Biomedical Engineering Reference
In-Depth Information
b
a
c
being the three inner angles of the triangle. With this definition, the equilateral
triangle has
,
,
0.
In order to measure the quality of the isotropic tetrahedral elements, we define
another quality measure
γ
κ
=
1 and degenerated (zero surface) triangles have
γ
κ
=
γ
τ
based also on the radius ratio of the mesh element (tetra-
hedron) [18, 19]:
6
√
6
V
τ
S
su
F
L
max
γ
τ
=
,
E
,
S
sum
F
V
τ
being the volume of the tetrahedron
τ
being the sum of the areas of the 4
faces of the tetrahedron, and
L
max
E
being the maximum edge length of the 6 edges of
the tetrahedron. This
γ
τ
quality measure lies in the interval
[
0
,
1
]
, an element with
γ
τ
=
0 being a sliver (zero volume).
We analyze the quality of the the lumen triangulations obtained with our remesh-
ing algorithm based on finite element conformal maps. The quality of the volume
meshes used for the simulations will be presented in the next section.
Fig. 13.4 shows two different steps in the parametrization-based remeshing al-
gorithm of an initial triangulation of an iliac artery bifurcation
4
First the initial
mesh is cut into different patches using the multiscale Laplacian partitioning method
(a)
(b)
(c)
Fig. 13.4.
Remeshing of an iliac bifurcation. The initial mesh is first split into two parts using
the multiscale Laplacian partitioning method described in [30] (a). Each of those two parts is then
mapped in the parametric space by computing a Laplacian harmonic map onto a unit disk (b) and
the presented conformal map with open boundaries (c)
4
It should be noted that the procedure is fully automatic as implemented in the open-source software
Gmsh.
