Computational homogenization for multiscale modeling of heterogeneous materials

Eduard Rohan*, Robert Cimrman, Vladimir Lukes

The modeling of heterogeneous materials means to find (compute)
global response of structures having characteristic size relevant
to the "macroscopic" scale, whereas the physical properties and
thereby the structure behaviour is given at the "microscopic"
scale characterized by a small parameter $\varepsilon \rightarrow
0$.  This setting of the problem requires to homogenize the
microstructural behaviour defined in terms of PDEs with
oscillating and possibly scale-dependent parameters reflecting
some sub-microscopic effects.

In the paper we explain a general multiscale modeling methodology
and present particular problems which we treated by
homogenization. These comprise fluid saturated porous media,
elastic waves in phononic materials, acoustic transmission on
perforated interfaces and large deforming poroelastic structures.

The multiscale homogenization-based models involve the limit
systems of PDEs possessing macroscopic fields of interest and the
local models which allow to 1) obtain macroscopic model
parameters and 2) to recover microscopic response for
a particular macroscopic loading. This "microscopic recovery"
procedure is more complicated for models with scale-dependent
material parameters, such as those governing the fluid-saturated
porous media with dual porosities; there the flow described by
the Darcy law is featured by heterogeneities (discontinuities) in
the hydraulic permeability, $K$, so that in some parts of the
microstructure $K$ depends on $\varepsilon$-square, which
corresponds to the dual porosity ($\varepsilon$ is the length
of the heterogeneity period). In general, the strong
heterogeneity results in a new quality of the macroscopic
constitutive laws in comparison with those defined at the
microscopic level. A similar phenomenon characterizes the
phononic materials where strong heterogeneities in elastic
coefficients are considered.

Numerical results illustrating the mentioned applications and
some particular numerical procedures implemented in the in-house
developed SfePy software will be presented.