A Multilevel Finite Element Approach to Study Pressure Ulcer Aetiology - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 286

2 Multi-level Modelling: The Concept
Multi-level modelling of continuum mechanics problems can be done in various
ways. The method that is described here, was originally developed to study the
behaviour of composite polymer materials and metals [ 10 , 11 ].
In our application a tissue is considered to consist of cells embedded in matrix
material. The microscopic length scale of the cells (lm) is orders of magnitude
smaller than the macroscopic dimensions (cm). Because of this, two models have
to be developed: one at the micro-level and one at the macro-level. The problem is
how to couple these models, or which boundary conditions have to be applied to
the micro-model. Furthermore, it is not trivial how the average mechanical state of
the micro-model can be transferred back to the macro model. The latter is called
the homogenization of the mechanical state of the micro-model.
There are several approaches for this coupling. One is, to use periodic boundary
conditions. When an external load is applied, the stress and strain fields in the
microstructure will show large gradients due to the microstructural heterogeneity.
However, due to the differences in scale, the microstructural deformation field
around a macroscopic point will be approximately the same as the deformation
field around neighbouring points. The repetitive deformations justify the
assumption of local periodicity, meaning that the microstructure can be thought of
as repeating itself near a macroscopic point (illustrated in Fig. 1 ). However, the
microstructure itself may differ from one macroscopic point to another. The
repetitive microstructural deformations suggest that macroscopic stresses and
strains around a certain macroscopic point can be found by averaging micro-
structural stresses and strains, in a small representative area of the microstructure
attributed to that point.
Now consider a two-dimensional representative volume element (RVE). The
periodic boundary condition implies:
1. The shapes of two opposite edges remain identical.
2. The stress vectors on opposite edges of the RVE are opposite to satisfy stress
continuity.
In the implementation of numerical codes this is implemented as tying the
displacements of nodes from one edge to corresponding nodes of the opposite
edge. The RVE deformations, orientation and overall dimensions are determined
by the positions of the three vertex points 1, 2 and 4 and the tying conditions. Next
to this the condition of opposite stress vectors on opposite boundaries have to be
satisfied. By averaging the deformations and the stresses in the RVE a coupling
can be made between the microscopic RVE and the macroscopic model.
In a finite element implementation of the theory in each integration point of the
elements of the macroscopic model a microscopic RVE is defined, which again is a
finite element model. The macroscopic deformations F macro are transferred to the
corresponding microscopic FE models, by prescribing the corner nodes and
applying periodic boundary conditions. The RVE problems are solved and in a full