JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 131, No. 1, pp. 89–113, October 2006 ( C  2006) DOI: 10.1007/s10957-006-9127-3 Arc-Length Method for Frictional Contact Problems Using Mathematical Programming with Complementarity Constraints Y. KANNO 1 AND J. A. C. MARTINS 2 Communicated by K. K. Choi Published Online: 1 December 2006 Abstract. A new formulation as well as a new solution technique is proposed for an equilibrium path-following method in two-dimensional quasistatic fric- tional contact problems. We consider the Coulomb friction law as well as a geometrical nonlinearity explicitly. Based on a criterion of maximum dissi- pation of energy, we propose a formulation as a mathematical program with complementarity constraints (MPEC) in order to avoid unloading solutions in which most contact candidate nodes become stuck. A regularization scheme for the MPEC is proposed, which can be solved by using a conventional non- linear programming approach. The equilibrium paths of various structures are computed in cases such that there exist some limit points and/or infinite number of successive bifurcation points. Key Words. Contact problems, Coulomb’s friction, arc-length method, mathematical program with complementarity constraints, maximum dissipation. 1. Introduction Various mathematical formulations (Ref. 1), as well as numerical approaches (Ref. 2), have been proposed for finite-dimensional frictional contact problems. However, both mathematical and algorithmic difficulties arise from the critical points along the equilibrium paths of frictional contact problems. This paper is concerned with an equilibrium path-following method for frictional contact prob- lems, which is applicable to cases in which there exist limit points and/or successive 1 Assistant Professor, Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan. 2 Professor, Instituto Superior T´ ecnico, Departamento de Engenharia Civil e Arquitectura and ICIST, Lisboa, Portugal. 89 0022-3239/06/1000-0089/0 C  2006 Springer Science+Business Media, Inc.