A scalable multi-level preconditioner for matrix-free µ-finite element analysis of human bone structures

The recent advances in microarchitectural bone imaging disclose the possibility to assess both the apparent density and the trabecular microstructure of intact bones in a single measurement. Coupling these imaging possibilities with microstructural finite element ( FE) analysis offers a powerful tool to improve bone stiffness and strength assessment for individual fracture risk prediction. Many elements are needed to accurately represent the intricate microarchitectural structure of bone; hence, the resulting FE models possess a very large number of degrees of freedom. In order to be solved quickly and reliably on state-of-the-art parallel computers, the FE analyses require advanced solution techniques. In this paper, we investigate the solution of the resulting systems of linear equations by the conjugate gradient algorithm, preconditioned by aggregation-based multigrid methods. We introduce a variant of the preconditioner that does not need assembling the system matrix but uses element-byelement techniques to build the multilevel hierarchy. The preconditioner exploits the voxel approach that is common in bone structure analysis, and it has modest memory requirements, at the same time robust and scalable. Using the proposed methods, we have solved in 12min a model of trabecular bone composed of 247 734 272 elements, yielding a matrix with 1 178 736 360 rows, using 1024 CRAY XT3 processors. The ability to solve, for the first time, large biomedical problems with over 1 billion degrees of freedom on a routine basis will help us improve our understanding of the influence of densitometric, morphological, and loading factors in the etiology of osteoporotic fractures such as commonly experienced at the hip, spine, and wrist. Copyright q 2007 John Wiley & Sons, Ltd