Numerical analysis of a strain-adaptive bone remodelling problem

In this paper we study from the numerical point of view a strain-adaptive bone remodelling that couples the displacements and the apparent density (the porosity) of the bone. The rate of this density at a particular location is described as an objective function, which depends on a particular stimulus at that location. The variational problem is written as a coupled system of a nonlinear variational equation for the displacement field and a nonlinear parabolic variational inequality for the apparent density. Then, fully discrete approximations are provided by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are proved, from which, under adequate regularity conditions, the linear convergence of the algorithm is deduced. Finally, some numerical simulations involving one- and two-dimensional test examples are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.