Biomedical Engineering Reference
In-Depth Information
Fig. 28.1
Schematic overview on nested dynamic multi-scale method for a given time step. The
deformation of the RVE boundary is prescribed according to the macroscopic deformation gradient.
After having solved the RVE problem, the macroscopic density, stress, and constitutive tensor are
determined by volume averaging. Nonlinearities on both scales are taken into account. For time
discretization of the dynamic macro-scale problem, the generalized alpha scheme is utilized
modeled using a homogenized, phenomenological constitutive law fitted to experi-
ments (Rausch et al.,
2011b
). At certain hotspots in our model, however, we want
to zoom in on the alveolar micro-structure in order to quantify local stresses and
strains relevant for VALI. To bridge the gap between the global parenchyma and the
local alveolar level, we have developed a novel computational multi-scale approach
based on the nested solution of the boundary value problems on both levels. The
benefit of this so-called FE
2
strategy is twofold; firstly, improved global properties
are derived due to the detailed modeling of the underlying complex micro-structure.
Secondly, the global parenchyma model serves as an 'embedding' of the locally re-
solved micro-structure, thereby providing physiologically reasonable boundary con-
ditions for alveolar simulations.
Our approach extends existing methods (Feyel and Chaboche,
2000
; Kouznetsova
et al.,
2001
; Miehe,
2003
; Geers et al.,
2010
; Peric et al.,
2010
) to coupled and
dynamic scenarios inherent to (mechanical) ventilation. To account for the tran-
sient effects, we have proposed to couple a dynamic simulation on the global level
with a quasi-static simulation of the discretized alveolar level (Wiechert and Wall,
2010
). This procedure enables us to investigate the time-dependent behavior of lung
parenchyma as a whole and local alveolar ensembles simultaneously without neces-
sitating to resolve the alveolar micro-structure completely. Instead, representative
volume elements (RVEs) of the underlying alveolar microstructure including inter-
