Temporal stability of node-based internal bone adaptation simulations

Jacobs et al. (1995, J. Biomechanics28, 449–459) introduced a new implementation of the remodeling theory developed by Beaupréet al. (1990, J. Orthop. Res.8, 651–661) that eliminates certain spurious spatial instabilities of previous implementations. Due to the highly nonlinear, coupled nature of multi-dimensional adaptation simulations, direct stability analyses of this method are not practical. In this manuscript, a linearized stability analysis was used to derive an expression for the critical time step for the stability of a simple model problem. In addition to accurately predicting the temporal response of the single degree-of-freedom problem, the analysis provided a conservative estimate of the critical time step for a more realistic, multiple degree-of-freedom adaptation simulation.