Mortar Methods for Single- and Multi-Field Applications in Computational Mechanics

  • Alexander Popp
  • Michael W. Gee
  • Wolfgang A. Wall
Conference paper


Mortar finite element methods are of great relevance as a non-conforming discretization technique in various single-field and multi-field applications. In computational contact analysis, the mortar approach allows for a variationally consistent treatment of non-penetration and frictional sliding constraints despite the inevitably non-matching interface meshes. Other single-field and multi-field problems, such as fluid–structure interaction (FSI), also benefit from the increased modeling flexibility provided by mortar methods. This contribution gives a review of the most important aspects of mortar finite element discretization and dual Lagrange multiplier interpolation for the aforementioned applications. The focus is on parallel efficiency, which is addressed by a new dynamic load balancing strategy and tailored parallel search algorithms for computational contact mechanics. For validation purposes, simulation examples from solid dynamics, contact dynamics and FSI will be discussed.


Dynamic Load Balance Lagrange Multiplier Vector Mortar Method Slave Side Nonlinear Solid Mechanic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The support of the first author (A.P.) by the TUM Graduate School is gratefully acknowledged.


  1. 1.
    Ben Belgacem, F.: The mortar finite element method with Lagrange multipliers. Numerische Mathematik 84(2), 173–197 (1999)Google Scholar
  2. 2.
    Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. In: H. Brezis, J. Lions (eds.) Nonlinear partial differential equations and their applications, pp. 13–51. Pitman/Wiley: London/New York (1994)Google Scholar
  3. 3.
    Chung, J., Hulbert, G.M.: A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-α method. Journal of Applied Mechanics 60, 371–375 (1993)Google Scholar
  4. 4.
    Ehrl, A., Popp, A., Gravemeier, V., Wall, W.A.: A mortar approach with dual Lagrange multipliers within a variational mutiscale finite element method for incompressible flow. Computer Methods in Applied Mechanics and Engineering, submitted (2012)Google Scholar
  5. 5.
    Gitterle, M., Popp, A., Gee, M.W., Wall, W.A.: Finite deformation frictional mortar contact using a semi-smooth newton method with consistent linearization. International Journal for Numerical Methods in Engineering 84(5), 543–571 (2010)Google Scholar
  6. 6.
    Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Thornquist, H.K., Tuminaro, R.S., Willenbring, J.M., Williams, A., Stanley, K.S.: An overview of the Trilinos project. ACM Transactions on Mathematical Software 31(3), 397–423 (2005)Google Scholar
  7. 7.
    Karypis, G., Kumar, V.: A parallel algorithm for multilevel graph partitioning and sparse matrix ordering. Journal of Parallel and Distributed Computing 48(1), 71–95 (1998)Google Scholar
  8. 8.
    Klöppel, T., Popp, A., Küttler, U., Wall, W.A.: Fluid–structure interaction for non-conforming interfaces based on a dual mortar formulation. Computer Methods in Applied Mechanics and Engineering 200(45–46), 3111–3126 (2011)Google Scholar
  9. 9.
    Mayer, U.M., Popp, A., Gerstenberger, A., Wall, W.A.: 3D fluid–structure–contact interaction based on a combined XFEM FSI and dual mortar contact approach. Computational Mechanics 46(1), 53–67 (2010)Google Scholar
  10. 10.
    Popp, A., Gee, M.W., Wall, W.A.: A finite deformation mortar contact formulation using a primal-dual active set strategy. International Journal for Numerical Methods in Engineering 79(11), 1354–1391 (2009)Google Scholar
  11. 11.
    Popp, A., Gitterle, M., Gee, M.W., Wall, W.A.: A dual mortar approach for 3D finite deformation contact with consistent linearization. International Journal for Numerical Methods in Engineering 83(11), 1428–1465 (2010)Google Scholar
  12. 12.
    Popp, A., Wohlmuth, B.I., Gee, M.W., Wall, W.A.: Dual quadratic mortar finite element methods for 3D finite deformation contact. SIAM Journal on Scientific Computing, accepted (2012)Google Scholar
  13. 13.
    Puso, M.A.: A 3D mortar method for solid mechanics. International Journal for Numerical Methods in Engineering 59(3), 315–336 (2004)Google Scholar
  14. 14.
    Seshaiyer, P., Suri, M.: hp submeshing via non-conforming finite element methods. Computer Methods in Applied Mechanics and Engineering 189(3), 1011–1030 (2000)Google Scholar
  15. 15.
    Simo, J.C., Tarnow, N.: The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Zeitschrift für Angewandte Mathematik und Physik 43(5), 757–792 (1992)Google Scholar
  16. 16.
    Stupkiewicz, S.: Finite element treatment of soft elastohydrodynamic lubrication problems in the finite deformation regime. Computational Mechanics 44(5), 605–619 (2009)Google Scholar
  17. 17.
    Wall, W.A., Gee, M.W.: BACI: A multiphysics simulation environment. Tech. rep., Technische Universität München (2012)Google Scholar
  18. 18.
    Wohlmuth, B.I.: A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM Journal on Numerical Analysis 38(3), 989–1012 (2000)Google Scholar
  19. 19.
    Wohlmuth, B.I.: Discretization methods and iterative solvers based on domain decomposition. Springer-Verlag Berlin Heidelberg (2001)Google Scholar
  20. 20.
    Wohlmuth, B.I., Popp, A., Gee, M.W., Wall, W.A.: An abstract framework for a priori estimates for contact problems in 3D with quadratic finite elements. Computational Mechanics, DOI: 10.1007/s00466-012-0704-z (2012)Google Scholar
  21. 21.
    Wriggers, P.: Computational contact mechanics. John Wiley & Sons (2002)Google Scholar
  22. 22.
    Yang, B., Laursen, T.A.: A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulations. Computational Mechanics 41(2), 189–205 (2008)Google Scholar
  23. 23.
    Yang, B., Laursen, T.A.: A large deformation mortar formulation of self contact with finite sliding. Computer Methods in Applied Mechanics and Engineering 197(6–8), 756–772 (2008)Google Scholar
  24. 24.
    Yang, B., Laursen, T.A.: A mortar-finite element approach to lubricated contact problems. Computer Methods in Applied Mechanics and Engineering 198(47–48), 3656–3669 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander Popp
    • 1
  • Michael W. Gee
    • 2
  • Wolfgang A. Wall
    • 1
  1. 1.Institute for Computational MechanicsTechnische Universität MünchenGarchingGermany
  2. 2.Mechanics and High Performance Computing GroupTechnische Universität MünchenGarchingGermany

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