Biomedical Engineering Reference
In-Depth Information
efficiently solved using a direct sparse symmetric-positive-definite solver such as
TAUCS 2 .
It is necessary to impose appropriate boundary conditions to guarantee that the
discrete minimization problem has a unique solution and that this unique solution de-
fines a one-to-one mapping (and hence avoids the degenerate solution u
constant).
For least square conformal maps, the mapping (13.9) has full rank only when the
number of pinned vertices 3 is greater or equal to 2 [28]. Pinning down two vertices
will set the translation, rotation and scale of the solution when solving the linear sys-
tem L C U
=
=
0 and will lead to what is called a free-boundary parametrization. It was
independently found by the authors of the LSCM [28] and the DCP [1] that pick-
ing two boundary vertices the farthest from each other seems to give good results in
general.
13.2.2 Volume meshes with boundary layers
We have implemented an advancing layer method [10, 17, 24] for the generation of
boundary layers. Those boundary layer meshes are attractive since they present high
aspect ratio, orthogonal and possibly graded elements at the wall. The method starts
from a surface mesh on which a boundary layer must be grown. From each surface
node a direction is picked for placing the nodes of the boundary layer mesh. The
direction is either computed using an estimate to the surface normal at the node using
Gouraud shading, or specified directly as a three-dimensional vector field - obtained
e.g. as the solution of a partial differential equation. The nodes are connected to form
layers of prisms that are subsequently subdivided into tetrahedra. There technique
is quite efficient in terms of computational time but cannot guarantee that there will
not be any overlap at tight corners. Therefore, the user has to take care to produce
elements of acceptable shape at sharp corners and to prevent element overlap in
regions of tight corners.
As explained in the introduction, for cardiovascular simulations, there is a double
necessity for boundary layer meshes: one for the viscous boundary layer mesh and
one for the arterial wall of given thickness. These boundary layer meshes can be
built by extruding outward and inward the lumen surface. Then a three-dimensional
Delaunay mesh generator is called to fill the remaining of the lumen volume with
isotropic tetrahedra. Fig. 13.3 shows an example of volume mesh with boundary
layers that is well suited for blood flow simulations in compliant vessels.
It should be noted however that for realistic blood flow simulations, the thickness
of the viscous boundary layer mesh and the mesh resolution for the inner tetrahedra
are often unknown prior to the computation. An effective approach to overcome
this difficulty is to start from the pre-defined boundary layer meshes as depicted in
Fig. 13.3 and to apply an adaptive procedure [9, 36, 37] where the distribution of
the spatial discretization errors are estimated and controlled by modifying the mesh
resolution. For example, in the case of unsteady blood flow simulations, one could
2
http://www.tau.ac.il/ stoledo/taucs/.
3
A pinned vertex i is a vertex for which we have fixed the values of the mapping u i and v i .
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