Numerical analysis for a new kind of obstacle problem

In this paper, we consider a new kind of obstacle problem for the elastic membrane. The is made rigid body covered by soft layer that deformable and allows penetration. It assigns reactive normal pressure, which depends on interpenetration membrane obstacle, during contact process. Three equivalent descriptions are derived, namely energy form, variational inequality form differential equation form. existence uniqueness solution proved. Based derive an optimal order error estimate finite element approximate under appropriate regularity assumptions. We also introduce series penalized problems prove its convergence result as penalty parameter converges to infinity. Numerical examples reported using linear elements solve problem, simulation results in good agreement with theoretical analysis.