On the old and new techniques for enforcing contact constraint conditions

Enforcement of contact conditions is usually performed using either the Penalty or the Lagrange Multipliers techniques. Both involves well-known advantages and shortcomings. The fact that both are still on the market in the context of Computational Mechanics is a clear evidence that such competition does not have winner. Considering the Penalty Method, the most interesting characteristics are related to its simplicity, and to the fact that the number of unknowns of the global problem is not affected. The well-known shortcomings are related to the penetration that affects the solution, and to the ill-conditioned problem resulting from trying to reduce it, increasing the penalty factor. Basically, in trying to simplify the problem, the Penalty Method corresponds to the insertion of a zero-length spring among the two contacting points. It is then clear that such spring will generate a penetration due to the contact forces. The related amount of penetration will depend on the spring stiffness, and will tend to zero (i.e. the exact solution) only when the spring stiffness will tend to infinity. My main efforts devoted to deal with the limits of the Penalty method, and to take advantage of some of its characteristics can be here resumed as:  The physical meaning that can be attributed to the penalty parameter with a suitable tuning, both for the mechanical and thermal problems [1], and for the electrical ones [2];  The acceleration of the classical augmentation techniques devoted to the improvement of the solution [3, 4];  The regularization with a smooth transition among the open and the closed contact status using the cross-constraints method [5];  The improvements of the penalty solution achieved using the Nitshe method to enforce contact constraints [6];  The discretization improvements obtained introducing the virtual slave-nodes [7];  The very promising shifted penalty technique [8]. References [1] Zavarise G, Wriggers P, Stein E, Schrefler BA. A numerical model for thermomechanical contact based on microscopic interface laws, Mech. Res. Commun., Vol. 19-(3), 173-182, 1992. [2] Boso DP, Zavarise G, Schrefler BA. A formulation for electrostatic-mechanical contact and its numerical solution. Int. J. Numer. Methods Eng., Vol. 64-(3), 382-400, 2005. DOI: 10.1002/nme.1371. [3] Zavarise G, Wriggers P, Schrefler BA. On augmented lagrangian algorithms for thermomechanical contact problems with friction, Int. J. Numer. Methods Eng., Vol. 38-(17), 2929-2949, 1995. [4] Zavarise G, De Lorenzis L. An augmented Lagrangian algorithm for contact mechanics based on linear regression, . Int. J. Numer. Methods Eng., Vol. 91, 825-842, 2012. DOI: 10.1002/nme.4294. [5] Zavarise G, Wriggers P, Schrefler BA. A method for solving contact problems, Int. J. Numer. Methods Eng., Vol. 42-(3), 473-498, 1998. [6] Wriggers P, Zavarise G. A formulation for frictionless contact problems using a weak form introduced by Nitsche. Comput. Mech., Vol. 41-(3), 407-420, 2008. DOI: 10.1007/s00466-007-0196-4. [7] Zavarise G, De Lorenzis L. A modified node-to-segment algorithm passing the contact patch test. Int. J. Numer. Methods Eng., Vol. 79, 379-416, 2009. DOI: 10.1002/nme.2559. [8] Zavarise G. The shifted penalty method, Comput. Mech., Vol. 56, 1-17, 2015. DOI: 10.1007/s00466-015-1150-5.