Biomedical Engineering Reference
In-Depth Information
Table 13.1.
Quality of the surface mesh using different planar mesh generators for the remeshing
of the parametric space computed with the conformal map. The qualities we look at are the the
minimum aspect ratio
min
κ
γ
and the mean aspect ratio
γ
κ
Mesh generator
Surface quality
γ
min
κ
γ
κ
Delaunay
0.18
0.966
MeshAdapt
0.36
0.972
Frontal
0.55
0.978
niques based respectively on the harmonic mapping [13, 34] and the convex com-
bination map of Floater [14]) and one direct remeshing algorithm based on mesh
adaptations (implemented in VMTK [2, 3]). As can be seen in Fig. 13.4, the har-
monic map (and also the convex combination map) is a parametrization with fixed
boundaries mapped on a unit circle in contrast with the conformal mapping which
is a mapping with open boundaries. The quality histograms of Fig 13.6 show that
our remeshing procedure based on conformal maps renders the highest quality map-
ping and has less small elements than the two other parametrization-based remeshing
methods have. Furthermore, it performs as well as the direct remeshing algorithm of
VMTK. Our surface remeshing algorithm is however more robust since it allows to
remesh any kind of surface. For example, VMTK is not able to remesh the skull and
pelvic surfaces but it is possible with our parametrization-based techniques [29].
An important element in the surface remeshing algorithm is the choice of
the planar mesh generator to remesh the parametrized surface (see Figs. 13.1(2)
and 13.1(3)). In Table 13.1, we compare the quality of the iliac surface meshes using
three different planar mesh generators implemented in Gmsh: a Frontal-Delaunay
algorithm [33], a planar Delaunay algorithm [18] and an algorithm based on local
mesh adaptation (called MeshAdapt, see [19] for more details). Table 13.1 shows
clearly that the best planar mesh generator for the conformal mapping is the Gmsh's
Frontal-Delaunay algorithm. This is not a surprise: frontal techniques tend to pro-
duce meshes that are aligned with principal directions. If the planar domain that has
to be meshed is equipped with a metric that conserves angles (i.e. when the mapping
is conformal), then the angle between the principal directions is conserved. The use
of conformal mapping helps therefore to obtain better results from the mesh gener-
ator, enabling us to produce high quality meshes.
13.4 Results
The numerical simulations aim at studying the effects of the mesh quality and the
mesh algorithm on an important clinical indicator such as the Wall-Shear Stress
(WSS). The comparison on different meshes shows how the mesh quality, and in
particular wether accounting or not for the fluid boundary layer, affects the reliability
of the simulations.
