A gradient-enhanced bone remodelling approach to avoid the checkerboard phenomenon

Numerical simulation of bone remodelling enables the investigation of short- and long-term stability of bone implants and thus can be an essential tool for surgical planning. The first development of related mathematical models dates back to the early 90's, and these models have been continuously refined since then. However, one issue which has been under discussion since those early days concerns a numerical instability known as checkerboarding. A literature review of recent approaches guided us to adopt a technique established in damage mechanics and topology optimisation, where similar mesh dependencies and instabilities occur. In our investigations, the so-called gradient enhancement is used to regularise the internal variable field, representing the evolution of the bone mass density. For this, a well-established mathematical model for load-adaptive bone remodelling is employed. A description of the constitutive model, the gradient enhancement extension and the implementation into an open-access Abaqus user element subroutine is provided. Parametric studies on the robustness of the approach are demonstrated using two benchmark examples. Finally, the presented approach is used to simulate a detailed femur model.

Automated summary

Numerical simulation of bone remodelling is a crucial tool for investigating the stability of bone implants and can aid in surgical planning. This paper introduces a numerical approach using gradient enhancement technique to regularize the internal variables in the bone density evolution model. The method's robustness is demonstrated through parametric studies on benchmark examples, and its effectiveness is further validated through a detailed femur model simulation.