There are five types of smoothing elements ((i) split smoothing element: there is one Gauss point
on each boundary segment for split smoothing element; (ii) split-blending smoothing element: one Gauss point
on each boundary segment is sufficient; (iii) tip smoothing element: five Gauss points
on a segment of smoothing element are sufficient; (iv) tip-blending smoothing element: five Gauss points
on each boundary segment are sufficient; (v) standard smoothing element: one Gauss point
on each boundary segment is sufficient) being used for numerical integration as mentioned in .
Take the 2 x 2 Gauss point
integration as an example, as indicated in Figure 6; the local Cartesian coordinate systems with the origins at the four Gauss points
can be established in a similar way as shown in Figure 4.
As we explained in previous sections, the RKEM shape functions are piecewise rational shape functions, therefore, we need to use many Gauss point
to evaluate the integrals such that we can obtain a well conditioned, non singular system of equations (15).
It can produce a graph between variables like, load and deflection, stress and strain, convergence details, single and multiple Gauss point
Then, at each Gauss point
and at the reinforcement positions, strain and stress elastic increments are computed.
i], which are evaluated at the Gauss point
, are multiplied by the radii at their respective points.
To begin the remapping, Gauss point
values are mapped onto the nodes of the old mesh using a least-squares algorithm.
The results shown were calculated at the Gauss point
just above the center-line.
At each element, the smoothed value for each stress component considered is obtained computing the average of the corresponding Gauss point
values at FS line and PS line, respectively.
Four types of finite elements are distinguished in these examples according to their positions with the crack, the standard element contains 3x3 Gauss points
, the element that having tip enriched control points contains 5 x 5 Gauss points
and the sub-triangle technique is used for the tip-element by 7 Gauss points
in each triangle, however the split element contains 6 x 6 Gauss points
for the double edge crack problem and the sub-triangle technique is used by 7 Gauss points
in each triangle for the cracked disk problem.
Analogous theorems hold for weight functions supported on any compact subset [DELTA] of (-1, 1), in which case the (normalized) Gauss points
approach the reciprocal density of the equilibrium measure of [DELTA].
pi] on the Gauss points
of the column finite element, the calculated concentrations vary greatly with the increase of the number of finite elements (of course, when the number of finite elements is big enough, the calculation results will approach to a limit).