12.2. Integration Point Locations

The ANSYS program 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.

12.2.1. Lines (1, 2, or 3 Points)

The standard 1-D 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 PointsIntegration Point Locations (xi)Weighting Factors (Hi)
10.00000.00000.000002.00000.00000.00000
2±0.57735 02691 896261.00000.00000.00000
3±0.77459 66692 414830.55555 55555 55556
0.00000.00000.000000.88888 88888 88889

For some integrations of multi-dimensional regions, the method of Equation 12–1 is simply expanded, as shown below.

12.2.2. Quadrilaterals (2 x 2 or 3 x 3 Points)

The numerical integration of 2-D 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.

Figure 12.19:  Integration Point Locations for Quadrilaterals

Integration Point Locations for Quadrilaterals

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.

12.2.3. Bricks and Pyramids (2 x 2 x 2 Points)

The 3-D 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.

Figure 12.20:  Integration Point Locations for Bricks and Pyramids

Integration Point Locations for Bricks and Pyramids

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.

12.2.4. Triangles (1, 3, or 6 Points)

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

TypeIntegration Point LocationWeighting 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 RuleCorner 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

Figure 12.21:  Integration Point Locations for Triangles

Integration Point Locations for Triangles

12.2.5. Tetrahedra (1, 4, 5, or 11 Points)

The integration points used for tetrahedra are given in Table 12.8: Numerical Integration for Tetrahedra.

Table 12.8:  Numerical Integration for Tetrahedra

TypeIntegration Point LocationWeighting Factor
1 Point RuleCenter PointL1=L2=L3=L4=.25000 00000 000001.00000 00000 00000
4 Point RuleCorner 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 RuleCenter PointL1=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 RuleCenter PointL1=L2=L3=L4=.25000 00000 000000.01315 55555 55555
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.

Figure 12.22:  Integration Point Locations for Tetrahedra

Integration Point Locations for Tetrahedra

12.2.6. Triangles and Tetrahedra (2 x 2 or 2 x 2 x 2 Points)

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.

Figure 12.23:  Integration Point Locations for Triangles and Tetrahedra

Integration Point Locations for Triangles and Tetrahedra

3x3 and 3x3x3 cases are handled similarly.

12.2.7. Wedges (3 x 2 or 3 x 3 Points)

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.

Figure 12.24:  6 and 9 Integration Point Locations for Wedges

6 and 9 Integration Point Locations for Wedges

12.2.8. Wedges (2 x 2 x 2 Points)

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.

Figure 12.25:  8 Integration Point Locations for Wedges

8 Integration Point Locations for Wedges

12.2.9. Bricks (14 Points)

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

TypeIntegration Point LocationWeighting Factor
14 Point RuleCorner Points

s = ±.75868 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.

Figure 12.26:  Integration Point Locations for 14 Point Rule

Integration Point Locations for 14 Point Rule

12.2.10. Nonlinear Bending (5 Points)

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

TypeIntegration Point Location[1]Weighting Factor
5±0.5000.1250000
±0.3000.5787036
0.0000.5925926

  1. Thickness coordinate going from -0.5 to 0.5.

These locations are shown in Figure 12.27: Nonlinear Bending Integration Point Locations.

Figure 12.27:  Nonlinear Bending Integration Point Locations

Nonlinear Bending Integration Point Locations

12.2.11. General Axisymmetric Elements

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)


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