Computing and Visualization in Science

, Volume 16, Issue 3, pp 119–138 | Cite as

Towards an efficient numerical simulation of complex 3D knee joint motion

  • Oliver SanderEmail author
  • Corinna Klapproth
  • Jonathan Youett
  • Ralf Kornhuber
  • Peter Deuflhard


We present a time-dependent finite element model of the human knee joint of full 3D geometric complexity together with advanced numerical algorithms needed for its simulation. The model comprises bones, cartilage and the major ligaments, while patella and menisci are still missing. Bones are modeled by linear elastic materials, cartilage by linear viscoelastic materials, and ligaments by one-dimensional nonlinear Cosserat rods. In order to capture the dynamical contact problems correctly, we solve the full PDEs of elasticity with strict contact inequalities. The spatio-temporal discretization follows a time layers approach (first time, then space discretization). For the time discretization of the elastic and viscoelastic parts we use a new contact-stabilized Newmark method, while for the Cosserat rods we choose an energy-momentum method. For the space discretization, we use linear finite elements for the elastic and viscoelastic parts and novel geodesic finite elements for the Cosserat rods. The coupled system is solved by a Dirichlet–Neumann method. The large algebraic systems of the bone–cartilage contact problems are solved efficiently by the truncated non-smooth Newton multigrid method.


Biomechanics Time-dependent contact problem Contact-stabilized Newmark method Domain decomposition Energy-momentum method Geodesic finite elements Knee model 


  1. 1.
    Abdel-Rahman, E., Hefzy, M.S.: A two-dimensional dynamic anatomical model of the human knee joint. J. Biomech. Eng. 115, 357–365 (1993)CrossRefGoogle Scholar
  2. 2.
    Ahn, J., Stewart, D.E.: Dynamic frictionless contact in linear viscoelasticity. IMA J. Numer. Anal. 29(1), 43–71 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Antman, S.S.: Nonlinear Problems Of Elasticity, Volume 107 of Applied Mathematical Sciences. Springer, Berlin (1991)Google Scholar
  4. 4.
    Arbenz, P., van Lenthe, G.H., Mennel, U., Müller, R., Sala, M.: A scalable multi-level preconditioner for matrix-free \(\mu \)-finite element analysis of human bone structures. Int. J. Numer. Meth. Eng. 73(7), 927–947 (2008)Google Scholar
  5. 5.
    Bastian, P., Birken, K., Johannsen, K., Lang, S., Neuß, N., Rentz-Reichert, H., Wieners, C.: UG—a flexible software toolbox for solving partial differential equations. Comp. Vis. Sci. 1, 27–40 (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic interface for adaptive and parallel scientific computing. Part II: implementation and tests in DUNE. Computing 82(2–3), 121–138 (2008)Google Scholar
  7. 7.
    Bastian, P., Buse, G., Sander, O.: Infrastructure for the coupling of Dune grids. In: Proceedings of the ENUMATH 2009, pp. 107–114. Springer, Berlin (2010)Google Scholar
  8. 8.
    Bei, Y., Fregly, B.J.: Multibody dynamic simulation of knee contact mechanics. Med. Eng. Phys. 26(9), 777–789 (2004)CrossRefGoogle Scholar
  9. 9.
    Blankevoort, L., Huiskes, H.: Ligament-bone interaction in a three-dimensional model of the knee. J. Biomech. Eng. 113(3), 263–269 (1991)CrossRefGoogle Scholar
  10. 10.
    Currey, J.D.: Bones: Structure and Mechanics. Princeton University Press, Princeton (2002)Google Scholar
  11. 11.
    Deuflhard, P., Krause, R., Ertel, S.: A contact-stabilized Newmark method for dynamical contact problems. Int. J. Numer. Methods Eng. 73(9), 1274–1290 (2007)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Deuflhard, P., Weiser, M.: Adaptive Numerical Solution of PDEs. de Gruyter, Berlin (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Donahue, T.H., Hull, M.L., Rashid, M.M., Jacobs, C.R.: A finite element model of the human knee joint for the study of tibio-femoral contact. J. Biomech. Eng. 124(3), 273–280 (2002)CrossRefGoogle Scholar
  14. 14.
    Eck, C.: Existenz und Regularität der Lösungen für Kontaktprobleme mit Reibung. PhD thesis, Universität Stuttgart (1996)Google Scholar
  15. 15.
    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1987)Google Scholar
  16. 16.
    Gräser, C., Kornhuber, R.: Multigrid methods for obstacle problems. J. Comp. Math. 27(1), 1–44 (2009)zbMATHGoogle Scholar
  17. 17.
    Gräser, C., Sack, U., Sander, O.: Truncated nonsmooth Newton multigrid methods for convex minimization problems. In: Proceedings of the DD18, pp. 129–136 (2009)Google Scholar
  18. 18.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  19. 19.
    Heller, M., König, C., Graichen, H., Hinterwimmer, S., Ehrig, R., Duda, G., Taylor, W.: A new model to predict in vivo human knee kinematics under physiological-like muscel activation. J. Biomech. 40, 45–53 (2007)CrossRefGoogle Scholar
  20. 20.
    Huang, C., Mow, V., Ateshian, G.: The role of flow-independent viscoelasticity in the biphasic tensile and compressive responses of articular cartilage. J. Biomech. Eng. 123(5), 410–7 (2001)CrossRefGoogle Scholar
  21. 21.
    Kehrbaum, S.: Hamiltonian Formulations of the Equilibrium Conditions Governing Elastic Rods: Qualitative Analysis and Effective Properties. PhD thesis, University of Maryland (1997)Google Scholar
  22. 22.
    Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity. SIAM, Philadelphia (1988)CrossRefzbMATHGoogle Scholar
  23. 23.
    Klapproth, C.: Adaptive numerical integration for dynamical contact problems. PhD thesis, Freie Universität Berlin (2011); also published at Cuvillier Verl. GöttingenGoogle Scholar
  24. 24.
    Klapproth, C., Deuflhard, P., Schiela, A.: A perturbation result for dynamical contact problems. Numer. Math. Theor. Meth. Appl. 2(3), 237–257 (2009)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Klapproth, C., Schiela, A., Deuflhard, P.: Consistency results on Newmark methods for dynamical contact problems. Numer. Math. 116(1), 65–94 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Klapproth, C., Schiela, A., Deuflhard, P.: Adaptive timestep control for the contact-stabilized Newmark method. Numer. Math. 119(1), 49–81 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kornhuber, R., Krause, R.: Adaptive multigrid methods for Signorini’s problem in linear elasticity. Comp. Vis. Sci. 4(1), 9–20 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Kornhuber, R., Krause, R., Sander, O., Deuflhard, P., Ertel, S.: A monotone multigrid solver for two body contact problems in biomechanics. Comp. Vis. Sci. 11(1), 3–15 (2008)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Kornhuber, R.: Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. B.G Teubner, Stuttgart (1997)Google Scholar
  30. 30.
    Krause, R., Sander, O.: Automatic construction of boundary parametrizations for geometric multigrid solvers. Comp. Vis. Sci. 9, 11–22 (2006)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Krause, R., Schittler, D., Reiter, M., Waldherr, S., Allgöwer, F., Karastoyanova, D., Leymann, F., Markert, B., Ehlers, W.: Bone remodelling: a combined biomechanical and systems-biological challenge. PAMM 11, 99–100 (2011)CrossRefGoogle Scholar
  32. 32.
    Lai, W., Hou, J., Mow, V.: A triphasic theory for the swelling and deformation behaviours of articular cartilage. J. Biomech. Eng. 113, 245–58 (1991)CrossRefGoogle Scholar
  33. 33.
    Laursen, T.A., Chawla, V.: Design of energy conserving algorithms for frictionless dynamic contact problems. Int. J. Numer. Methods Eng. 40(5), 863–886 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Laursen, T.A.: Computational Contact and Impact Mechanics. Springer, Berlin (2003)CrossRefGoogle Scholar
  35. 35.
    Machado, M., Flores, P., Claro, J., Ambrósio, J., Silva, M., Completo, A., Lankarani, H.: Development of a planar multibody model of the human knee joint. Nonlinear Dyn. 60, 459–478 (2010)CrossRefzbMATHGoogle Scholar
  36. 36.
    McLean, S.G., Su, A., van den Bogert, A.J.: Development and validation of a 3-D model to predict knee joint loading during dynamic movement. J. Biomech. Eng. 125, 864–875 (2003)CrossRefGoogle Scholar
  37. 37.
    Nackenhorst, U.: Computational methods for studies on the biomechanics of bones. Found. Civil Environ. Eng. 11, 99–100 (2006)Google Scholar
  38. 38.
    Penrose, M.T., Holt, G.M., Beaugonin, M., Hose, D.R.: Development of an accurate three-dimensional finite element knee model. Comp. Meth. Biomech. Biomed. Eng. 5(4), 291–300 (2002)CrossRefGoogle Scholar
  39. 39.
    Piazza, S.J., Delp, S.L.: Three-dimensional dynamic simulation of total knee replacement motion during a step-up task. J. Biomech. Eng. 123, 599–607 (2001)CrossRefGoogle Scholar
  40. 40.
    Putz, R., Pabst, R., ed.: Sobotta—Atlas der Anatomie des Menschen. Urban & Fischer (2000)Google Scholar
  41. 41.
    Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, Oxford (1999)zbMATHGoogle Scholar
  42. 42.
    Sander, O.: Multidimensional Coupling in a Human Knee Model. PhD thesis, Freie Universität Berlin (2008)Google Scholar
  43. 43.
    Sander, O., Schiela, A.: Energy Minimizers of the Coupling of a Cosserat Rod to an Elastic Continuum. Technical Report 903, Matheon (2012); submittedGoogle Scholar
  44. 44.
    Sander, O.: Geodesic finite elements for Cosserat rods. Int. J. Numer. Methods Eng. 82(13), 1645–1670 (2010)zbMATHMathSciNetGoogle Scholar
  45. 45.
    Sander, O.: The Psurface Library. Technical Report 708, Matheon (2010); to appear in CVSGoogle Scholar
  46. 46.
    Sander, O.: Coupling Geometrically Exact Cosserat Rods and Linear Elastic Continua. Technical Report 772, Matheon (2011); to appear in Proc. DD20Google Scholar
  47. 47.
    Schechter, E.: Handbook of Analysis and its Foundations. Academic Press, San Diego (1997)zbMATHGoogle Scholar
  48. 48.
    Simo, J.C., Tarnow, N.: The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. J. Appl. Math. Phys. 43, 757–792 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    Simo, J.C., Tarnow, N., Doblaré, M.: Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. Int. J. Numer. Methods Eng. 38, 1431–1473 (1995)CrossRefzbMATHGoogle Scholar
  50. 50.
    Suh, J., Spilker, R.: Indentation analysis of biphasic articular cartilage: nonlinear phenomena under finite deformation. J. Biomech. Eng. 116(1), 1–9 (1994)CrossRefGoogle Scholar
  51. 51.
  52. 52.
    van Eijden, T.M., Kouwenhoven, E., Verburg, J., Weijs, W.A.: A mathematical model of the patellofemoral joint. J. Biomech. 19, 219–229 (1986)CrossRefGoogle Scholar
  53. 53.
    Weiss, J., Gardiner, J.C.: Computational modelling of ligament mechanics. Crit. Rev. Biomed. Eng. 29(4), 1–70 (2001)Google Scholar
  54. 54.
    Wilson, W., van Donkelaar, C., van Rietbergen, R., Huiskes, R.: The role of computational models in the search for the mechanical behaviour and damage mechanisms of articular cartilage. Med. Eng. Phys. 27, 810–826 (2005)CrossRefGoogle Scholar
  55. 55.
    Wilson, W., van Donkelaar, C., van Rietbergen, R., Ito, K., Huiskes, R.: Stresses in the local collagen network of articular cartilage: a poroviscoelastic fibril-reinforced finite element study. J. Biomech. Eng. 37(3), 357–66 (2004)CrossRefGoogle Scholar
  56. 56.
    Wohlmuth, B.: Discretization Methods and Iterative Solvers based on Domain Decomposition. LNCSE, vol. 17. Springer, Berlin (2001)CrossRefGoogle Scholar
  57. 57.
    Wohlmuth, B.: Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numerica 20, 569–734 (2011)Google Scholar
  58. 58.
    Wohlmuth, B., Krause, R.: Monotone methods on nonmatching grids for nonlinear contact problems. SIAM J. Sci. Comput. 25(1), 324–347 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    Yao, J., Salo, A.D., Lee, J., Lerner, A.L.: Sensitivity of tibio-menisco-femoral joint contact behavior to variations in knee kinematics. J. Biomech. Eng. 41(2), 390–398 (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Oliver Sander
    • 1
    Email author
  • Corinna Klapproth
    • 2
  • Jonathan Youett
    • 1
  • Ralf Kornhuber
    • 1
  • Peter Deuflhard
    • 2
  1. 1.Freie Universität Berlin Institut für MathematikBerlinGermany
  2. 2.Zuse Institute Berlin (ZIB)BerlinGermany

Personalised recommendations