Biomedical Engineering Reference
In-Depth Information
Fig. 3.14 a Voronoï cell and b Respective hexahedrons. c Initial hexahedron. d Hexahedron
isoparameterization
and
determination
of
the
quadrature
integration
points.
e
Quadrature
integration points in Cartesians coordinates
39
,
40
], which increases the method computational cost, reducing consequently the
meshless method efficiency.
3.3.2.2 3D Nodal Based Integration
The previously presented two-dimensional numerical integration scheme can be
directly extended for the three-dimensional space. If irregular nodal distributions
are used to discretize the problem domain, then the 3D Voronoï cell is subdivided
in hexahedral sub-cells. In opposition, if the problem domain is discretized in a
regular mesh, then the 3D Voronoï cell is subdivided in tetrahedral sub-cells.
Afterwards, all the numerical integration schemes presented in
Sect. 3.3.2.1
can be
applied. In Fig.
3.14
it is illustrated the 3D procedure.
First the three-dimensional Voronoï cells are obtained for each node belonging
to the set of field nodes discretizing the problem domain. In Fig.
3.14
a it is pre-
sented an example of a three-dimensional Voronoï cell. The Voronoï cell is par-
titioned in hexahedral or tetrahedral sub-cells. In Fig.
3.14
b it is shown the
