Biomedical Engineering Reference
In-Depth Information
Fig. 3.1
a Solid domain. b Regular nodal discretization example. c Irregular nodal discretization
example
Similarly with mesh dependent discretization numerical methods, in meshless
methods the nodal density of the discretization, as well as the nodal spatial dis-
tribution, affects the method performance. A fine nodal distribution leads generally
to more accurate results, however the computational cost grows with the increase
of the total number of nodes. The unbalanced distribution of the nodes discretizing
the problem domain can lead to a lower accuracy. Generally locations with pre-
dictable stress concentrations, such as: domain discontinuities; convex boundaries;
crack tips; essential boundaries; natural boundaries; etc., should present a higher
nodal density when compared with locations in which smooth stress distributions
are expected [
1
,
2
], Fig.
3.1
c.
After the nodal discretization a background integration mesh is constructed,
nodal dependent or nodal independent many numerical methods require it in order
to numerically integrate the weak form equations governing the physic phenom-
enon. The integration mesh can have the size of the problem domain or even a
larger one, without affecting too much the final results [
3
].
It is common to use Gaussian integration meshes, as in the FEM, fitted to the
problem domain, Fig.
3.2
a, however other approaches in meshless methods are
also valid, Fig.
3.2
b. Another way to integrate the weak form equations is using
the nodal integration, which resorts to the Voronoï diagrams in order to obtain the
integration weight on each node, Fig.
3.2
c.
As it is perceptible, using the nodal integration, the nodal distribution will
additionally serve as an integration mesh. Nonetheless, the nodal integration
generally leads to the decrease of accuracy, being necessary to implement a sta-
bilization method which increases the computational cost [
4
-
6
].
