4. Volumic elements#

4.1. Tetrahedra: ELREFE TE4, T10#

_images/100012360000182F0000170C3B52F571AC5ECFD5.svg

Node coordinates:

\(x\)

\(y\)

\(z\)

\(\mathrm{N1}\)

\(\mathrm{N2}\)

\(\mathrm{N3}\)

\(\mathrm{N4}\)

\(\mathrm{N5}\)

0.5

0.5

\(\mathrm{N6}\)

0.5

\(\mathrm{N7}\)

0.5

\(\mathrm{N8}\)

0.5

0.5

0.5

\(\mathrm{N9}\)

0.5

0.5

\(\mathrm{N10}\)

0.5

Shape functions:

4-knot formula

\(\{\begin{array}{}{w}_{1}(x,y,z)=y\\ {w}_{2}(x,y,z)=z\\ {w}_{3}(x,y,z)=1-x-y-z\\ {w}_{4}(x,y,z)=x\end{array}\)

10-knot formula

(4.7)#\[\begin{split}\begin{array}{}{w}_{1}=y(2y-1)\\ {w}_{2}=z(2z-1)\\ {w}_{3}=(1-x-y-z)(1-2x-2y-2z)\\ {w}_{4}=x(2x-1)\\ {w}_{5}=4yz\end{array}\end{split}\]

Numerical integration formula:

Formula with 1 point, order 1 in \(x,y,z\): (FPG1)

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(1/4\)

\(1/4\)

\(1/4\)

\(1/6\)

Formula with 4 points, order 2 in \(x,y,z\): (FPG4)

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(a\)

\(a\)

\(a\)

\(1/24\)

2

\(a\)

\(a\)

\(b\)

\(1/24\)

3

\(a\)

\(b\)

\(a\)

\(1/24\)

4

\(b\)

\(a\)

\(a\)

\(1/24\)

with: \(a=\frac{5-\sqrt{5}}{\text{20}}\), \(b=\frac{5+3\sqrt{5}}{\text{20}}\)

Formula with 5 points, order 3 in \(x,y,z\): (FPG5)

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(a\)

\(a\)

\(a\)

\(-2/15\)

2

\(b\)

\(b\)

\(b\)

\(3/40\)

3

\(b\)

\(b\)

\(c\)

\(3/40\)

4

\(b\)

\(c\)

\(b\)

\(3/40\)

5

\(c\)

\(b\)

\(b\)

\(3/40\)

With: \(a=0\text{.}\text{25}\), \(b=\frac{1}{6}\), \(c=0\text{.}5\)

Formula with 15 points, order 5 in \(x,y,z\): (FPG15)

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(a\)

\(a\)

\(a\)

\(8/405\)

2 3 4 5

\({b}_{1}\) \({b}_{1}\) \({b}_{1}\) \({c}_{1}\)

\({b}_{1}\)

\({b}_{1}\) \({c}_{1}\) \({b}_{1}\)

\({b}_{1}\)

\({c}_{1}\) \({b}_{1}\) \({b}_{1}\)

\(\frac{2\text{665}-\text{14}\sqrt{\text{15}}}{\text{226}\text{800}}\)

6 7 8 9

\({b}_{2}\) \({b}_{2}\) \({b}_{2}\) \({c}_{2}\)

\({b}_{2}\)

\({b}_{2}\) \({c}_{2}\) \({b}_{2}\)

\({b}_{2}\)

\({c}_{2}\) \({b}_{2}\) \({b}_{2}\)

\(\frac{2\text{665}+\text{14}\sqrt{\text{15}}}{\text{226}\text{800}}\)

10 11 12 13 14 15

\(d\) \(d\) \(e\) \(d\) \(e\) \(e\)

\(d\)

\(e\) \(d\) \(e\) \(d\) \(e\)

\(e\)

\(d\) \(d\) \(e\) \(e\) \(d\)

\(\frac{5}{\text{567}}\)

with:

\(a=0\text{.}\text{25}\)

\(\begin{array}{}{b}_{1}=\frac{7+\sqrt{\text{15}}}{\text{34}}\\ {b}_{2}=\frac{7-\sqrt{\text{15}}}{\text{34}}\end{array}\)

\(\begin{array}{}{c}_{1}=\frac{\text{13}-3\sqrt{\text{15}}}{\text{34}}\\ {c}_{2}=\frac{\text{13}+3\sqrt{\text{15}}}{\text{34}}\end{array}\)

\(\begin{array}{}d=\frac{5-\sqrt{\text{15}}}{\text{20}}\\ e=\frac{5+\sqrt{\text{15}}}{\text{20}}\end{array}\)

4.2. Pentahedra: ELREFE PE6, P15, P15, P18, P21#

N19 ROAD

N20 ROAD

N21 ROAD

_images/Shape1.gif

Node coordinates:

\(x\)

\(y\)

\(z\)

\(\mathrm{N1}\)

-1.

\(\mathrm{N2}\)

-1.

\(\mathrm{N3}\)

-1.

\(\mathrm{N4}\)

\(\mathrm{N5}\)

\(\mathrm{N6}\)

\(\mathrm{N7}\)

-1.

0.5

0.5.

\(\mathrm{N8}\)

-1.

0.5.

\(\mathrm{N9}\)

-1.

0.5

\(\mathrm{N10}\)

\(\mathrm{N11}\)

\(\mathrm{N12}\)

\(\mathrm{N13}\)

0.5

0.5

\(\mathrm{N14}\)

0.5

\(\mathrm{N15}\)

0.5

\(\mathrm{N16}\)

0.5

0.5

\(N17\)

0.5

\(\mathrm{N18}\)

0.5

\(N19\)

-1.

1/3

1/3

\(N20\)

1/3

1/3

\(N21\)

1/3

1/3

Shape functions:

6-knot formula

(4.7)#\[\begin{split}\begin{array}{}{w}_{1}=\frac{1}{2}y(1-x)\\ {w}_{2}=\frac{1}{2}z(1-x)\\ {w}_{3}=\frac{1}{2}(1-y-z)(1-x)\end{array}\end{split}\]

15-knot formula

(4.7)#\[\begin{split}\begin{array}{}{w}_{1}=y(1-x)(2y-2-x)/2\\ {w}_{2}=z(1-x)(2z-2-x)/2\\ {w}_{3}=(x-1)(1-y-z)(x+2y+2z)/2\\ {w}_{4}=y(1+x)(2y-2+x)/2\\ {w}_{5}=z(1+x)(2z-2+x)/2\\ {w}_{6}=(-x-1)(1-y-z)(-x+2y+2z)/2\\ {w}_{7}=2yz(1-x)\\ {w}_{8}=2z(1-y-z)(1-x)\end{array}\end{split}\]

18 knot formula

(4.7)#\[\begin{split}\begin{array}{}{w}_{1}=xy(x-1)(2y-1)/2\\ {w}_{2}=xz(x-1)(2z-1)/2\\ {w}_{3}=x(x-1)(z+y-1)(2z+2y-1)/2\\ {w}_{4}=xy(x+1)(2y-1)/2\\ {w}_{5}=xz(x+1)(2z-1)/2\\ {w}_{6}=x(x+1)(z+y-1)(2z+2y-1)/2\\ {w}_{7}=2xyz(x-1)\\ {w}_{8}=-2xz(x-1)(z+y-1)\\ {w}_{9}=-2xy(x-1)(z+y-1)\end{array}\end{split}\]

6-point numerical integration formulas (order 3 in \(x\), order 2 in \(y\) and \(z\)) (FPG6)

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(-1/\sqrt{3}\)

0.5

0.5

\(1/6\)

2

\(-1/\sqrt{3}\)

0.5

\(1/6\)

3

\(-1/\sqrt{3}\)

0.5

\(1/6\)

4

\(1/\sqrt{3}\)

0.5

0.5

\(1/6\)

5

\(1/\sqrt{3}\)

0.5

\(1/6\)

6

\(1/\sqrt{3}\)

0.5

\(1/6\)

8-point numerical integration formula: (FPG8)

2 Gauss points in \(x\) (order 3).

4 Hammer points in \(y\) and \(z\) (3rd order).

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(-a\)

\(1/3\)

\(1/3\)

\(-27/96\)

2

\(-a\)

0.6

0.2

\(25/96\)

3

\(-a\)

0.2

0.6

\(25/96\)

4

\(-a\)

0.2

0.2

\(25/96\)

5

\(+a\)

\(1/3\)

\(1/3\)

\(-27/96\)

6

\(+a\)

0.6

0.2

\(25/96\)

7

\(+a\)

0.2

0.6

\(25/96\)

8

\(+a\)

0.2

0.2

\(25/96\)

With \(a=0.577350269189626\)

21-point numerical integration formula: (FPG21)

3 Gauss points in \(x\) (order 5).

7 Hammer points in \(y\) and \(z\) (order 5 in \(y\) and \(z\)).

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(-\alpha\)

\(1/3\)

\(1/3\)

\({c}_{1}\frac{9}{80}\)

2 3 4

\(-\alpha\) \(-\alpha\) \(-\alpha\)

\(a\)

\(1-\mathrm{2a}\) \(a\)

\(a\)

\(a\) \(1-\mathrm{2a}\)

\({c}_{1}(\frac{155+\sqrt{15}}{2400})\)

5 6 7

\(-\alpha\) \(-\alpha\) \(-\alpha\)

\(b\)

\(1-\mathrm{2b}\) \(b\)

\(b\)

\(b\) \(1-\mathrm{2b}\)

\({c}_{1}(\frac{155-\sqrt{15}}{2400})\)

8

\(1/3\)

\(1/3\)

\({c}_{2}\frac{9}{80}\)

9 10 11

0. 0. 0.

\(a\)

\(1-\mathrm{2a}\) \(a\)

\(a\)

\(a\) \(1-\mathrm{2a}\)

\({c}_{2}(\frac{155+\sqrt{15}}{2400})\)

12 13 14

0. 0. 0.

\(b\)

\(1-\mathrm{2b}\) \(b\)

\(b\)

\(b\) \(1-\mathrm{2b}\)

\({c}_{2}(\frac{155-\sqrt{15}}{2400})\)

15

\(\alpha\)

\(1/3\)

\(1/3\)

\({c}_{1}\frac{9}{80}\)

16 17 18

\(\alpha\) \(\alpha\) \(\alpha\)

\(b\)

\(1-\mathrm{2a}\) \(a\)

\(a\)

\(a\) \(1-\mathrm{2a}\)

\({c}_{1}(\frac{155+\sqrt{15}}{2400})\)

19 20 21

\(\alpha\) \(\alpha\) \(\alpha\)

\(b\)

\(1-\mathrm{2b}\) \(b\)

\(b\)

\(b\) \(1-\mathrm{2b}\)

\({c}_{1}(\frac{155-\sqrt{15}}{2400})\)

with:

\(\alpha =\sqrt{\frac{3}{5}}\)

\({c}_{1}=\frac{5}{9}\)

\({c}_{2}=\frac{8}{9}\)

\(a=\frac{6+\sqrt{15}}{21}\)

\(b=\frac{6-\sqrt{15}}{21}\)

27-point numerical integration formula (FPG27): see [bib3].

Point

\(x\)

\(y\)

\(z\)

Weight

1

0.0

0.895512822481133

0.052243588759434

0.027191062410231

2

0.0

0.052243588759434

0.052243588759434

0.895512822481133

0.027191062410231

3

0.0

0.052243588759434

0.052243588759434

0.027191062410231

4

0.0

0.198304865473555

0.270635256143164

0.040636041641641220

5

0.0

0.198304865473555

0.531059878383280

0.040636041641641220

6

0.0

0.270635256143164

0.531059878383280

0.040636041641641220

7

0.0

0.531059878383280

0.270635256143164

0.040636041641641220

8

0.0

0.531059878383280

0.198304865473555

0.040636041641641220

9

0.0

0.270635256143164

0.198304865473555

0.040636041641641220

10

0.936241512371697

0.3333333333333333333

0.33333333333333333

0.05027512371697

0.050275140937507

11

0.948681147283254

0.841699897299232

0.079150051350384

0.011774414962347

12

0.948681147283254

0.079150051350384

0.841699897299232

0.011774414962347

13

0.948681147283254

0.079150051350384

0.079150051350384

0.011774414962347

14

0.600638052820557

0.054831294873304

0.308513201856883

0.041951149272741

15

0.600638052820557

0.054831294873304

0.636655503269814

0.041951149272741

16

0.600638052820557

0.308513201856883

0.636655503269814

0.041951149272741

17

0.600638052820557

0.636655503269814

0.308513201856883

0.041951149272741

18

0.600638052820557

0.636655503269814

0.054831294873304

0.041951149272741

19

0.600638052820557

0.308513201856883

0.054831294873304

0.041951149272741

20

-0.93624151212371697

0.3333333333333333333

0.33333333333333333

0.05027512371697

0.050275140937507

21

-0.948681147283254

0.841699897299232

0.079150051350384

0.011774414962347

22

-0.948681147283254

0.079150051350384

0.841699897299232

0.011774414962347

23

-0.948681147283254

0.079150051350384

0.079150051350384

0.011774414962347

24

-0.600638052820557

0.054831294873304

0.308513201856883

0.041951149272741

25

-0.600638052820557

0.054831294873304

0.636655503269814

0.0419511492820557

0.041951149272741

26

-0.600638052820557

0.308513201856883

0.636655503269814

0.0419511492820557

0.041951149272741

27

-0.600638052820557

0.636655503269814

0.308513201856883

0.041951149272741

28

-0.600638052820557

0.636655503269814

0.054831294873304

0.041951149272741

29

-0.600638052820557

0.308513201856883

0.054831294873304

0.041951149272741

4.3. Hexahedra: ELREFE HE8, H20, H27#

_images/Shape2.gif _images/Shape3.gif

Node coordinates:

\(x\)

\(y\)

\(z\)

N1

-1.

-1.

-1.

N2

-1.

-1.

N3

-1.

N4

-1.

-1.

N5

-1.

-1.

N6

-1.

N7

N8

-1.

N9

-1.

-1.

N10

-1.

N11

-1.

N12

-1.

-1.

N13

-1.

-1.

N14

-1.

N15

N16

-1.

N17

-1.

N18

N19

N20

-1.

N21

-1.

N22

-1.

N23

N24

N25

-1.

N26

N27

Shape functions:

8-knot formula

(4.7)#\[\begin{split}\begin{array}{}{w}_{1}=\frac{1}{8}(1-x)(1-y)(1-z)\\ {w}_{2}=\frac{1}{8}(1+x)(1-y)(1-z)\\ {w}_{3}=\frac{1}{8}(1+x)(1+y)(1-z)\\ {w}_{4}=\frac{1}{8}(1-x)(1+y)(1-z)\end{array}\end{split}\]

20-knot formula

(4.7)#\[\begin{split}\begin{array}{}{w}_{1}=\frac{1}{8}(1-x)(1-y)(1-z)(-2-x-y-z)\\ {w}_{2}=\frac{1}{8}(1+x)(1-y)(1-z)(-2+x-y-z)\\ {w}_{3}=\frac{1}{8}(1+x)(1+y)(1-z)(-2+x+y-z)\\ {w}_{4}=\frac{1}{8}(1-x)(1+y)(1-z)(-2-x+y-z)\\ {w}_{5}=\frac{1}{8}(1-x)(1-y)(1+z)(-2-x-y+z)\\ {w}_{6}=\frac{1}{8}(1+x)(1-y)(1+z)(-2+x-y+z)\\ {w}_{7}=\frac{1}{8}(1+x)(1+y)(1+z)(-2+x+y+z)\\ {w}_{8}=\frac{1}{8}(1-x)(1+y)(1+z)(-2-x+y+z)\\ {w}_{9}=\frac{1}{4}(1-{x}^{2})(1-y)(1-z)\\ {w}_{10}=\frac{1}{4}(1-{y}^{2})(1+x)(1-z)\end{array}\end{split}\]

27-knot formula

(4.7)#\[\begin{split}\begin{array}{}{w}_{1}=\frac{1}{8}x(x-1)y(y-1)z(z-1)\\ {w}_{2}=\frac{1}{8}x(x+1)y(y-1)z(z-1)\\ {w}_{3}=\frac{1}{8}x(x+1)y(y+1)z(z-1)\\ {w}_{4}=\frac{1}{8}x(x-1)y(y+1)z(z-1)\\ {w}_{5}=\frac{1}{8}x(x-1)y(y-1)z(z+1)\\ {w}_{6}=\frac{1}{8}x(x+1)y(y-1)z(z+1)\\ {w}_{7}=\frac{1}{8}x(x+1)y(y+1)z(z+1)\\ {w}_{8}=\frac{1}{8}x(x-1)y(y+1)z(z+1)\\ {w}_{9}=\frac{1}{4}(1-{x}^{2})y(y-1)z(z-1)\\ {w}_{\text{10}}=\frac{1}{4}x(x+1)(1-{y}^{2})z(z-1)\\ {w}_{\text{11}}=\frac{1}{4}(1-{x}^{2})y(y+1)z(z-1)\\ {w}_{\text{12}}=\frac{1}{4}x(x-1)(1-{y}^{2})z(z-1)\\ {w}_{\text{13}}=\frac{1}{4}x(x-1)y(y-1)(1-{z}^{2})\\ {w}_{\text{14}}=\frac{1}{4}x(x+1)y(y-1)(1-{z}^{2})\end{array}\end{split}\]

Gauss quadrature formula with 2 points in each direction (order 3) (FPG8)

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(-1/\sqrt{3}\)

\(-1/\sqrt{3}\)

\(-1/\sqrt{3}\)

2

\(-1/\sqrt{3}\)

\(-1/\sqrt{3}\)

\(1/\sqrt{3}\)

3

\(-1/\sqrt{3}\)

\(1/\sqrt{3}\)

\(-1/\sqrt{3}\)

4

\(-1/\sqrt{3}\)

\(1/\sqrt{3}\)

\(1/\sqrt{3}\)

5

\(1/\sqrt{3}\)

\(-1/\sqrt{3}\)

\(-1/\sqrt{3}\)

6

\(1/\sqrt{3}\)

\(-1/\sqrt{3}\)

\(1/\sqrt{3}\)

7

\(1/\sqrt{3}\)

\(1/\sqrt{3}\)

\(-1/\sqrt{3}\)

8

\(1/\sqrt{3}\)

\(1/\sqrt{3}\)

\(1/\sqrt{3}\)

Gauss quadrature formula with 3 points in each direction (order 5): (FPG27)

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(-\alpha\)

\(-\alpha\)

\(-\alpha\)

\({c}_{1}^{3}\)

2

\(-\alpha\)

\(-\alpha\)

\({c}_{1}^{2}{c}_{2}\)

3

\(-\alpha\)

\(-\alpha\)

\(\alpha\)

\({c}_{1}^{3}\)

4

\(-\alpha\)

\(-\alpha\)

\({c}_{1}^{2}{c}_{2}\)

5

\(-\alpha\)

\({c}_{1}{c}_{2}^{2}\)

6

\(-\alpha\)

\(\alpha\)

\({c}_{1}^{2}{c}_{2}\)

7

\(-\alpha\)

\(\alpha\)

\(-\alpha\)

\({c}_{1}^{3}\)

8

\(-\alpha\)

\(\alpha\)

\({c}_{1}^{2}{c}_{2}\)

9

\(-\alpha\)

\(\alpha\)

\(\alpha\)

\({c}_{1}^{3}\)

10

\(-\alpha\)

\(-\alpha\)

\({c}_{1}^{2}{c}_{2}\)

11

\(-\alpha\)

\({c}_{1}{c}_{2}^{2}\)

12

\(-\alpha\)

\(\alpha\)

\({c}_{1}^{2}{c}_{2}\)

13

\(-\alpha\)

\({c}_{1}{c}_{2}^{2}\)

14

\({c}_{2}^{3}\)

15

\(\alpha\)

\({c}_{1}{c}_{2}^{2}\)

16

\(\alpha\)

\(-\alpha\)

\({c}_{1}^{2}{c}_{2}\)

17

\(\alpha\)

\({c}_{1}{c}_{2}^{2}\)

18

\(\alpha\)

\(\alpha\)

\({c}_{1}^{2}{c}_{2}\)

19

\(\alpha\)

\(-\alpha\)

\(-\alpha\)

\({c}_{1}^{3}\)

20

\(\alpha\)

\(-\alpha\)

\({c}_{1}^{2}{c}_{2}\)

21

\(\alpha\)

\(-\alpha\)

\(\alpha\)

\({c}_{1}^{3}\)

22

\(\alpha\)

\(-\alpha\)

\({c}_{1}^{2}{c}_{2}\)

23

\(\alpha\)

\({c}_{1}{c}_{2}^{2}\)

24

\(\alpha\)

\(\alpha\)

\({c}_{1}^{2}{c}_{2}\)

25

\(\alpha\)

\(\alpha\)

\(-\alpha\)

\({c}_{1}^{3}\)

26

\(\alpha\)

\(\alpha\)

\({c}_{1}^{2}{c}_{2}\)

27

\(\alpha\)

\(\alpha\)

\(\alpha\)

\({c}_{1}^{3}\)

with:

\(\alpha =\sqrt{\frac{3}{5}}\)

\({c}_{1}=\frac{5}{9}\)

\({c}_{2}=\frac{8}{9}\)

4.4. Pyramids: ELREFE PY5, P13, P19#

_images/10000201000003E8000003215D64B9F61CD54C22.png

The blue nodes are in the middle of the faces, the red one is in the middle of the cell.

The square base is made up of the quadrangle \({N}_{1}{N}_{2}{N}_{3}{N}_{4}\) and \({N}_{5}\) is the top of the pyramid.

\(x\)

\(y\)

\(z\)

\({N}_{1}\)

\({N}_{2}\)

\({N}_{3}\)

—1.

\({N}_{4}\)

—1.

\({N}_{5}\)

\({N}_{6}\)

0.5

0.5

0.5

\({N}_{7}\)

—0.5

0.5

\({N}_{8}\)

—0.5

—0.5

\({N}_{9}\)

0.5

—0.5

\({N}_{10}\)

0.5

0.5

\({N}_{11}\)

0.5

0.5

\({N}_{12}\)

—0.5

0.5

\({N}_{13}\)

—0.5

0.5

\({N}_{14}\)

0

\({N}_{15}\)

1/3

1/3

1/3

\({N}_{16}\)

-1/3

1/3

1/3

\({N}_{17}\)

-1/3

-1/3

1/3

\({N}_{18}\)

1/3

-1/3

1/3

\({N}_{19}\)

0

0

0

0.2

Shape functions:

5-knot formula

\(\begin{array}{}{w}_{1}=\frac{(-x+y+z-1)(-x-y+z-1)}{4(1-z)}\\ {w}_{2}=\frac{(-x-y+z-1)(x-y+z-1)}{4(1-z)}\\ {w}_{3}=\frac{(x+y+z-1)(x-y+z-1)}{4(1-z)}\\ {w}_{4}=\frac{(x+y+z-1)(-x+y+z-1)}{4(1-z)}\\ {w}_{5}=z\end{array}\)

13-knot formula

\(\begin{array}{}{w}_{1}=\frac{(-x+y+z-1)(-x-y+z-1)(x-0\text{.}5)}{2(1-z)}\\ {w}_{2}=\frac{(-x-y+z-1)(x-y+z-1)(y-0\text{.}5)}{2(1-z)}\\ {w}_{3}=\frac{(x-y+z-1)(x+y+z-1)(-x-0\text{.}5)}{2(1-z)}\\ {w}_{4}=\frac{(x+y+z-1)(-x+y+z-1)(-y-0\text{.}5)}{2(1-z)}\\ {w}_{5}=\mathrm{2z}(z-0\text{.}5)\\ {w}_{6}=-\frac{(-x+y+z-1)(-x-y+z-1)(x-y+z-1)}{2(1-z)}\\ {w}_{7}=-\frac{(-x-y+z-1)(x-y+z-1)(x+y+z-1)}{2(1-z)}\end{array}\)

\(\begin{array}{}{w}_{8}=-\frac{(x-y+z-1)(x+y+z-1)(-x+y+z-1)}{2(1-z)}\\ {w}_{9}=-\frac{(x+y+z-1)(-x+y+z-1)(-x-y+z-1)}{2(1-z)}\\ {w}_{\text{10}}=\frac{z(-x+y+z-1)(-x-y+z-1)}{1-z}\\ {w}_{\text{11}}=\frac{z(-x-y+z-1)(x-y+z-1)}{1-z}\\ {w}_{\text{12}}=\frac{z(x-y+z-1)(x+y+z-1)}{1-z}\\ {w}_{\text{13}}=\frac{z(x+y+z-1)(-x+y+z-1)}{1-z}\end{array}\)

Numerical integration formula with 5 points of order 2 (FPG5):

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(\mathrm{0,5}\)

\(0\)

\({h}_{1}\)

\({p}_{1}\)

2

\(0\)

\(\mathrm{0,5}\)

\({h}_{1}\)

\({p}_{1}\)

3

\(–\mathrm{0,5}\)

\(0\)

\({h}_{1}\)

\({p}_{1}\)

4

\(0\)

\(–\mathrm{0,5}\)

\({h}_{1}\)

\({p}_{1}\)

5

\(0\)

\(0\)

\({h}_{1}\)

\({p}_{1}\)

with:

\({h}_{1}=0.1531754163448146\)

\({h}_{2}=0.6372983346207416\)

\({p}_{1}=\frac{2}{15}\)

Numerical integration formula with 6 order 3 points (FPG6):

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(0\)

\(0\)

\({h}_{1}\)

\({p}_{1}\)

2

\(0\)

\(0\)

\({h}_{2}\)

\({p}_{2}\)

3

\(–a\)

\(0\)

\({h}_{3}\)

\({p}_{3}\)

4

\(0\)

\(–a\)

\({h}_{3}\)

\({p}_{3}\)

5

\(0\)

\(a\)

\({h}_{3}\)

\({p}_{3}\)

6

\(a\)

\(0\)

\({h}_{3}\)

\({p}_{3}\)

With:

\(a=0.5610836110587396\)

\({p}_{1}=0.1681372559485071\)

\({p}_{2}=0.07500000404404333\)

\({p}_{3}=0.1058823516685291\)

\({h}_{1}=0.1681372559485071\)

\({h}_{2}=0.00000000567585\)

\({h}_{3}=0.1058823516685291\)

10-point numerical integration formula (FPG10) of order 4, see [2]:

Point

\(x\)

\(y\)

\(z\)

Weight

1

\(0\)

\(0\)

\({h}_{1}\)

\({w}_{1}\)

2

\(0\)

\(0\)

\({h}_{2}\)

\({w}_{2}\)

3

\(-a\)

\(-a\)

\({h}_{3}\)

\({w}_{3}\)

4

\(-a\)

\(a\)

\({h}_{3}\)

\({w}_{3}\)

5

\(a\)

\(a\)

\({h}_{3}\)

\({w}_{3}\)

6

\(a\)

\(-a\)

\({h}_{3}\)

\({w}_{3}\)

7

\(-b\)

\(0\)

\({h}_{4}\)

\({w}_{4}\)

8

\(0\)

\(-b\)

\({h}_{4}\)

\({w}_{4}\)

9

\(0\)

\(b\)

\({h}_{4}\)

\({w}_{4}\)

10

\(b\)

\(0\)

\({h}_{4}\)

\({w}_{4}\)

With:

\(a=0.3252907781991163\)

\(b=0.65796699712169\)

\({h}_{1}=0.6772327888861374\)

\({h}_{2}=0.1251369531087465\)

\({h}_{3}=0.3223841495782137\)

\({h}_{4}=0.0392482838988154\)

\({w}_{1}=0.07582792211376127\)

\({w}_{2}=0.1379222683930349\)

\({w}_{3}=0.07088305859288367\)

\({w}_{4}=0.04234606044708394\)

Numerical integration formula with 15 Gauss points (FPG15) of order 5:

Point

\(x\)

\(y\)

\(z\)

Weight

1

0.0

0.0

0.0

0.7298578807825067

0.0456235799393942674

2

0.0

0.0

0.0

0.300401020813769

0.112931409661816

3

0.0

0.0

0.0

0.0000000064917722

0.03913635721904967

4

-0.3532630157731623

-0.3532630157731623

0.125

0.050960862086209874681

5

-0.3532630157731623

0.3532630157731623

0.125

0.050960862086209874681

6

0.3532630157731623

0.3532630157731623

0.125

0.05096086209874681

7

0.3532630157731623

-0.3532630157731623

0.125

0.050960862086209874681

8

-0.705117122727788277

0.531059878383280

0.061111907062023

0.0264472678827788277

0.02644726771976367

9

0.0

-0.705117121227788277

0.061111907062023

0.02644726771976367

10

0.0

0.705117121227788277

0.061111907062023

0.02644726771976367

11

0.705117122727788277

0.0

0.061111907062023

0.02644726771976367

12

-0.432882864103541

0.0

0.4236013371197248

0.011774414962347

13

0.0

-0.432882864103541

0.4236013371197248

0.011774414962347

14

0.0

0.432882864103541

0.4236013371197248

0.041951149272741

15

0.432882864103541

0.0

0.4236013371197248

0.041951149272741

Numerical integration formula with 24 Gauss points (FPG24) of order 6:

Point

\(x\)

\(y\)

\(z\)

Weight

1

0.0

0.0

0.0

0.8076457976939595

0.01697526244176133

2

0.0

0.0

0.0

0.0017638088528196

0.0107023421167942

3

0.0

0.0

0.0

0.1382628064637306

0.0797197029683492

4

0.0

0.0

0.0

0.4214239119356371

0.0687071134661012

5

-0.4172976755573542

-0.4172976755573542

0.097447341025462

0.024633755573542

0.02463375557353542

0.024633755573542

6

-0.4172976755573542

0.4172976755573542

0.097447341025462

0.024633755573542

0.02463375557353542

0.4172976755573542

0.097447341025462

0.024633755573542

7

0.4172976755573542

0.4172976755573542

0.097447341025462

0.024633755573542

0.02463375557353542

0.4172976755573542

0.097447341025462

0.024633755573542

8

0.4172976755573542

-0.4172976755573542

0.097447341025462

0.024633755573542

0.02463375557353542

0.024633725573542

9

-0.2169627046883496

-0.2169627046883496

0.5660745906233009

0.02105846883496

0.02105838883496

0.0210583863632544886

10

-0.2169627046883496

0.2169627046883496

0.5660745906233009

0.0210583846883496

0.02105838883496

0.0210583863632544886

11

0.2169627046883496

0.2169627046883496

0.5660745906233009

0.02105838883496

0.0210583868632544886

12

0.2169627046883496

-0.2169627046883496

0.5660745906233009

0.02105838883496

0.0210583868632544886

13

-0.565680854444256755

0.0

0.0294777308457207

0.0248000862596322

14

0.0

-0.565680858544256755

0.0294777308457207

0.0248000862596322

15

0.0

0.565680858544256755

0.0294777308457207

0.0248000862596322

16

0.565680854444256755

0.0

0.0294777308457207

0.0248000862596322

17

-0.498079091780705

0.0

0.2649158632121295

0.049254923117951295125

0.04925492311795127

18

0.0

-0.498079079091780705

0.2649158632121295

0.04925492311795127

19

0.0

0.498079099091780705

0.2649158632121295

0.04925492311795127

20

0.498079091780705

0.0

0.2649158632121295

0.04925492311795125

0.04925492311795127

21

-0.9508994872144825

0.0

0.0

0.048249070631936

0.0028934404244966

22

0.0

-0.950899484872144825

0.048249070631936

0.0028934404244966

23

0.0

0.950899484872144825

0.048249070631936

0.0028934404244966

24

0.9508994872144825

0.0

0.048249070631936

0.0028934404244966