Learning Outcomes:
The basis of the geometrically exact theory for contact interaction is to build the computational algorithms in the proper selected coordinate system in order to describe the contact interaction in all its geometrical details. This results to the special structure of the computational mechanics course - study in applied differential geometry, kinematics of contact, formulation of a weak form and linearization in a special coordinate system in a covariant form. Afterward, most popular methods to enforce contact conditions are formulated consequently, first for 1D and then for 2D systems finally leading to examples in 3D. The closed form results are applied for the finite element discretization. The structure of contact elements for these methods is studied in detail and all numerical algorithms are derived in a ready for implementation form.

Hands on training in implementation of the derived contact algorithms are presented with the institutes research code FEAP-MeKa.

Content:
• Continuum formulation of a contact problem (Signorini's problem): weak and strong formulation.
• Necessary information from the differential geometry of curves and surfaces
• Curvilinear coordinate systems necessary for the various contact types
• Geometry and kinematics for arbitrary two body contact problem in a covariant form
• Abstract form of formulations in computational mechanics.
• Weak formulation in a covariant form
• Various methods of enforcement contact constraints in a covariant and in operator form
• Consistent linearization in a covariant form: normal and tangential parts
• Various discretization techniques of both the weak form and its linearization: residual and tangent matrix
• A set of analytical solution used for verification of the implemented contact algorithms (Hertz solution, contact patch tests for non-frictional and frictional cases
• Modelling of frictional contact: elastoplastic analogy, return-mapping scheme
• A possible way of generalization of Coulomb friction law

Literature:
[1] Johnson K. L. Contact Mechanics. Cambridge University Press. 1987.
[2] Kikuchi N., Oden J. T. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM. 1988.
[3] Konyukhov A., Schweizerhof K. 2012 Computational Contact Mechanics Geometrically Exact Theory for Arbitrary Shaped Bodies. Springer. 2012.
[4] Laursen T. Computational Contact and Impact Mechanics Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis. Springer, Berlin. 2002.
[5] Sofonea M., Matei A. Mathematical Models in Contact Mechanics. Cambridge University Press. 2012.
[6] Taylor R.L. FEAP electronic resourcesa aa http://www.ce.berkeley.edu/projects/feap/
[7] Wriggers P. Computational Contact Mechanics. John Wiley and Sons. 2002.
[8] Yastrebov A. Numerical Methods in Contact Mechanics. Wiley-ISTE. 2013.