Biomedical Engineering Reference
In-Depth Information
Fig. 3.2 a Fitted Gaussian integration mesh. b General Gaussian integration mesh. c Voronoï
diagram for nodal integration
With the nodal distribution defined and the integration mesh constructed the
nodal connectivity can be imposed. In the FEM the nodal connectivity is prede-
fined by the used of 'elements', however in the meshless methods there are no
elements. Thus, for each interest point x
I
of the problem domain concentric areas
or volumes are defined, and the nodes inside these areas or volumes belong to the
influence-domain of node x
i
. In the majority of the meshless methods the interest
points are equivalent to the integration points from the background integration
mesh, however there are meshless methods using the collocation point method-
ology or the nodal integration scheme, in which the interest points are the nodes of
the nodal discretization. The shape and size of the influence-domain, which
depends on the relative position of the interest point, affects the quality of the
results, Fig.
3.3
a. It is recommend that all influence-domains possess approxi-
mately the same number of nodes inside, as so the size of the influence-domain
should be dependent on the nodal density around the interest point. In Fig.
3.3
bit
is shown a bad choice in the influence-domain strategy.
Afterwards the field variables can be obtained with the approximation or
interpolation function. Take for example the displacement field u as the field
variable under consideration. In meshless methods the displacement components
u
I
= (u
x
u
y
u
z
) at any interest point x
I
within the problem domain are approximated
or interpolated using the nodal displacement of the nodes inside the influence-
domain of such interest point x
I
,
u
ð
x
I
Þ
¼
X
n
u
i
ð
x
I
Þ
u
ð
x
i
Þ
ð
3
:
1
Þ
i¼1
being n the number of nodes inside the influence-domain of the interest point x
I
,
u(x
i
) are the displacement components of each node within the influence-domain
