Multi-level Micro-Finite Element Analysis for Human Bone Structures

Peter Arbenz

email: arbenz@inf.ethz.ch
Postal address: Prof. Dr. Peter Arbenz, Institute of Computational Science Universitaetsstrasse 6, ETH Zentrum CAB G69.3, 8092 Zurich, Switzerland

Using microarchitectural bone imaging, it is possible to assess both the apparent density and the trabecular microstructure of intact bones in a single measurement.  In combination with microstructural finite element (microFE) analysis this could provide a powerful tool to improve strength assessment and individual fracture risk prediction.  However, the resulting microFE models are very large and require dedicated solution techniques.  So, in this paper we investigate the efficient solution of the resulting large systems of linear equations by the conjugate gradient algorithm with smoothed aggregation multilevel preconditioning.  We introduce a variant of smoothed aggregation preconditioning in which the matrix of the finest level is stored in an element-by-element fashion.  In this way, a large fraction of the overall storage requirement is saved.  The implementation is based on the Trilinos framework and, in particular, on the multilevel solver package ML.  We report on numerical results showing that our approach leads to a fully parallel finite element solver.

Our numerical results show that a human bone model of about 5 million elements can be solved in about a minute.  These short solution times will allow to assess the mechanical quality of bone in vivo on a routine basis.  Furthermore, our highly scalable solution methods make it possible to analyze the very large models of whole bones measured in vitro, which can have up to 1 billion degrees of freedom.