# hpfem/esco2012-boa

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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}