Numerical analysis of a contact problem including bone remodeling

In this work, a contact problem between an elastic body and a deformable obstacle is numerically studied. The bone remodeling of the material is also taken into account in the model and the contact is modeled using the normal compliance contact condition. The variational problem is written as a nonlinear variational equation for the displacement field, coupled with a first-order ordinary differential equation to describe the physiological process of bone remodeling. An existence and uniqueness result of weak solutions is stated. Then, fully discrete approximations are introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained, from which the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, some 2D numerical results are presented to demonstrate the behavior of the solution.