Numerical integration

The coordinates and the weight factors for numerical integration for several standard element topologies are available. Specific numerical integration is defined by its code number. Numerical integration is available under all supported environments as a part of supplementary routines. The integration points are constructed accordingly to the given integration code. Codes for the basic one two and three dimensional numerical integration rules are presented in tables below.  Basic integration codes can be combined in order to get more complicated multi-dimensional integrational rules. The combined code is given as a function of up to three basic codes as follows:

 

code=codeA+1000*codeB+1000000*codeC

 

, where codeA, codeB and codeC are any of the basic integration codes. For example, 2×2×5 Gauss integration can be represented with the code 24002 or equivalent code 24021021. Numbering of the integration points is depicted below.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


One dimensional

Code

Description

No. of points

Disposition

20

1 point Gauss

1

21

2 point Gauss

2

22

3 point Gauss

3

23

4 point Gauss

4

24

5 point Gauss

5

 

Triangle

Code

Description

No. of points

Disposition

12

1 point integration

1

13

3 point integration - nodes

3

14

3 point integration

3

16

4 point integration

4

17

7 point integration

7

Quadrilateral

Code

Description

No. of points

Disposition

1

1 point integration

1

2

2´2 Gauss integration

4

3

3´3 Gauss integration

9

4

5 point special rule

5

5

points in nodes

4

Tetrahedra

Code

Description

No. of points

Disposition

15

1 point integration

1

18

4 point integration

4

19

5 point integration

5

 

Hexahedra

Code

Description

No. of points

Disposition

6

1 point integration

1

7

2´2´2  Gauss integration

8

8

3´3´3  Gauss integration

27

9

4´4´4  Gauss integration

64

10

9 point special rule

9

11

points in nodes

8