Keywords: porous medium, perfusion, large deformation, Biot model, homogenization, multiscale modeling.

In this paper we explain our homogenization-based approach to the multiscale modeling of large deforming fluid saturated materials. Such problem arises in many engineering applications involving the mechanics of soils or tissue biomechanics, namely in the tissue blood perfusion problem.

We consider a fluid-saturated poroelastic medium with large heterogeneities in the elastic and permeability coefficients. In the linear-elastic case, analysis of such medium was treated by homogenization so that the effective material parameters involved in the macroscopic model were obtained in advance for given specific microstructures using solutions of auxiliary microscopic problems [1]. In the nonlinear case the microstructure changes with progressing deformation, so that it must be updated in time, as also discussed in [2,3]. We propose to combine the linear homogenization approach and the time discretization with the local reference microstructures updated from their previous deformation history. The new macroscopic response of the time increment is obtained using a homogenized linear subproblem.

The computational algorithm can be characterized as the cycle of the following steps:

1. for a given reference microstructure, the local configuration (LC), 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 the MR and update the LC.

This multiscale analysis leads to the split of the computational procedure into two levels: at the macroscopic level information from all local configurations is collected and the computation of the macroscopic response is performed, at the microscopic level the auxiliary problems are solved independently for each LC. We provide a numerical illustration of this approach.

1

E. Rohan, R. Cimrman, "Two-scale modelling of tissue perfusion problem using homogenization of dual porous media", International Journal for Multiscale Computational Engineering, 8(1), 81-102, 2010. doi:10.1615/IntJMultCompEng.v8.i1.70

2

E. Rohan, "Modelling large deformation induced microflow in soft biological tissues", Theor. and Comp. Fluid Dynamics, 20, 251-276, 2006. doi:10.1007/s00162-006-0020-3

3

E. Rohan, R. Cimrman, V. Lukes, "Numerical modelling and homogenized constitutive law of large deforming fluid saturated heterogeneous solids", Computers and Structures, 84, 1095-1114, 2006. doi:10.1016/j.compstruc.2006.01.008

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