Numerical stability and convergence analysis of bone remodeling model

• Most of remodeling models are lacking of an analysis of its stability and convergence.
• A novel modification is proposed, assuming the reference equilibrium stimulus as time-dependent.
• This reference equilibrium stimulus is also considered to be dependent on the local loading history.
• This modification improves the convergence and the numerical stability in the longterm.

Bone remodeling is the mechanism that regulates the relationship between bone morphology and its external mechanical loads. It is based on the fact that bone adapts itself to the mechanical conditions to which it is exposed. The first phenomenological law that qualitatively described this mechanism is generally known as Wolff’s law. During recent decades, a great number of numerically implemented mathematical laws have been proposed, but most of them have not presented a full analysis of stability and convergence. In this paper, we revisit the Stanford bone remodeling theory where a novel assumption is proposed, which considers that the reference equilibrium stimulus is dependent on the loading history. Fully discrete approximations are introduced by using the finite element method and the explicit Euler scheme. Some a priori error estimates are proved, from which the linear convergence of the algorithm is deduced under additional regularity conditions. Numerical simulations are presented to demonstrate the behavior of the solution. This modification improves the convergence of the solution, clearly leading to its numerical stability in the long-term.