Computing pointwise contact between bodies: a class of formulations based on master–master approach

  • Alfredo Gay NetoEmail author
  • Peter Wriggers
Original Paper


In the context of pointwise contact interaction between bodies, a formulation based on surface-to-surface description (master–master) is employed. This leads to a four-variable local contact problem, which solution is associated with general material points on contact surfaces, where contact mechanical action-reaction are represented. We propose here a methodology that permits, according to necessity, a selective dimension reduction of this local contact problem. Thus, the formulation includes curve-to-curve, point-to-surface, curve-to-surface or other contact descriptions as particular degenerations of the surface-to-surface approach. This is done by assuming convective coordinates in the original local contact problem. An operator for performing the so-called “local contact problem degeneration” is presented. It modifies automatically the dimension of the local contact problem and related requirements for its solution. The proposed method is particularly useful for handling singularity scenarios. It also creates a possibility for representing conformal contact by pointwise actions on a non-uniqueness scenario. We present applications and examples that demonstrate benefits for beam-to-beam contact. Ideas and developments, however, are general and may be applied to other geometries of contacting bodies.


Contact Master–master Degeneration Optimization 



The first author acknowledges FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) under the Grant 2016/14230-6 and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) under the Grant 304680/2018-4.


  1. 1.
    Wriggers P (2002) Computational contact mechanics. Wiley, West SussexzbMATHGoogle Scholar
  2. 2.
    Laursen TA (2003) Computational contact and impact mechanics fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer, BerlinzbMATHGoogle Scholar
  3. 3.
    Francavilla A, Zienkiewicz OC (1975) A note on numerical computation of elastic contact problems. Int J Numer Methods Eng 9:913–924CrossRefGoogle Scholar
  4. 4.
    Stadter JT, Weiss RO (1979) Analysis of contact through finite element gaps. Comput Struct 10:867–873CrossRefzbMATHGoogle Scholar
  5. 5.
    Wriggers P, Rust WT, Reddy BD (2016) A virtual element method for contact. Comput Mech 58(6):1039–1050MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Wriggers P, Van TV, Stein E (1990) Finite-element-formulation of large deformation impact-contact problems with friction. Comput Struct 37:319–333CrossRefzbMATHGoogle Scholar
  7. 7.
    Simo JC, Wriggers P, Taylor RL (1985) A perturbed Lagrangian formulation for the finite element solution of contact problems. Comput Methods Appl Mech Eng 50:163–180MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bandeira AA, Pimenta PM, Wriggers P (2004) Numerical derivation of contact mechanics interface laws using a finite Element approach for large 3D deformation. Int J Numer Meth Eng 59:173–195MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bathe KJ, Chaudhary AB (1985) A solution method for planar and axisymmetric contact problems. Int J Numer Methods Eng 21:65–88CrossRefzbMATHGoogle Scholar
  10. 10.
    Puso MA (2004) A 3D mortar method for solid mechanics. Int J Numer Methods Eng 59(3):315–336MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Puso MA, Laursen TA (2004) A mortar segment-to-segment contact method for large deformation solid mechanics. Comput Methods Appl Mech Eng 193:601–629CrossRefzbMATHGoogle Scholar
  12. 12.
    Fischer KA, Wriggers P (2005) Frictionless 2d contact formulations for finite deformations based on the mortar method. Comput Mech 36:226–244CrossRefzbMATHGoogle Scholar
  13. 13.
    De Lorenzis L, Wriggers P, Zavarise G (2012) A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method. Comput Mech 49(1):1–20MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Popp A et al (2010) A dual mortar approach for 3D finite deformation contact with consistent linearization. Int J Numer Meth Eng 83:1428–1465MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Popp A, Gee WM, Wall WA (2011) Finite deformation contact based on a 3D dual mortar and semi-smooth newton approach. In: Zavarise G, Wriggers P (eds) Trends in computational contact mechanics. Springer, Berlin, pp 57–77CrossRefGoogle Scholar
  16. 16.
    Wriggers P, Zavarise G (1997) On contact between three-dimensional beams undergoing large deflections. Commun Numer Methods Eng 13:429–438MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zavarise G, Wriggers P (2000) Contact with friction between beams in 3-D space. Int J Numer Methods Eng 49:977–1006CrossRefzbMATHGoogle Scholar
  18. 18.
    Gay Neto A, Pimenta PM, Wriggers P (2015) Self-contact modeling on beams experiencing loop formation. Comput Mech 55(1):193–208MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Litewka P, Wriggers P (2002) Frictional contact between 3D beams. Comput Mech 28:26–39CrossRefzbMATHGoogle Scholar
  20. 20.
    Litewka P (2007) Hermite polynomial smoothing in beam-to-beam frictional contact. Comput Mech 40:815–826CrossRefzbMATHGoogle Scholar
  21. 21.
    Konyukhov A, Schweizerhof K (2010) Geometrically exact covariant approach for contact between curves. Comput Methods Appl Mech Eng 199:2510–2531MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Konyukhov A, Schweizerhof K (2013) Computational contact mechanics. Springer, BerlinCrossRefzbMATHGoogle Scholar
  23. 23.
    Gay Neto A, Pimenta PM, Wriggers P (2014) Contact between rolling beams and flat surfaces. Int J Numer Methods Eng 97:683–706MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zavarise G, Wriggers P (1998) A segment-to-segment contact strategy. Math Comput Model 28(4–8):497–515CrossRefzbMATHGoogle Scholar
  25. 25.
    Zavarise G, de Lorenzis L (2009) A modified node-to-segment algorithm passing the contact patch test. Int J Numer Methods Eng 79:379–416CrossRefzbMATHGoogle Scholar
  26. 26.
    Gay Neto A, Pimenta PM, Wriggers P (2016) A Master-surface to master-surface formulation for beam to beam contact. Part I: frictionless interaction. Comput Methods Appl Mech Eng 303:400–429MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gay Neto A, Pimenta PM, Wriggers P (2017) A master-surface to master-surface formulation for beam to beam contact. Part II: frictional interaction. Comput Methods Appl Mech Eng 319:146–174MathSciNetCrossRefGoogle Scholar
  28. 28.
    Litewka P (2013) Enhanced multiple-point beam-to-beam frictionless contact finite element. Comput Mech 52:1365–1380MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Litewka P (2015) Frictional beam-to-beam multiple-point contact finite element. Comput Mech 56:243–264MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Durville D (2012) Contact-friction modeling within elastic beam assemblies: an application to knot tightening. Comput Mech 49:687–707MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Durville D (2010) Simulation of the mechanical behaviour of woven fabrics at the scale of fibers. Int J Mater Form 3(Suppl. 2):S1241–S1251CrossRefGoogle Scholar
  32. 32.
    Chamekh M, Mani-Aouadi S, Moakher M (2014) Stability of elastic rods with self-contact. Comput Methods Appl Mech Eng 279:227–246MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Chamekh M, Mani-Aouadi S, Moakher M (2009) Modeling and numerical treatment of elastic rods with frictionless self-contact. Comput Methods Appl Mech Eng 198:3751–3764MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Meier C, Popp A, Wall WA (2016) A finite element approach for the line-to-line contact interaction of thin beams with arbitrary orientation. Comput Methods Appl Mech Eng 308:377–413MathSciNetCrossRefGoogle Scholar
  35. 35.
    Meier C, Wall WA, Popp A (2017) A unified approach for beam-to-beam contact. Comput Methods Appl Mech Eng 315(1):972–1010MathSciNetCrossRefGoogle Scholar
  36. 36.
    Konyukhov A (2015) Geometrically exact theory of contact interactions—applications with various methods FEM and FCM. J Appl Math Phys 3:1022–1031CrossRefGoogle Scholar
  37. 37.
    Gay Neto A, Pimenta PM, Wriggers P (2018) Contact between spheres and general surfaces. Comput Methods Appl Mech Eng 328:686–716MathSciNetCrossRefGoogle Scholar
  38. 38.
    Gay Neto A (2017) Giraffe user’s manual—generic interface readily accessible for finite elements. Disponivel em Accessed 25 Jan 2019
  39. 39.
    Gay Neto A (2016) Dynamics of offshore risers using a geometrically-exact beam model with hydrodynamic loads and contact with the seabed. Eng Struct 125:438–454CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Polytechnic School at University of São PauloSão PauloBrazil
  2. 2.Leibniz Universität-HannoverHannoverGermany

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