p-FEMs in biomechanics: Bones and arteries

The p-version of the finite element method (p-FEM) is extended to problems in the field of biomechanics: the mechanical response of bones and arteries. These problems are extremely challenging, partly because the constitutive models governing these materials are very complex and have not been investigated by sufficiently rigorous methods. Furthermore, these biological structures have a complex geometrical description (substructures with high aspect ratios), undergo finite deformations (arteries), are anisotropic and almost incompressible (arteries). The intrinsic verification capabilities and high convergence rates demonstrated for linear problems are being exploited and enhanced here, so that validation of the results can be easily conducted by comparison to experimental observations. In the first part of the paper we present p-FE models for patient-specific femurs generated semi-automatically from quantitative computed tomography (qCT) scans with inhomogeneous linear elastic material assigned directly from the qCT scan. The FE results are being verified and thereafter validated on a cohort of 17 fresh-frozen femurs which were defrosted, qCT-scanned, and tested in an in vitro setting. The complex combined passive–active mechanical response of human arteries is considered in the second part and the enhancement of p-FEMs to these non-linear problems is detailed. We apply a new ‘p-prediction’ algorithm in the iterative scheme and demonstrate the efficiency of p-FEMs compared to traditional commercial h-FEMs as Abaqus (in respect of both degrees of freedom and CPU times). The influence of the active response is shown to be crucial if a realistic mechanical response of an artery is sought.

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