Contact, Friction, Discrete Mechanical Structures and Mathematical Programming

  • A. Klarbring
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 384)


These lecture notes contain three main parts. The first part is the mathematical modeling of the discrete finite-dimensional, small displacement, quasistatic, frictional contact problem.

The second part contains four sections, where four different contact problems are formulated as Linear Complementarity Problems (LCPs): the frictionless problem, the steady sliding problem, the rate problem and the incremental problem. These are all derived from the quasistatic problem formulated in the first part. The four problems are discussed with respect to existence and multiplicity of solutions.

The third and final part of the lecture notes concerns structural optimization in contact problems. We go through the main peculiarities related to optimum mechanical design of structures involving unilateral frictionless contact and give examples of results obtained for the so called gap design problem.


Contact Force Contact Problem Linear Complementarity Problem Normal Contact Force Quasivariational Inequality 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Alart, “Critères d’injectivité et de surjectivité pour certaines applications de Rn dans lui-même; application a la mécanique du contact”, Mathematical Modelling and Numerical Analysis 27 (1993) 203–222.zbMATHMathSciNetGoogle Scholar
  2. [2]
    A. Alart, F. Lebon, F. Quittau and K. Rey, “Frictional contact problem in elas-tostatics: revisiting the uniqueness condition”, in Contact Mechanics, eds. M. Raous, M. Jean and J.J. Moreau, Plenum Press, New York (1995) 63–70.CrossRefGoogle Scholar
  3. [3]
    A.M. Al-Fahed and P.D. Panagiotopoulos, “Multifingered frictional robot grippers: a new type of numerical implementation”, Computers & Structures 42 (1992) 555–562.CrossRefGoogle Scholar
  4. [4]
    A.M. Al-Fahed, G.E. Stavrolakis and P.D. Panagiotopoulos, “Hard and soft fingered robot grippers. The linear complementarity approach”, Zeitschrift für Angewandte Mathematik und Mechanik 71 (1991) 257–265.ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    L.-E. Andersson, “Quasistatic frictional contact problems with finitely many degrees of freedom”, manuscript.Google Scholar
  6. [6]
    L.-E. Andersson, “A quasistatic frictional problem with normal compliance”, Non-linear Analysis 16 (1991) 347–369.CrossRefzbMATHGoogle Scholar
  7. [7]
    L.-E. Andersson, “A global existence result for a quasistatic contact problem with friction”, Advances in Mathematical Sciences and Applications 5 (1995) 249–286.zbMATHMathSciNetGoogle Scholar
  8. [8]
    C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley, Chichester (1984).zbMATHGoogle Scholar
  9. [9]
    R.L. Benedict, J.E. Taylor, “Optimal design for elastic bodies in contact”, in Optimization of distributed-parameter structures, eds. E.J. Haug and J. Cea, Sijhoff and Noordhoff, Alphen aan den Rijn, Holland (1981), 1443–1599.Google Scholar
  10. [10]
    J.F. Besseling, “Finite element methods”, in Trends in Solid Mechanics, eds. J.F. Besseling and Van der Heijden, Sijthoff and Noordhoff (1979) 53–78Google Scholar
  11. [11]
    D. Chan and J.S. Pang, “The generalized quasi-variational inequality problem”, Mathematics of Operations Research 7 (1982) 211–222.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    P.W. Christensen, A. Klarbring, J.-S. Pang and N. Strömberg, “Formulation and Comparison of Algorithms for Frictional Contact Problems”, International Journal for Numerical Methods in Engineering, forthcoming.Google Scholar
  13. [13]
    P.W. Christensen and J.-S. Pang, “Frictional contact algorithms based on semis-mooth Newton methods”, Reformulation- Nonsmooth, Piecewise Smooth, Semis-mooth and Smoothing Methods, forthcoming.Google Scholar
  14. [14]
    M. Cocu, “Existence of solutions of Signorini problems with friction”, International Journal of Engineering Science 22 (1984) 567–575.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    T.F. Conry and A. Seireg, “A mathematical programming method for design of elastic bodies in contact”, Journal of Applied Mechanics 2(1971) 387–392.ADSCrossRefGoogle Scholar
  16. [16]
    R.W. Cottle, J.-S. Pang and R.E. Stone, The Linear Complementarity Problem, Academic Press, Boston (1992).zbMATHGoogle Scholar
  17. [17]
    L. Demkowicz and J.T. Oden, “On some existence and uniqueness results in contact problems with nonlocal friction”, Nonlinear Analysis 6 (1982) 1075–1093.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    I. Doudoumis, E. Mitsopoulou and N. Charalambakis, “The influence of the friction coefficients on the uniqueness of the solution of the unilateral contact problem”, in Contact Mechanics, eds. M. Raous, M. Jean and J.J. Moreau, Plenum Press, New York (1995) pp. 63–70.Google Scholar
  19. [19]
    G. Duvaut, “Equilibre d’un solide élastique avec contact unilatéral et frottement de Coulomb”, Comptes Rendues Academie Sciences, Paris/A 290 (1980) 263–265.zbMATHMathSciNetGoogle Scholar
  20. [20]
    G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer, Berlin (1976).CrossRefzbMATHGoogle Scholar
  21. [21]
    G. Fichera, “Boundary value problems in elasticity with unilateral constraints”, in Encyclopedia of Physics, ed. S. Flügge, Vol. VI a/2, Springer Verlag, Berlin (1972).Google Scholar
  22. [22]
    C. Fleury, “First and second order convex approximation strategies in structural optimization”, Structural Optimization 1 (1989) 3–10.CrossRefGoogle Scholar
  23. [23]
    C. Fleury, “CONLIN: An efficient dual optimizer based on convex approximation concepts”, Structural Optimization 1 (1989) 81–89CrossRefGoogle Scholar
  24. [24]
    C. Fleury, “Sequential convex programming for structural optimization problems”, In: Optimization of Large Structural Systems, Proceedings of the NATO/DFG ASI Conference, Berchtesgaden, Germany, September23 October 4, ed. G.I.N. Rozvany, Kluwer, Dordrecht (1993) 531–555.Google Scholar
  25. [25]
    F. Gastaldi, Remarks on a noncoercive contact problem with friction in elasto-statics, Istituto di analisi numerica, Pubblicazioni N. 649, Pavia (1988).Google Scholar
  26. [26]
    F. Gastaldi, M.D.P. Monteiro Marques and J.A.C. Martins, “A two degree-of-freedom quasistatic contact problem with viscous damping”, Advances in Mathematical Sciences and Applications, forthcoming.Google Scholar
  27. [27]
    F. Gastaldi, M.D.P. Monteiro Marques, and J.A.C. Martins, “Mathematical analysis of a two degree-of-freedom frictional contact problem with discontinuous solutions” Nonlinear Differential Equations and Applications, forthcoming.Google Scholar
  28. [28]
    J. Haslinger and A. Klarbring, Shape optimization in unilateral problems using generalized reciprocal energy as objective functional, Nonlinear Analysis, Methods & Applications, Vol. 21, No. 11, pp. 815–834, 1993.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material and Topology Design, Second Edition, Wiley 1996.zbMATHGoogle Scholar
  30. [30]
    E.J. Haug and B.M. Kwak, “Contact stress minimization by contour design”, International Journal for Numerical Methods in Engineering, 12 (1978) 917–939.ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    D. Hilding, in preparation.Google Scholar
  32. [32]
    D. Hilding, A. Klarbring, J. Petersson, “Optimization of structures in unilateral contact”, Applied Mechanics Reviews, forthcoming.Google Scholar
  33. [33]
    V. Janovsky, Catastrophic features of Coulomb friction model, Technical Report KNM-0105044/80, Charles University, Prague (1980).Google Scholar
  34. [34]
    V. Janovsky, “Catastrophic features of Coulomb friction model”, in The Mathematics of Finite Elements and Applications, ed. J.R. Whiteman, Academic Press, London (1981) 259–264.Google Scholar
  35. [35]
    J. Jarusek, “Contact problems with friction. Coercive case”, Czechoslovak Mathematical Journal 33 (1983) 237–261.MathSciNetGoogle Scholar
  36. [36]
    J. Jarusek, “Contact problems with friction. Semicoercive case”, Czechoslovak Mathematical Journal 34 (1984) 619–629.MathSciNetGoogle Scholar
  37. [37]
    L. Johansson and A. Klarbring, “The rigid punch problem with friction using variational inequalities and linear complementarity”, Mechanics of Structures and Machines 20(3) (1992) 293–319.CrossRefMathSciNetGoogle Scholar
  38. [38]
    Y. Kato, “Signorini’s problem with friction in linear elasticity”, Japan Journal of Applied Mathematics 4 (1987) 237–268.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [39]
    A. Klarbring, Quadratic Programs in Frictionless Contact Problems. International Journal of Engineering Science, Vol. 22, No. 7, 1986, pp 1207–1217CrossRefMathSciNetGoogle Scholar
  40. [40]
    A. Klarbring, “A mathematical programming approach to three-dimensional contact problems with friction”, Computational Methods in Applied Mechanics and Engineering 58 (1986) 175–200.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  41. [41]
    A. Klarbring, “Contact problems with friction by linear complementarity”, in Unilateral Problems in Structural Analysis, Vol. 2 (CISM Courses and Lectures, No. 304), eds. G. Del Piero, F. Maceri, Springer, Wien (1987) 197–219.Google Scholar
  42. [42]
    A. Klarbring, “Derivation and analysis of rate boundary-value problems with friction”, European Journal of Mechanics/A 1 (1990) 211–226.MathSciNetGoogle Scholar
  43. [43]
    A. Klarbring, “Examples of non-uniqueness and non-existence of solutions to qua-sistatic contact problems with friction”, Ingenieur-Archiv 60 (1990) 529–541.Google Scholar
  44. [44]
    A. Klarbring, “Mathematical programming and augmented Lagrangian methods for frictional contact problems”, in Proceedings Contact Mechanics International Symposium, ed. A. Curnier, Presses Polytechniques et Universitaires Romandes (1992) 409–422.Google Scholar
  45. [45]
    A. Klarbring, On the problem of optimizing contact force distributions, Journal of Optimization Theory and Applications, Vol. 74, pp. 131–150, 1992.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [46]
    A. Klarbring, “Mathematical programming in contact problems”, Chapter 7 in Computational Methods in Contact Mechanics, eds. M.H. Aliabadi and C. A. Brebbia, Computational Mechanics Publications, Southampton (1993) 233–263.Google Scholar
  47. [47]
    A. Klarbring, Steady sliding and linear complementarity, in Complementarity and Variational Problems, eds. M.C. Ferris, J.-S. Pang, SIAM (1997) 132–147.Google Scholar
  48. [48]
    A. Klarbring and G. Björkman, “A mathematical programming approach to contact problems with friction and varying contact surface”, Computers & Structures 30 (1988) 1185–1198.CrossRefzbMATHGoogle Scholar
  49. [49]
    A. Klarbring and J. Haslinger, On almost constant contact stress distributions by shape optimization, Structural Optimization, Vol. 5, pp. 213–216, 1993.CrossRefGoogle Scholar
  50. [50]
    A. Klarbring, A. Mikelic and M. Shillor, “Frictional contact problems with normal compliance”, International Journal of Engineering Science 26 (1988) 811–832.CrossRefzbMATHMathSciNetGoogle Scholar
  51. [51]
    A. Klarbring, A. Mikelic and M. Shillor, “On friction problems with normal compliance”, Nonlinear Analysis 13 (1989) 935–955.CrossRefzbMATHMathSciNetGoogle Scholar
  52. [52]
    A. Klarbring, A. Mikelic and M. Shillor, “A global existence result for the qua-sistatic frictional contact problem with normal compliance”, International Series of Numerical Mathematics 101, Birkhüser Verlag, Basel (1991) 85–111.Google Scholar
  53. [53]
    A. Klarbring and J.S. Pang, “Existence of solutions to discrete semicoercive frictional contact problems”, SIAM Journal on Optimization, 8(2) (1998) 414–442.CrossRefzbMATHMathSciNetGoogle Scholar
  54. [54]
    A. Klarbring, J. Petersson and M. Rönnquist, Truss topology optimization involving unilateral contact, Journal of Optimization Theory and Applications, Vol. 87, pp. 1–31,1995.CrossRefzbMATHMathSciNetGoogle Scholar
  55. [55]
    A. Klarbring and M. Rönnqvist, “Nested approach to structural optimization in nonsmooth mechanics”, Struct. Opt., 10 (1995) 79–86.CrossRefGoogle Scholar
  56. [56]
    A. Klarbring and J.S. Pang, “The discrete steady sliding problem”, ZAMM, forthcoming.Google Scholar
  57. [57]
    T. Larsson and M. Rönnqvist “A method for structural optimization which combines second-order approximations and dual techniques”, Structural Optimization 5 (1993) 225–232.CrossRefGoogle Scholar
  58. [58]
    Z.-Q. Luo, J.-S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press 1997.zbMATHGoogle Scholar
  59. [59]
    Z.-Q. Luo and P. Tseng, A decomposition property for a class of square matrices, Applied Mathematics Letters 4(5) (1991) 67–69.CrossRefzbMATHMathSciNetGoogle Scholar
  60. [60]
    J.A.C. Martins, M.D.P. Monteiro Marques, and F. Gastaldi, “On an example of non-existence of solutions to a quasistatic frictional contact problem”, European Journal of Mechanics /A 13 (1994) 113–133.zbMATHMathSciNetGoogle Scholar
  61. [61]
    J.A.C. Martins, F.M.F. Simões, F. Gastaldi, and M.D.P. Monteiro Marques, “Dis-sipative graph solutions for a 2 degree-of-freedom quasistatic frictional contact problem”, International Journal of Engineering Science 33 (1995) 1959–1986.CrossRefzbMATHMathSciNetGoogle Scholar
  62. [62]
    J. Nečas, J. Jarušek, and J. Haslinger, “On the solution of the variational inequality to the Signorini problem with small friction”, Bolletino UMI 5 (1980) 796–811.Google Scholar
  63. [63]
    P.D. Panagiotopoulos, “Convex analysis and unilateral static problems”, Ingenieur-Archiv 45 (1976) 55–68.CrossRefzbMATHGoogle Scholar
  64. [64]
    J.S. Pang and J.C. Trinkle, “Complementarity formulations and existence of solutions of multi-rigid-body contact problems with Coulomb friction”, Mathematical Programming, forthcoming.Google Scholar
  65. [65]
    J. Petersson, Behaviorally constrained contact force optimization, Structural Optimization, Vol. 9, pp. 189–193, 1995.CrossRefGoogle Scholar
  66. [66]
    J. Petersson and A. Klarbring, Saddle point approach to stiffness optimization of discrete structures including unilateral contact, Control and Cybernetics, Vol. 3, pp. 461–479, 1994.MathSciNetGoogle Scholar
  67. [67]
    J. Petersson, On stiffness maximization of variable thickness sheet with unilateral contact, Quarterly of Applied Mathematics 54(3) (1996) 541–550.zbMATHMathSciNetGoogle Scholar
  68. [68]
    J. Petersson and J. Haslinger, An approximation theory for optimum sheets in unilateral contact, Quarterly of Applied Mathematics, forthcoming.Google Scholar
  69. [69]
    J. Petersson and M. Patriksson, Topology optimization of sheets in contact by a subgradient method, International Journal for Numerical Methods in Engineering 40(7) (1997) 1295–1321.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  70. [70]
    R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton 1972.Google Scholar
  71. [71]
    D.E. Stewart and J.C. Trinkle, “An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction”, International Journal for Numerical Methods in Engineering 39 (1996) 2673–2691.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  72. [72]
    N. Strömberg, L. Johansson, A. Klarbring, “Derivation and analysis of a generalized standard model for contact, friction and wear”, International Journal of Solids and Structures 33(13) (1996) 1817–1836.CrossRefzbMATHMathSciNetGoogle Scholar
  73. [73]
    N. Strömberg, “An augmented Lagrangian method for fretting problems”, European Journal of Mechanics/A 16 (1997) 573–593.zbMATHGoogle Scholar
  74. [74]
    K. Svanberg, “The method of moving asymptotes - A new method for structural optimization”, International Journal for Numerical Methods in Engineering 24 (1987) 359–373.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  75. [75]
    E. Tonti, “On the mathematical structure of a large class of physical theories” Rendiconti Academia Nationale Dei Lincei LII (1972) 48–56, 350–356.MathSciNetGoogle Scholar
  76. [76]
    J.C. Trinkle, J.S. Pang, S. Sudarsky and G. Lo, “On dynamic multi-rigid-body contact problems with Coulomb friction”, to appear in Zeitschrift für Angewandte Mathematik und Mechanik.Google Scholar

Copyright information

© Springer-Verlag Wien 1999

Authors and Affiliations

  • A. Klarbring
    • 1
  1. 1.Linköping UniversityLinköpingSweden

Personalised recommendations