Numerical analysis of a bone remodeling model with damage

Abstract

Bone tissue is a material with a complex structure and mechanical properties. Repetitive loads or diseases can cause microfractures to appear in the bone tissue, which results in a deterioration of its mechanical properties. On the other hand, bone is a constantly evolving tissue, adapting its density to the loading conditions it is subjected to. In this work, we study, from the numerical point of view, a strain-adaptive bone remodeling model coupled with a damage model. The variational problem is written as a coupled system consisting of a nonlinear variational equation for the displacement field and nonlinear parabolic variational inequalities for the apparent density and damage. Then, fully discrete approximations are introduced, using the finite element method and a hybrid combination of Euler schemes. A priori error estimates are proved under adequate regularity conditions, and the linear convergence of the algorithm is deduced. Finally, some one- and two-dimensional numerical simulations of test examples are presented, to demonstrate the accuracy of the approximations and the behavior of the solutions.