Dual analysis of frictionless problems by displacement and equilibrium finite elements

The paper deals with the duality of the frictionless contact problem in the sense of convex analysis, contact between an elastic body and a rigid foundation under small deformation conditions. A new method using equilibrium finite elements is proposed. Virtual work theorems are first shown under variational equality forms. Fenchel-Rockafellar duality and Arrow-Hurwicz duality are used to deduce four variational principles of elastic bodies in contact. The boundary contact finite elements are then formulated from these theoretical foundations of convex analysis. A new interpretation of equilibrium finite elements and their associated boundary contact finite elements is described. Lastly, the different contact finite elements, their properties and numerical results are described. Agreement with analytical solutions is satisfactory.