Biomedical Engineering Reference
In-Depth Information
Fig. 3.9 a Initial quadrilateral from the grid-cell. b Transformation of the initial quadrilateral
into an isoparametric square shape and application of the 2 9 2 quadrature point rule. c Return to
the initial quadrilateral shape
3.3.1 Gaussian Quadrature Integration
Recently a meshless method, based on the RPI, using a stabilized nodal integration
[
40
,
41
] was successfully implemented and the obtained results proved to be better
than the other meshless RPI approaches based on Gauss-Legendre integration
schemes [
9
,
42
]. However the extra time spent in stabilizing the nodal integration
does not pay the increased accuracy of the final solution. Within the Gauss-
Legendre integration, the solid domain is divided in a regular grid, as Fig.
3.2
aor
b indicates. Then each grid-cell is filled with integration points, respecting the
Gauss-Legendre quadrature rule. The detailed description of the Gauss-Legendre
integration procedure, which is beyond the scope of the present topic, can be found
in the literature [
32
,
34
]. Nevertheless a simple example is illustrated. Assume the
grid-cell present in Fig.
3.9
a. The initial quadrilateral is transform in an isopara-
metric square, Fig.
3.9
b, then Gauss-Legendre quadrature points are distributed
inside the isoparametric square, in Fig.
3.9
b it is used a 2 9 2 quadrature. Using
isoparametric interpolation functions the Cartesian coordinates of the quadrature
points are obtained, Fig.
3.9
c. The integration weight of the quadrature point is
obtained multiplying the isoparametric weight of the quadrature point with the
inverse of the Jacobian matrix determinant of the respective grid-cell.
If the grid fits the solid domain no pos-treatment is required, Fig.
3.2
a, however
if the grid is larger than the solid domain all the integration points outside the solid
domain have to be removed, Fig.
3.2
b.
In Fig.
3.2
b a blind regular quadrature integration mesh was constructed, in this
case all the integration points in the grey area are removed. Although fitted
integration meshes present a higher computational cost in comparison with blind
regular quadrature meshes, it also produces more stable and accurate results.
Generally meshless methods use regular quadrature integration meshes because it
is simpler to apply.
