Mechanical APDL makes use of both standard and nonstandard numerical integration formulas. The particular integration scheme used for each matrix or load vector is given with each element description in Element Library. Both standard and nonstandard integration formulas are described in this section. The numbers after the subsection titles are labels used to identify the integration point rule. For example, line (1, 2, or 3 points) represents the 1, 2, and 3 point integration schemes along line elements. Midside nodes, if applicable, are not shown in the figures in this section.
The standard 1D numerical integration formulas which are used in the element library are of the form:
 | (12–1) |
where:
| f(x) = function to be integrated |
| Hi = weighting factor (see Table 12.6: Gauss Numerical Integration Constants) |
| xi = locations to evaluate function (see Table 12.6: Gauss Numerical Integration Constants; these locations are usually the s, t, or r coordinates) |
 = number of integration
(Gauss) points |
Table 12.6: Gauss Numerical Integration Constants
| No. Integration Points | Integration Point Locations (xi) | Weighting Factors (Hi) |
|---|---|---|
| 1 | 0.00000.00000.00000 | 2.00000.00000.00000 |
| 2 | ±0.57735 02691 89626 | 1.00000.00000.00000 |
| 3 | ±0.77459 66692 41483 | 0.55555 55555 55556 |
| 0.00000.00000.00000 | 0.88888 88888 88889 |
For some integrations of multi-dimensional regions, the method of Equation 12–1 is simply expanded, as shown below.
The numerical integration of 2D quadrilaterals gives:
 | (12–2) |
and the integration point locations are shown in Figure 12.19: Integration Point Locations for Quadrilaterals. The locations and weighting factors can be calculated using Table 12.6: Gauss Numerical Integration Constants two times.
One element models with midside nodes (such as PLANE183) using a 2 x 2 mesh of integration points have been seen to generate spurious zero energy (hourglassing) modes.
The 3D integration of bricks and pyramids gives:
 | (12–3) |
and the integration point locations are shown in Figure 12.20: Integration Point Locations for Bricks and Pyramids. The locations and weighting factors can be calculated using Table 12.6: Gauss Numerical Integration Constants three times.
One element models with midside nodes using a 2 x 2 x 2 mesh of integration points have been seen to generate spurious zero energy (hourglassing) modes.
The integration points used for these triangles are given in Table 12.7: Numerical Integration for Triangles and appear as shown in Figure 12.21: Integration Point Locations for Triangles. L varies from 0.0 at an edge to 1.0 at the opposite vertex.
Table 12.7: Numerical Integration for Triangles
| Type | Integration Point Location | Weighting Factor | |
|---|---|---|---|
| 1 Point Rule |
L1=L2=L3=.3333333 | 1.000000 | |
| 3 Point Rule |
L1=.66666 66666 66666 L2=L3=.16666 66666 66666 Permute L1, L2, and L3 for other locations) | 0.33333 33333 33333 | |
| 6 Point Rule | Corner Points |
L1=0.81684 75729 80459 L2=L3=0.09157 62135 09661 Permute L1, L2, and L3 for other locations) | 0.10995 17436 55322 |
| Edge Center Points |
L1=0.10810 30181 6807 L2=L3=0.44594 84909 15965 Permute L1, L2, and L3 for other locations) | 0.22338 15896 78011 | |
The integration points used for tetrahedra are given in Table 12.8: Numerical Integration for Tetrahedra.
Table 12.8: Numerical Integration for Tetrahedra
| Type | Integration Point Location | Weighting Factor | |
|---|---|---|---|
| 1 Point Rule | Center Point | L1=L2=L3=L4=.25000 00000 00000 | 1.00000 00000 00000 |
| 4 Point Rule | Corner Points |
L1=.58541 01966 24968 L2=L3=L4=.13819 66011 25010 Permute L1, L2, L3, and L4 for other locations) | 0.25000 00000 00000 |
| 5 Point Rule | Center Point | L1=L2=L3=L4=.25000 00000 00000 | -0.80000 00000 00000 |
| Corner Points |
L1=.50000 00000 00000 L2=L3=L4=.16666 66666 66666 Permute L1, L2, L3, and L4 for other locations) | 0.45000 00000 00000 | |
| 11 Point Rule | Center Point | L1=L2=L3=L4=.25000 00000 00000 | -0.01315 55555 55556 |
| Corner Point |
L1=L2=L3=.0714285714285714 L4=.78571 42857 14286 (Permute L1, L2, L3 and L4 for other three locations) | 0.00762 22222 22222 | |
| Edge Center Points |
L1=L2=0.39940 35761 66799 L3=L4=0.10059 64238 33201 Permute L1, L2, L3 and L4 such that two of L1, L2, L3 and L4 equal 0.39940 35761 66799 and the other two equal 0.10059 64238 33201 for other five locations | 0.02488 88888 88888 | |
These appear as shown in Figure 12.22: Integration Point Locations for Tetrahedra. L varies from 0.0 at a face to 1.0 at the opposite vertex.
These elements use the same integration point scheme as for 4-node quadrilaterals and 8-node solids, as shown in Figure 12.23: Integration Point Locations for Triangles and Tetrahedra. The locations and weighting factors can be calculated using Table 12.6: Gauss Numerical Integration Constants two or three times.
3x3 and 3x3x3 cases are handled similarly.
These wedge elements use an integration scheme that combines linear and triangular integrations, as shown in Figure 12.24: 6 and 9 Integration Point Locations for Wedges. The locations and weighting factors can be calculated using Table 12.6: Gauss Numerical Integration Constants and Table 12.7: Numerical Integration for Triangles.
These wedge elements use the same integration point scheme as for 8-node solid elements as shown by two orthogonal views in Figure 12.25: 8 Integration Point Locations for Wedges. The locations and weighting factors can be calculated using Table 12.6: Gauss Numerical Integration Constants three times.
The 20-node solid uses a different type of integration point scheme. This scheme places points close to each of the 8 corner nodes and close to the centers of the 6 faces for a total of 14 points. These locations are given in Table 12.9: Numerical Integration for 20-Node Brick:
Table 12.9: Numerical Integration for 20-Node Brick
| Type | Integration Point Location | Weighting Factor | |
|---|---|---|---|
| 14 Point Rule | Corner Points |
s = ±.75878 69106 39328 t = ±.75878 69106 39329 r = ±.75878 69106 39329 | .33518 00554 01662 |
| Center Points |
s = ±.79582 24257 54222, t=r=0.0 t = ±.79582 24257 54222, s=r=0.0 r = ±.79582 24257 54222, s=t=0.0 | .88642 65927 97784 | |
and are shown in Figure 12.26: Integration Point Locations for 14 Point Rule.
Both beam and shell elements that have nonlinear materials must have their effects accumulated thru the thickness. This uses nonstandard integration point locations, as both the top and bottom surfaces have an integration point in order to immediately detect the onset of the nonlinear effects.
Table 12.10: Thru-Thickness Numerical Integration
| Type | Integration Point Location[1] | Weighting Factor |
|---|---|---|
| 5 | ±0.500 | 0.1250000 |
| ±0.300 | 0.5787036 | |
| 0.000 | 0.5925926 |
These locations are shown in Figure 12.27: Nonlinear Bending Integration Point Locations.
The numerical integration of general axisymmetric elements gives:
 | (12–4) |
Hi and Hj are weighting factors on the rz plane, as shown in Figure 11.20: General Axisymmetric Solid Elements (when Nnp = 3). The values are shown in Table 12.6: Gauss Numerical Integration Constants. In circumferential direction θ:
 | (12–5) |