Abstract
The method presented here is a variation of the classical penalty one, suited to reduce penetration of the contacting surfaces. The slight but crucial modification concerns the introduction of a shift parameter that moves the minimum point of the constrained potential toward the exact value, without any penalty increase. With respect to the classical augmentation procedures, the solution improvement is embedded within the original penalty contribution. The problem is almost consistently linearized, and the shift is updated before each Newton’s iteration. However, adding few iterations, with respect to the original penalty method, a reduction of the penetration of several orders of magnitude can be achieved. The numerical tests have shown very attractive characteristics and very stable solution paths. This permits to foresee a wide area of applications, not only in contact mechanics, but for any problem, like e.g. incompressible materials, where a penalty contribution is required.
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References
Zavarise G, De Lorenzis LA (2009) A modified node-to-segment algorithm passing the contact patch test. Int J Numer Methods Eng 79:379–416
Luenberger DG, Ye Y (2008) Linear and nonlinear programming, 3rd edn. Springer, New York
Bertsekas DP (1988) Constrained optimization and Lagrange multiplier methods. Computer science and applied mechanics series. Academic Press, California
Bazaraa MS, Sheraly HD, Shetty CM (1993) Nonlinear programming: theory and algorithms. Wiley, New York
Wriggers P, Simo JC (1985) A note on tangent stiffness for fully nonlinear contact problems. Commun Appl Numer Methods 1:199–203
Alart P, Curnier A (1991) A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput Methods Appl Mech Eng 92:353–375
Simo JC, Laursen TA (1992) An augmented Lagrangian treatment of contact problems involving friction. Comput Struct 37:319–331
Wriggers P, Zavarise G (1993) Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interface. Commun Numer Methods Eng 9:815–824
Zavarise G, Wriggers P, Schrefler BA (1995) On augmented Lagrangian algorithms for thermomechanical contact problems with friction. Int J Numer Methods Eng 38:2929–2950
Zavarise G, Wriggers P (1998) A superlinear convergent Lagrangian procedure for contact problems. Eng Comput 16(1): 88–119
Pietrzak G, Curnier A (1999) Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangian treatment. Comput Methods Appl Mech Eng 177:351–381
Hüeber S, Wohlmuth B (2005) A primal-dual active set strategy for non-linear multibody contact problems. Comput Methods Appl Mech Eng 194:3147–3155
Hüeber S, Stadler G, Wohlmuth B (2008) A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction. SIAM J Sci Comput 30:572–596
Zavarise G De, Lorenzis L (2012) An augmented Lagrangian algorithm for contact mechanics based on linear regression. Int J Numer Methods Eng 91:825–842
Zavarise G, Wriggers P, Schrefler BA (1998) A method for solving contact problems. Int J Numer Methods Eng 42(3):473–498
Zavarise G, Wriggers P, Stein E, Schrefler BA (1992) Real contact mechanisms and finite-element formulation—a coupled thermomechanical approach. Int J Numer Methods Eng 35(4):767–785
Wriggers P (2006) Computational contact mechanics. Springer, Berlin
Gentle JE (2007) Matrix algebra—theory, computations, and applications in statistics. Springer, New York
Wriggers P, Zavarise G (2008) A formulation for frictionless contact problems using a weak form introduced by Nitsche. Comput Mech 41(3):407–420
Johnson KL (1985) Contact mechanics. Cambridge University Press, London
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Zavarise, G. The shifted penalty method. Comput Mech 56, 1–17 (2015). https://doi.org/10.1007/s00466-015-1150-5
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DOI: https://doi.org/10.1007/s00466-015-1150-5