Temporal stability of node-based internal bone adaptation simulations

Jacobs et al. (1995, J. Biomechanics 28, 449-459) introduced a new implementation of the remodeling theory developed by Beaupre et 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.