An equilibrium finite element method for contact problem with application to strict error estimation

Dual analysis, requiring a pair of compatible and equilibrated solutions, has been well used for strict error estimation of many problems, including contact problems. Compatible solution is easily obtained by conventional displacement-based finite element method (FEM), while getting the equilibrated solution is not straightforward. To this end, an equilibrium finite element method (EFEM) is developed in this paper for equilibrated solution of contact problems. In doing so, there are two key steps. The first is the establishment of complementary energy principle for contact problems, along with inequality pressure constraint on contact force. In the second step, a traction-based equilibrium element is developed to construct the discrete equilibrated stress field, with which the complementary energy formulation turns into a quadratic programming problem. How to get strict bounds for the discretization error in compatible or equilibrated solution by dual analysis is also addressed. Numerical examples are finally conducted to see the performance and the availability to strict error estimation of the proposed EFEM.