Biomedical Engineering Reference
In-Depth Information
The height of the liquid film was arbitrarily defined as 3:5 m, which means
by the hypothesis h2 that the adhesion starts at a distance between membranes of
d
2
10
6
m. The distance of the roughness interference (hypothesis h5) was
also chosen arbitrarily as d
1
D
D
7
10
6
m. The value of surface tension was chosen
3
10
3
N=m, which is a physiological value [
15
].
as 7:5
3.4
Numerical Stability
The algorithm used to search the non-linear solution was the Newton-Raphson
method [
23
]. Because the system has limit points, it was necessary to employ a
continuation method to prevent divergence close to these points. The continuation
method here applied was the arc-length method [
24
] linearised by Schweizerhof
and Wriggers [
25
].
4R su s
From Figs.
8
-
16
a sequence of configurations related to different transmural
pressures p
tm
is shown. The thick lines represent the truss elements current position,
the dash lines their initial positions and the circles represent the nodes. The fixed
thin line indicate the opposed surface and the dash-dot line the distance d
2
from the
opposed surface, or the position where the adhesion forces start and cease to affect
the above membrane. The application of the arc-length algorithm makes it possible
to find states of unstable equilibrium. Otherwise, an unstable degree of freedom
could present big variations between iterations, which would induce numerical
diversion.
It is interesting to observe the free nodes vertical trajectory, following the
numerical steps. In Fig.
17
the trajectory of half-symmetric structure is shown.
Figure
18
shows the adhesion of the central nodes in detail.
Figure
19
is a graph of pressure to cross-sectional area of the alveolus (analogous
to the volume in excised lungs), which clearly shows the hysteresis caused by the
adhesion. The magnitude of the cross-sectional area was calculated by the trapezoid
integration method. Each point in evidence in Fig.
19
is a configuration of a pre-set
transmural pressure. Most of these configurations are shown in Figs.
8
-
16
.
Figure
20
shows the equilibrium trajectory got by the arc-length method. The
arc-length method prevent numerical snap-through and snap-back, which would
destabilize the simulation.
5
Discussion
The sequel from Figs.
8
-
16
rebuilds the intuitively expected mechanics. The curve
in Fig.
19
has the same pattern of hysteresis observed experimentally for whole
lung, see Fig.
2
. The difference of inclination level between the recruitment paths
