Computers & Stnomres Vol. 40, No. 2, pp. 203-209, 1991 Printed in Great Britain. 004s7949/91 $3.00 + 0.00 Pergamon Press plc ON THE TREATMENT OF INEQUALITY CONSTRAINTS ARISING FROM CONTACT CONDITIONS IN FINITE ELEMENT ANALYSIS A. L. ETEROVIC and K. J. BATHE Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-356, Cambridge, MA 02139, U.S.A. Abstract--Existing methods for the analysis of contact problems deal with the inequality constraints arising from contact conditions by means of an implicit iteration on all constraints. This paper presents a formulation for contact problems with friction for large deformations where all inequality constraints are enforced explicitly. A robust solution technique for the resulting system of nonlinear equations can then be used. This approach admits the use of line search procedures to enlarge the region of convergence. 1. ~ODU~ON Much attention has been devoted over the past years to the development of solution methods for the analysis of contact problems. Since the earlier studies [l-4], a number of different approaches have been proposed and researched. Contact problems range from frictionless contact in small-strain elastic analysis, to contact with friction in general large-strain inelastic analysis. Although conceptually related, these cases differ sig- nificantly in the way they may be formulated and solved. Problems that admit an energy functional, such as general hyperelastic behavior with contact con- ditions that do not include frictional effects, can be formulated as unilaterally constrained optimization problems. In this case a number of optimization methods with global convergence properties are available [5-71, and algorithms based on mathemati- cal programming have been proposed [8]. The situation is much more complex when the problem includes inelastic material behavior or nonconservative frictional conditions are present. In this case global convergence is much harder to achieve. Among the most commonly used methods for contact analysis are the Lagrange multiplier method [9-121, the penalty method 113-181, the perturbed Lagrangian method [19-211 and the augmented Lagrangian method [22]. The penalty method has the advantage that the contact constraints are taken into account with no increase in the number of degrees of freedom. Indeed, using the displacement-based finite element formu- lation the number of unknowns equals the number of displacement degrees of freedom, independent of the number of constraints. In the classical Lagrange multiplier method contact tractions are considered as additional degrees of freedom. The perturbed Lagrangian method can be considered as a generalization of the Lagrange multiplier method where an additional term involving the contact tractions is added to the variational equations. The classical Lagrange multiplier method is then obtained as a limiting case, while the penalty method is recovered by solving for the contact trac- tions and eliminating them from the equilibrium equations, It should be noted that in both the penalty method and the perturbed Lagrangian method an approxi- mate solution to the original problem is obtained, which depends on an arbitrary penalty parameter, and in practice this parameter can dr~ti~lly affect the results. The augmented Lagrangian method is a widely used method in optimization where a function of displacements and Lagrange multipliers is con- structed so that its minimum corresponds to the solution of the constrained problem. In order to apply the above procedures to contact inequality constraints, methods are used in which in each iteration only those constraints that are active contribute to the incremental equations. There are two undesirable consequences of this approach. First, the quadratic convergence of Newton’s method may be seriously affected if there are frequent changes of the active set, and second, few global convergence results are available. It has been observed that the local convergence properties of the Newton iteration are hard to achieve, and emphasis has been given to the use of consistent tangent stiffness matrices for nonlinear contact analysis [20,23,24]. However, for fully quadratic convergence the tangent stiffness matrix has to satisfy regularity conditions that are being violated by the most commonly used definitions of the gap or interpenetration functions, The objective of this paper is to present a method for finite element analysis of contact problems in 203