Finite element simulation of adaptive bone remodelling: A stability criterion and a time stepping method

Adaptive bone remodelling simulations which use finite element analysis can potentially aid in the design of orthopedic implants and can provide examples which test specific bone remodelling hypotheses in a quantitative manner. By concentrating on remodelling algorithms in which the geometry is fixed but the tissue stiffness changes based on strain energy density, we have predicted stability conditions for bone remodelling and we have tested the applicability of these conditions using numerical simulations. The stability requirements arrived at using a finite element formulation are similar to the requirements arrived at in an earlier analytical study. In order to test the stability conditions, we have developed an Euler backward time stepping technique which uses the derivation for stability. These simulations arrived at solutions which were impossible using Euler forward time stepping as applied in this study. Cases in which a simplified version of the derived Euler backward method are unstable or marginally stable have also been seen, but when the Euler backward method is applied using the full derived matrices, no instabilities are apparent. The results of the stability tests indicate that the converged density distributions in the examples studied are stable. Although a priori conditions which ensure stability are not found, a test for stability is provided, given an assumed density distribution.