author = {Rumpf, Martin and Schwen, Lars Ole and Wilke, Hans-Joachim and Wolfram,
  title = {Numerical Homogenization of Trabecular Bone Specimens using Composite
	Finite Elements},
  booktitle = {Multiphysics Simulations -- Advanced Methods for Industrial Engineering},
  year = {2010},
  pages = {127--143},
  organization = {Fraunhofer},
  publisher = {Multi-Science Publishing},
  abstract = {Numerical homogenization is a tool to determine effective macroscopic
	material properties for microstructured materials. This tool is tailored
	and applied to ensembles of young and elder human and of porcine
	and bovine vertebral bone specimens. On the microscale of the spongiosa
	a linearized Lam{\'{e}}--Navier type elasticity model is assumed
	and the computed macroscopic material properties are represented
	by a general elasticity tensor. The computation is based on a suitable
	set of microscopic simulations on the cubic specimens for macroscopic
	strain scenarios. The subsequent evaluation of the effective stresses
	is used to determine effective linear elasticity tensors. A Composite
	Finite Element discretization is taken into account to resolve the
	complicated domain. The classical strain--stress and a corresponding
	variational homogenization approach are compared. In case of an (artificial)
	periodic microstructure, a fundamental cell is easily identified
	and a macroscopic unit strain can be imposed using affine-periodic
	boundary conditions. In contrast, statistically periodic structures
	require the identification of statistically representative prototype
	cells. Unit macroscopic strains are then imposed only in an approximate
	sense using displacement boundary conditions. The impact of the resulting
	boundary artifacts on the solution are compensated for via restricting
	the evaluation of effective stress to a suitably selected smaller
	subset of the cubic specimen. Furthermore, an optimization approach
	is used to identify possible axes of orthotropy of the resulting
	linear elasticity tensor. Finally, the different specimens of human,
	porcine and bovine spongiosa are analyzed statistically.},
  pdf = { 1},
  isbn = {978-1-907132-36-0},
  journal = {International Journal of Multiphysics}