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M. Stipsitz, D. H. Pahr:
"An efficient solver for CT based nonlinear microFE simulations of trabecular structures";
Talk: 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Vienna; 2019-02-18 - 2019-02-22.

Computer tomography (CT) becomes a widely used non-destructive method to image microstructured materials like bone. Simulation models generated from this uCT images - so called micro finite element (uFE) models - have resolutions of around 15-30um and allow studying the failure mechanisms at the micro-level. The high resolution results in large uFE models (1-2000 mio DOFs), which require efficient parallel solvers. The open source solver ParOSol [1] was designed for such tasks but is limited to linear problems. In this work, a nonlinear extension - ParOSolNL - is presented and applied to capture the apparent behavior of trabecular biopsies.

Methods
The linear-elastic uFE solver ParOSol was extended to include damage based material nonlinearity. The simple material model is able to capture material degradation and material failure via a scalar damage variable. The tension-compression asymmetry of bone tissue is taken into account via asymmetric damage onset stresses. The material parameters are identified using CT imaged trabecular bone biopsies from different body locations, showing a variety of different densities. Finally, 21 trabecular samples were simulated under compression or tension and compared to experimental results obtained by [2].

Results
Good agreement between the nonlinear simulations and the experiments was found in the apparent stress-strain curves of the trabecular biopsies. Linear regression analysis shows that the apparent modulus, apparent 0.2%-offset yield stress and ultimate stress are highly correlated between the experiments and the simulations. The failed regions obtained in the simulations look plausible; showing diffuse damage regions as well as distinct fractures. The FE models have 3-15 mio DOFs and resulted in average physical simulation times of 1.2 hours using 28 CPUs.

Conclusion
Although a very simple nonlinear material model and solving algorithm is used, the apparent behavior of trabecular biopsies can be captured quite well. The local failure regions look reasonable; however, the reliability of local results needs to be further investigated. Additionally, it has been shown that ParOSolNL can be applied to whole bone structures with sufficiently high resolution. Thus, we hope that ParOSolNL will enable more insight into the failure mechanism of bone structures.

References:
[1] C. Flaig and P. Arbenz; Parallel Computing (2011) 37:846-854
[2] J. Schwiedrzik, T. Gross et al.; Int. J. Numer. Meth. Biomed. Engng. (2016) e02739