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\title{Modeling of Heterogeneous Large Deforming Perfused Media Using Homogenization-based Preconditioning}
\tocauthor{E. Rohan} \author{} \institute{}
\maketitle
\begin{center}
{\large Eduard Rohan}\\
New Technologies Research Center, Pilsen\\
{\tt rohan@kme.zcu.cz}
\end{center}
\section*{Abstract}
We propose a computational homogenization framework for two-scale
modeling of fluid-saturated porous media with material and kinematical
nonlinearity, cf. \cite{Rohan2009-draft,RohanCimrman2009} for small deformation. In \cite{rohan-lukes-ect2010} we derived homogenized
model for large deforming media using an incremental formulation based
on the updated Lagrangian configuration where, due to assumed local
periodicity, microstructures are represented by local reference cells
(LRCs) associated with the finite element approximation of the
homogenized problem. Two-scale algorithm comprises the following
steps: 1) for given local microscopic configurations compute the local
response functions and the effective constitutive parameters, 2)
compute the macroscopic response (MR) for given external loads, 3)
compute the deformation and stresses at each reference microstructure
using MR and update the LRCs. This algorithm is implemented in the
Sfepy code \cite{sfepy}. The homogenized model is an approximate tool to solve
the direct nonlinear problem which, for a given size of
heterogeneities, typically results in vast computations on very fine
meshes. We study a ``physical'' preconditioner for solving the ``direct
problem''. Its construction is based on the two-scale framework:
homogenized parameters and local solution recovery using
characteristic responses of the microstructure. We discuss modeling approximation properties of the homogenized model for a given size of microstructures.
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{rohan-lukes-ect2010}
{\sc E. Rohan and V. Luke\v{s}}. {Homogenization of perfusion in large-deforming medium using the Updated Lagrangean Formulation.}. In Proc. 7th Int. Conf. ECT 2010, Saxe-Coburg Publ., Edinburgh, Paper 83.
\bibitem{Rohan2009-draft}
E. Rohan, R. Cimrman.
Multiscale FE simulation of diffusion-deformation processes in
homogenized dual-porous media.
Math. and Comp. Simul., In Press (2011)
\bibitem{RohanCimrman2009}
E. Rohan, R. Cimrman.
Two-scale modelling of tissue perfusion problem using homogenization
of dual porous media.
Int. Jour. Multiscale Comput. Eng., 8 (2010), 81-102.
\bibitem{sfepy}
R. Cimrman et~al. Software, finite element code and applications, \emph{SfePy} home page.
{\tt http://sfepy.org}, 2009.
\end{thebibliography}