Comparative analysis of numerical integration schemes of density equation for a computational model of bone remodelling

Abstract

In this study, a computational model of bone remodelling problem as proposed by Weinans et al. (1992) is described and solved by other temporal integration techniques different from the Euler scheme. This model considers three types of numerical integration schemes of the evolution of the material density during the remodelling: Euler, Heun and Runge–Kutta methods. Also the strain and the density field are obtained inside each element, at Gauss points or at the nodes of the mesh. A square plate with 1.00 m of side subjected to non-uniform pressure is simulated with two meshes of quadrilateral element with size and  m. Two increments time size: and days are used. The results show that Euler, Heun and Runge–Kutta's methods correctly approached the problem of bone remodelling and that there were no appreciable differences in the patterns obtained by the mesh and time step used. In contrast, using an element-based approach and node-based approach, substantial differences were produced in bone remodelling density pattern. ‘Chess board’ type discontinuities were found in the element approach near the applied pressure area, as were well-defined columns away from this. The node-based approach showed continuity in density distribution. These patterns were well represented by the methods for resolving the density equation. This study concluded that any method of time integration could be used for these meshes and time steps size.

Keywords:

• numerical modelling
• bone remodelling
• Wolff's law

Acknowledgements

D.A. Garzón-Alvarado was funded by the División de Investigación de Bogotá (DIB), Universidad Nacional de Colombia through the support to the project ‘Modeling and Simulation of Processes in Mechanical and Biomedical Engineering: Second Phase’. DIB code 8008170.