The focus of the current
contribution is on the development of the unified geometrical formulation of
contact algorithms in a covariant form for various geometrical situations of
contacting bodies leading to contact pairs: surface-to-surface,
line-to-surface, point-to-surface, line-to-line, point-to-line, point-to-point.
The construction of the corresponding computational contact algorithms are
considered in accordance with the geometry of contact bodies in a covariant
form. These forms can be easily discredited within finite element methods
independently of order of approximation and, therefore, the result is
straightforwardly applied within iso-geometric finite element methods. This
approach is recently became known asgeometrically exact theory of
contact interaction . Application for contact between bodies with iso- and
anisotropic surface, for contact between cables and curvilinear beams as well
as recent development for contact between cables and bodies isstraightforward. Recent developments
include the improvement of the curve-to-surface (deformable) contact algorithm.
Cite this paper
A. Konyukhov and K. Schweizerhof, "Geometrically Exact Theory of Contact Interactions–Further Developments and Achievements," Open Journal of Applied Sciences, Vol. 3 No. 1, 2013, pp. 15-20. doi: 10.4236/ojapps.2013.31B1004.
 P. Alart and A. Heege, “Consistent Tangent Matrices of Curved Contact Operators Involving Anisotropic Friction,” Revue Europeenne des Elements Finis, Vol. 4, 1995, pp. 183-207.
 M. E. Gurtin, J. Weissmueller and F. Larche, “A General Theory of Curved Deformable Interfaces in Solids at Equilibrium,” Philosophical Magazine A, Vol. 78, No. 5, 1998, pp. 1093-1109.
 J. H. Heegaard and A. Curnier, “Geometric Properties of 2D and 3D Unilateral Large Slip Contact Operators,”Computer Methods in Applied Mechanics and Engineering, Vol. 131, No. 3-4, 1996, pp. 263-286.
 A. Heege and P. Alart, “A Frictional Contact Element for Strongly Curved Contact Problems,” International Journal for Numerical Methods in Engineering, Vol. 39, No. 1, 1996, pp. 165-184.
 R. E. Jones and P. Papadopoulos, “A Geometric Interpretation of Frictional Contact Mechanics,” Zeitschriftf¨ urAngewandteMathematik und Physik, Vol. 57, No. 6, 2006, pp. 1025-1041.
 A. Konyukhov and K. Schweizerhof, “Contact Formulation Via a Velocity Description Allowing Efficiency Improvements in Frictionless Contact Analysis,” Computational Mechanics, Vol. 33, No. 3, 2004, pp. 165-173.
 A. Konyukhov and K. Schweizerhof, “Covariant Description for Frictional Contact Problems,” Computational Mechanics, Vol. 35, No. 3, 2005, pp. 190-213.
 A. Konyukhov and K. Schweizerhof, “On the Solvability of Closest Point Projection Procedures in Contact Analysis: Analysis and Solution Strategy for Surfaces of Arbitrary Geometry,” Computer Methods in Applied Mechanics and Engineering, Vol. 197, No. 33-40, 2008, pp. 3045-3056.
 A. Konyukhov and K. Schweizerhof, “Geometrically Exact Covariant Approach for Contact between Curves Representing Beam and Cable Type Structures,” Computer Methods in Applied Mechanics and Engineering, Vol. 199, No. 37-40, 2010, pp. 2510-2531.
 A. Konyukhov and K. Schweizerhof, “Computational Contact Mechanics – Geometrically Exact Theory for Arbitrary Shaped Bodies,” Springer, Heidelberg, New York, Dordrecht, London, 2012.
 L. Krstulovic-Opara, P. Wriggers and J. Korelc, “A C1 Continuous Formulation for 3D Finite Deformation Frictional Contact,” Computational Mechanics, Vol. 29, No. 1, 2002, pp. 27-42. doi:10.1007/s00466-002-0317-z
 T. A. Laursen and J. C. Simo, “A continuum-based Finite Element Formulation for the Implicit Solution of Multi-body Large Deformation Frictional Contact Problems,” International Journal for Numerical Methods in Engineering, Vol. 35, 1993, pp. 3451-3485.
 P. Litewka and P. Wriggers, “Frictional Contact between 3D Beams,” Computational Mechanics, Vol. 28, No. 1, 2002, pp. 26-39. doi: 10.1007/s004660100266
 H. Parisch and C. Luebbing, “A Formulation of Arbitrarily Shaped Surface Elements for Three-dimensional Large Deformation Contact with Friction,” International Journal for Numerical Methods in Engineering, Vol. 40, No. 18, 1997, pp. 3359-3383.
 D. Peric and D. R. J. Owen, “Computational Model for 3D Contact Problems with Frictionbased on the Penalty Method,” International Journal for Numerical Methods in Engineering, Vol. 35, No. 6, 1992, pp. 1289-1309.
 P. Wriggers and J. C. Simo, “A Note on Tangent Stiffness for Fully Nonlinear Contact Problems,” Communications in Applied Numerical Methods, Vol. 1, No. 5, 1985, pp. 199-203. doi:10.1002/cnm.1630010503
 G. Zavarise and P. Wriggers, “Contact with Friction between Beams in 3D Space,” International Journal for Numerical Methods in Engineering, Vol. 49, No. 8, 2000, 977-1006.