton, A. 1998. A simple average nodal pressure tetrahedral element for incompressible and nearly incompressible dynamic explicit applications. Comm. Num. Meth. Eng. 14, 437–449.Google ScholarCross Ref
    5. Boroomand, B., and Khalilian, B. 2004. On using linear elements in incompressible plane strain problems: a simple edge based approach for triangles. Int. J. Num. Meth. Eng. 61, 1710–1740.Google ScholarCross Ref
    6. Bourguignon, D., and Cani, M. P. 2000. Controlling anisotropy in mass-spring systems. In Eurographics, Eurographics Assoc., 113–123.Google Scholar
    7. Bridson, R., Fedkiw, R., and Anderson, J. 2002. Robust treatment of collisions, contact and friction for cloth animation. ACM Trans. Graph. (SIGGRAPH Proc.) 21, 594–603. Google ScholarDigital Library
    8. Bridson, R., Marino, S., and Fedkiw, R. 2003. Simulation of clothing with folds and wrinkles. In Proc. of the 2003 ACM SIGGRAPH/Eurographics Symp. on Comput. Anim., 28–36. Google ScholarDigital Library
    9. Choi, S.-C. 2006. Iterative Methods for Singular Linear Equations and Least-Squares Problems. PhD thesis, Stanford Univ.Google Scholar
    10. Cockburn, B., Schötzau, D., and Wang, J. 2006. Discontinuous Galerkin methods for incompressible elastic materials. Comput. Methods Appl. Meth. Eng. 195, 3184–3204.Google ScholarCross Ref
    11. Cooper, L., and Maddock, S. 1997. Preventing collapse within mass-spring-damper models of deformable objects. In 5th Int. Conf. in Central Europe on Comput. Graph, and Vis.Google Scholar
    12. de Souza Neto, E., Pires, F. A., and Owen, D. 2005. F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. part i: formulation and benchmarking. Int. J. Num. Meth. Eng. 62, 353–383.Google ScholarCross Ref
    13. Desbrun, M., and Gascuel, M.-P. 1995. Animating soft substances with implicit surfaces. In Proc. SIGGRAPH 95, 287–290. Google ScholarDigital Library
    14. Dolbow, J., and Devan, A. 2004. Enrichment of enhanced assumed strain approximation for representing strong discontinuities: Addressing volumetric incompressibility and the discontinuous patch test. Int. J. Numer. Meth. Eng. 59, 47–67.Google ScholarCross Ref
    15. Fedkiw, R., Stam, J., and Jensen, H. 2001. Visual simulation of smoke. In Proc. of ACM SIGGRAPH 2001, 15–22. Google ScholarDigital Library
    16. Feldman, B. E., O’Brien, J. F., and Arikan, O. 2003. Animating suspended particle explosions. ACM Trans. Graph. (SIGGRAPH Proc.) 22, 3, 708–715. Google ScholarDigital Library
    17. Hong, M., Jung, S., Choi, M., and Welch, S. 2006. Fast volume preservation for a mass-spring system. IEEE Comput. Graph, and Appl. 26 (September), 83–91. Google ScholarDigital Library
    18. Irving, G., Teran, J., and Fedkiw, R. 2004. Invertible finite elements for robust simulation of large deformation. In Proc. of the ACM SIGGRAPH/Eurographics Symp. on Comput. Anim., 131–140. Google ScholarDigital Library
    19. Kharevych, L., Yang, W., Tong, Y., Kanso, E., Marsden, J., Schröder, P., and Desbrun, M. 2006. Geometric variational integrators for computer animation. ACM SIGGRAPH/Eurographics Symp. on Comput. Anim., 43–51. Google ScholarDigital Library
    20. Lahiri, S., Bonet, J., Peraire, J., and Casals, L. 2005. A variationally consistent fractional time-step integration method for incompressibility and nearly incompressible lagrangian dynamics. Int. J. Numer. Meth. Eng. 59, 1371–95.Google ScholarCross Ref
    21. Lasseter, R. 1987. Principles of traditional animation applied to 3D computer animation. Comput. Graph. (SIGGRAPH Proc.), 35–44. Google ScholarDigital Library
    22. Molino, N., Bridson, R., Teran, J., and Fedkiw, R. 2003. A crystalline, red green strategy for meshing highly deformable objects with tetrahedra. In 12th Int. Meshing Roundtable, 103–114.Google Scholar
    23. Nedel, L. P., and Thalmann, D. 1998. Real time muscle deformations using mass-spring systems. In Proc. Comput. Graph. Int., 156–165. Google ScholarDigital Library
    24. Nixon, D., and Lobb, R. 2002. A fluid-based soft-object model. IEEE Comput. Graph. Appl. 22, 4, 68–75. Google ScholarDigital Library
    25. Oñate, E., Rojek, J., Taylor, R., and Zienkiewicz, O. 2004. Finite calculus formulation for incompressible solids using linear triangles and tetrahedra. Int. J. Numer. Meth. Eng. 59, 1473–1500.Google ScholarCross Ref
    26. Picinbono, G., Delingette, H., and Ayache, N. 2001. Non-linear and anisotropic elastic soft tissue models for medical simulation. In IEEE Int. Conf. Robot, and Automation.Google Scholar
    27. Pires, F. A., de Souza Neto, E., and Owen, D. 2004. On the finite element prediction of damage growth and fracture initiation in finitely deforming ductile materials. Comput. Meth. in Appl. Mech. and Eng. 193, 5223–5256.Google ScholarCross Ref
    28. Platt, J. C., and Barr, A. H. 1988. Constraint methods for flexible models. Comput. Graph. (SIGGRAPH Proc.), 279–288. Google ScholarDigital Library
    29. Promayon, E., Baconnier, P., and Puech, C. 1996. Physically-based deformations constrained in displacements and volume. In Eurographics, vol. 15, Eurographics Assoc.Google Scholar
    30. Punak, S., and Peters, J. 2006. Localized volume preservation for simulation and animation. In Poster, SIGGRAPH Proc., ACM. Google ScholarDigital Library
    31. Rojek, J., Nate, E. O., and Taylor, R. 2006. Cbs-based stabilization in explicit solid dynamics. Int. J. Numer. Meth. Eng. 66, 1547–68.Google ScholarCross Ref
    32. Roth, S. H., Gross, M., Turello, M. H., and Carls, S. 1998. A Bernstein-Bézier based approach to soft tissue simulation. Comput. Graph. Forum (Proc. Eurographics) 17, 3, 285–294.Google ScholarCross Ref
    33. Selle, A., Su, J., Irving, G., and Fedkiw, R. 2007. Highly detailed folds and wrinkles for cloth simulation. In Proc. of ACM SIGGRAPH/Eurographics Symp. on Comput. Anim. (in review).Google Scholar
    34. Simo, J., and Taylor, R. 1991. Quasi-incompressible finite elasticity in principal stretches: continuum basis and numerical examples. Comput. Meth. in Appl. Mech. and Eng. 51, 273–310. Google ScholarDigital Library
    35. Teran, J., Blemker, S., Ng, V., and Fedkiw, R. 2003. Finite volume methods for the simulation of skeletal muscle. In Proc. of the 2003 ACM SIGGRAPH/Eurographics Symp. on Comput. Anim., 68–74. Google ScholarDigital Library
    36. Teran, J., Sifakis, E., Salinas-Blemker, S., Ng-Thow-Hing, V., Lau, C., and Fedkiw, R. 2005. Creating and simulating skeletal muscle from the visible human data set. IEEE Trans. on Vis. and Comput. Graph. 11, 3, 317–328. Google ScholarDigital Library
    37. Teschner, M., Heidelberger, B., Müller, M., and Gross, M. 2004. A versatile and robust model for geometrically complex deformable solids. In Proc. Comput. Graph. Int., 312–319. Google ScholarCross Ref
    38. Thoutireddy, P., Molinari, J., Repetto, E., and Ortiz, M. 2002. Tetrahedral composite finite elements. Int. J. Num. Meth. Eng. 53, 1337–1351.Google ScholarCross Ref
    39. von Funck, W., Theisel, H., and Seidel, H.-P. 2006. Vector field based shape deformations. ACM Trans. Graph. (SIGGRAPH Proc.) 25, 3, 1118–1125. Google ScholarDigital Library
    40. Weiss, J., Maker, B., and Govindjee, S. 1996. Finite-element implementation of incompressible, transversely isotropic hyperelasticity. Comput. Meth. in Appl. Mech. and Eng. 135, 107–128.Google ScholarCross Ref
    41. Yoon, S., and Kim, M. 2006. Sweep-based freeform deformations. In Eurographics, vol. 25, Eurographics Assoc., 487–96.Google Scholar

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