“Unified simulation of elastic rods, shells, and solids” by Martin, Kaufmann, Botsch, Grinspun and Gross

  • ©Sebastian Martin, Peter Kaufmann, Mario Botsch, Eitan Grinspun, and Markus Gross




    Unified simulation of elastic rods, shells, and solids



    We develop an accurate, unified treatment of elastica. Following the method of resultant-based formulation to its logical extreme, we derive a higher-order integration rule, or elaston, measuring stretching, shearing, bending, and twisting along any axis. The theory and accompanying implementation do not distinguish between forms of different dimension (solids, shells, rods), nor between manifold regions and non-manifold junctions. Consequently, a single code accurately models a diverse range of elastoplastic behaviors, including buckling, writhing, cutting and merging. Emphasis on convergence to the continuum sets us apart from early unification efforts.


    1. Adams, B., Ovsjanikov, M., Wand, M., Seidel, H.-P., and Guibas, L. J. 2008. Meshless modeling of deformable shapes and their motion. In Proc. of Symp. on Computer Animation, 77–86. Google ScholarDigital Library
    2. An, S. S., Kim, T., and James, D. L. 2008. Optimizing cubature for efficient integration of subspace deformations. ACM Trans. on Graphics 27, 5, 164:1–164:11. Google ScholarDigital Library
    3. Atluri, S. N., Cho, J. Y., and Kim, H.-G. 1999. Analysis of thin beams, using the meshless local Petrov-Galerkin method, with generalized moving least squares interpolations. Computational Mechanics, 24, 334–347.Google ScholarCross Ref
    4. Baraff, D., and Witkin, A. 1998. Large steps in cloth simulation. In Proc. of ACM SIGGRAPH, 43–54. Google ScholarDigital Library
    5. Bargteil, A. W., Wojtan, C., Hodgins, J. K., and Turk, G. 2007. A finite element method for animating large viscoplastic flow. ACM Trans. on Graphics 26, 3, 16.1–16.8. Google ScholarDigital Library
    6. Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., and Grinspun, E. 2008. Discrete elastic rods. ACM Trans. on Graphics 27, 3, 63:1–63:12. Google ScholarDigital Library
    7. Bridson, R., Marino, S., and Fedkiw, R. 2003. Simulation of clothing with folds and wrinkles. In Proc. of Symp. on Computer Animation, 28–36. Google ScholarDigital Library
    8. Chen, Y., Davis, T. A., Hager, W. W., and Rajamanickam, S. 2008. Algorithm 887: CHOLMOD, supernodal sparse cholesky factorization and update/downdate. ACM Trans. Math. Softw. 35, 3, 1–14. Google ScholarDigital Library
    9. Cirak, F., Ortiz, M., and Schröder, P. 2000. Subdivision surfaces: A new paradigm for thin-shell finite-element analysis. Int. J. Numer. Methods Eng. 47, 12, 2039–2072.Google ScholarCross Ref
    10. Delingette, H. 2008. Triangular springs for modeling nonlinear membranes. IEEE Trans. on Visualization and Computer Graphics 14, 2, 329–341. Google ScholarDigital Library
    11. Etzmuss, O., Gross, J., and Strasser, W. 2003. Deriving a particle system from continuum mechanics for the animation of deformable objects. IEEE Trans. on Visualization and Computer Graphics 9, 538–550. Google ScholarDigital Library
    12. Fries, T. P., and Matthies, H. G. 2004. Classification and overview of meshfree methods. Informatikbericht 2003-03, revised 2004, Institute of Scientific Computing, Technical University Braunschweig.Google Scholar
    13. Gerszewski, D., Bhattacharya, H., and Bargteil, A. W. 2009. A point-based method for animating elastoplastic solids. In Proc. of Symp. on Computer Animation, 133–138. Google ScholarDigital Library
    14. Grinspun, E., Hirani, A. N., Desbrun, M., and Schröder, P. 2003. Discrete shells. In Proc. of Symp. on Computer Animation, 62–67. Google ScholarDigital Library
    15. Guo, X., Li, X., Bao, Y., Gu, X., and Qin, H. 2006. Meshless thin-shell simulation based on global conformal parameterization. IEEE Trans. on Visualization and Computer Graphics 12, 3, 375–385. Google ScholarDigital Library
    16. Hadap, S., Cani, M.-P., Lin, M., Kim, T.-Y., Bertails, F., Marschner, S., Ward, K., and Kaĉić-Alesić, Z. 2007. Strands and hair: modeling, animation, and rendering. ACM SIGGRAPH 2007 Courses, 1–150. Google ScholarDigital Library
    17. Hauth, M., and Strasser, W. 2004. Corotational simulation of deformable solids. In Proc. of WSCG, 137–145.Google Scholar
    18. Hughes, T. J. R. 2000. The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Dover Publications.Google Scholar
    19. Irving, G., Schroeder, C., and Fedkiw, R. 2007. Volume conserving finite element simulations of deformable models. ACM Trans. on Graphics 26, 3, 13.1–13.6. Google ScholarDigital Library
    20. Kikuuwe, R., Tabuchi, H., and Yamamoto, M. 2009. An edge-based computationally efficient formulation of Saint Venant-Kirchhoff tetrahedral finite elements. ACM Trans. on Graphics 28, 1, 1–13. Google ScholarDigital Library
    21. Krysl, P. 2005. A Pragmatic Introduction to the Finite Element Method for Thermal and Stress Analysis. Pressure Cooker Press, San Diego.Google Scholar
    22. Lloyd, B., Székely, G., and Harders, M. 2007. Identification of spring parameters for deformable object simulation. IEEE Trans. on Visualization and Computer Graphics 13, 1081–1094. Google ScholarDigital Library
    23. Lloyd, S. 1957. Least squares quantization in PCM. Tech. rep., Tech. rep., Bell Telephone Laboratories, Murray Hill, NJ.Google Scholar
    24. Malvern, L. E. 1969. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
    25. Mezger, J., Thomaszewski, B., Pabst, S., and Strasser, W. 2008. Interactive physically-based shape editing. In Proc. of ACM Symp. on Solid and Physical Modeling, 79–89. Google ScholarDigital Library
    26. Molino, N., Bao, Z., and Fedkiw, R. 2004. A virtual node algorithm for changing mesh topology during simulation. ACM Trans. on Graphics 23, 3, 385–392. Google ScholarDigital Library
    27. Müller, M., Dorsey, J., McMillan, L., Jagnow, R., and Cutler, B. 2002. Stable real-time deformations. In Proc. of Symp. on Computer Animation, 163–170. Google ScholarDigital Library
    28. Müller, M., Keiser, R., Nealen, A., Pauly, M., Gross, M., and Alexa, M. 2004. Point-based animation of elastic, plastic and melting objects. In Proc. of Symp. on Computer Animation, 141–151. Google ScholarDigital Library
    29. Müller, M., Heidelberger, B., Hennix, M., and Ratcliff, J. 2007. Position based dynamics. Journal of Visual Communication and Image Representation 18, 2, 109–118. Google ScholarDigital Library
    30. Naghdi, P. 1972. Handbuch der Physik, Mechanics of Solids II, vol. VI a/2. Springer, Berlin.Google Scholar
    31. Nesme, M., Kry, P. G., Jeřábková, L., and Faure, F. 2009. Preserving topology and elasticity for embedded deformable models. ACM Trans. on Graphics 28, 3, 52:1–52:9. Google ScholarDigital Library
    32. O’Brien, J. F., and Hodgins, J. K. 1999. Graphical modeling and animation of brittle fracture. In Proc. of ACM SIGGRAPH, 137–146. Google ScholarDigital Library
    33. O’Brien, J. F., Bargteil, A. W., and Hodgins, J. K. 2002. Graphical modeling and animation of ductile fracture. ACM Trans. on Graphics 21, 3, 291–294. Google ScholarDigital Library
    34. Pai, D. K. 2002. STRANDS: interactive simulation of thin solids using Cosserat models. Comput. Graphics Forum 21, 3, 347–352.Google ScholarCross Ref
    35. Pauly, M., Keiser, R., Adams, B., Dutre, P., Gross, M., and Guibas, L. J. 2005. Meshless animation of fracturing solids. ACM Trans. on Graphics 24, 3, 957–964. Google ScholarDigital Library
    36. Provot, X. 1995. Deformation constraints in a mass-spring model to describe rigid cloth behavior. In Graphics Interface ’95, 147–154.Google Scholar
    37. Rubin, M. B. 1985. On the theory of a cosserat point and its application to the numerical solution of continuum problems. Journal of Applied Mechanics 52, 2, 368–372.Google ScholarCross Ref
    38. Selle, A., Lentine, M., and Fedkiw, R. 2008. A mass spring model for hair simulation. ACM Trans. on Graphics 27, 3, 64.1–64.11. Google ScholarDigital Library
    39. Sifakis, E., Shinar, T., Irving, G., and Fedkiw, R. 2007. Hybrid simulation of deformable solids. In Proc. of Symp. on Computer Animation, 81–90. Google ScholarDigital Library
    40. Spillmann, J., and Teschner, M. 2007. CoRdE: Cosserat rod elements for the dynamic simulation of one-dimensional elastic objects. In Proc. of Symp. on Computer Animation, 63–72. Google ScholarDigital Library
    41. Stam, J. 2009. Nucleus: Towards a unified dynamics solver for computer graphics. In IEEE International Conference on Computer-Aided Design and Computer Graphics, 1–11.Google ScholarCross Ref
    42. Steinemann, D., Otaduy, M. A., and Gross, M. 2006. Fast arbitrary splitting of deforming objects. In Proc. of Symp. on Computer Animation, 63–72. Google ScholarDigital Library
    43. Terzopoulos, D., and Fleischer, K. 1988. Modeling inelastic deformation: Viscoelasticity, plasticity, fracture. In Proc. of ACM SIGGRAPH, 269–278. Google ScholarDigital Library
    44. Terzopoulos, D., Platt, J., Barr, A., and Fleischer, K. 1987. Elastically deformable models. In Proc. of ACM SIGGRAPH, 205–214. Google ScholarDigital Library
    45. Van Gelder, A. 1998. Approximate simulation of elastic membranes by triangulated spring meshes. Journal of Graphics Tools 3, 2, 21–42. Google ScholarDigital Library
    46. Veubeke, B. F. D. 1976. The dynamics of flexible bodies. International Journal of Engineering Science 14, 895–913.Google ScholarCross Ref
    47. Volino, P., Magnenat-Thalmann, N., and Faure, F. 2009. A simple approach to nonlinear tensile stiffness for accurate cloth simulation. ACM Trans. on Graphics 28, 4, 1–16. Google ScholarDigital Library
    48. Wang, X., and Devarajan, V. 2005. 1D and 2D structured mass-spring models with preload. Visual Computer 21, 7, 429–448.Google ScholarCross Ref
    49. Wicke, M., Steinemann, D., and Gross, M. 2005. Efficient animation of point-sampled thin shells. Comput. Graphics Forum 24, 667–676.Google ScholarCross Ref
    50. Wojtan, C., and Turk, G. 2008. Fast viscoelastic behavior with thin features. ACM Trans. on Graphics 27, 3, 47:1–47:8. Google ScholarDigital Library
    51. Yang, H., Saigal, S., Masud, A., and Kapania, R. 2000. A survey of recent shell finite elements. Int. J. Numer. Methods Eng., 47, 101–127.Google ScholarCross Ref
    52. Zerbato, D., Galvan, S., and Fiorini, P. 2007. Calibration of mass spring models for organ simulations. In IEEE/RSJ International Conference on Intelligent Robots and Systems, 370–375.Google Scholar

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