An analysis of dual formulations for the finite element solution of two-body contact problems

This paper examines the convergence properties of dual finite element formulations of the two-dimensional frictionless two-body contact problem under the assumption of infinitesimal kinematics. The centerpiece of the proposed analysis is the well-known Babuška–Brezzi condition, suitably adapted to the present problem. It is demonstrated for certain canonical geometries that several widely used methods that employ pressure or force interpolations derived from the discretizations of both surfaces violate the Babuška–Brezzi condition, thus producing increasingly oscillatory solutions under mesh refinement. Alternative algorithms are proposed that circumvent this difficulty and are shown to yield convergent solutions.