4. Les éléments volumiques

4.1. Tétraèdres : ELREFE TE4, T10

Coordonnées des nœuds:

	\(x\)	\(y\)	\(z\)
\(\mathrm{N1}\)
\(\mathrm{N2}\)
\(\mathrm{N3}\)
\(\mathrm{N4}\)
\(\mathrm{N5}\)		0.5	0.5
\(\mathrm{N6}\)			0.5
\(\mathrm{N7}\)		0.5
\(\mathrm{N8}\)	0.5	0.5
\(\mathrm{N9}\)	0.5		0.5
\(\mathrm{N10}\)	0.5

Fonctions de forme:

Formule à 4 nœuds

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

Formule à 10 nœuds

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

Formule d’intégration numérique:

Formule à 1point, d’ordre1 en \(x,y,z\) : (FPG1)

Point	\(x\)	\(y\)	\(z\)	Poids
1	\(1/4\)	\(1/4\)	\(1/4\)	\(1/6\)

Formule à 4points, d’ordre2 en \(x,y,z\) : (FPG4)

Point	\(x\)	\(y\)	\(z\)	Poids
1	\(a\)	\(a\)	\(a\)	\(1/24\)
2	\(a\)	\(a\)	\(b\)	\(1/24\)
3	\(a\)	\(b\)	\(a\)	\(1/24\)
4	\(b\)	\(a\)	\(a\)	\(1/24\)

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

Formule à 5points, d’ordre3 en \(x,y,z\) : (FPG5)

Point	\(x\)	\(y\)	\(z\)	Poids
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	\(c\)	\(b\)	\(b\)	\(3/40\)

Avec: \(a=0.25\) , \(b=\frac{1}{6}\) , \(c=0.5\)

Formule à 15points, d’ordre5 en \(x,y,z\) : (FPG15)

Point	\(x\)	\(y\)	\(z\)	Poids
1	\(a\)	\(a\)	\(a\)	\(8/405\)
2 3 4 5	\({b}_{1}\) \({b}_{1}\) \({b}_{1}\) \({c}_{1}\)	\({b}_{1}\) \({b}_{1}\) \({c}_{1}\) \({b}_{1}\)	\({b}_{1}\) \({c}_{1}\) \({b}_{1}\) \({b}_{1}\)	\(\frac{2665-14\sqrt{15}}{226800}\)
6 7 8 9	\({b}_{2}\) \({b}_{2}\) \({b}_{2}\) \({c}_{2}\)	\({b}_{2}\) \({b}_{2}\) \({c}_{2}\) \({b}_{2}\)	\({b}_{2}\) \({c}_{2}\) \({b}_{2}\) \({b}_{2}\)	\(\frac{2665+14\sqrt{15}}{226800}\)
10 11 12 13 14 15	\(d\) \(d\) \(e\) \(d\) \(e\) \(e\)	\(d\) \(e\) \(d\) \(e\) \(d\) \(e\)	\(e\) \(d\) \(d\) \(e\) \(e\) \(d\)	\(\frac{5}{567}\)

avec:

\(a=0.25\)

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

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

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

4.2. Pentaèdres : ELREFE PE6, P15, P18,P21

N19

N20

N21

Coordonnées des nœuds:

	\(x\)	\(y\)	\(z\)
\(\mathrm{N1}\)	-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

Fonctions de forme:

Formule à 6 nœuds

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

Formule à 15 nœuds

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

Formule à 18 nœuds

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

Formules d’intégration numérique à 6 points (ordre3 en \(x\) , ordre2 en \(y\) et \(z\) ) (FPG6)

Point	\(x\)	\(y\)	\(z\)	Poids
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		\(1/6\)

Formule d’intégration numérique à 8 points: (FPG8)

2 points de Gauss en \(x\) (ordre 3).

4 points de Hammer en \(y\) et \(z\) (ordre3).

Point	\(x\)	\(y\)	\(z\)	Poids
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	\(25/96\)

Avec \(a=0.577350269189626\)

Formule d’intégration numérique à 21 points: (FPG21)

3 points de Gauss en \(x\) (ordre 5).

7 points de Hammer en \(y\) et \(z\) (ordre5 en \(y\) et \(z\) ).

Point	\(x\)	\(y\)	\(z\)	Poids
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	\(\alpha\) \(\alpha\) \(\alpha\)	\(b\) \(1-\mathrm{2b}\) \(b\)	\(b\) \(b\) \(1-\mathrm{2b}\)	\({c}_{1}(\frac{155-\sqrt{15}}{2400})\)

avec:

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

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

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

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

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

Formule d’intégration numérique à 27 points (FPG27): voir [bib3].

Point	\(x\)	\(y\)	\(z\)	Poids
1	0.0	0.895512822481133	0.052243588759434	0.027191062410231
2	0.0	0.052243588759434	0.895512822481133	0.027191062410231
3	0.0	0.052243588759434	0.052243588759434	0.027191062410231
4	0.0	0.198304865473555	0.270635256143164	0.040636041641220
5	0.0	0.198304865473555	0.531059878383280	0.040636041641220
6	0.0	0.270635256143164	0.531059878383280	0.040636041641220
7	0.0	0.531059878383280	0.270635256143164	0.040636041641220
8	0.0	0.531059878383280	0.198304865473555	0.040636041641220
9	0.0	0.270635256143164	0.198304865473555	0.040636041641220
10	0.936241512371697	0.333333333333333	0.333333333333333	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	-0.936241512371697	0.333333333333333	0.333333333333333	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	0.041951149272741
26	-0.600638052820557	0.308513201856883	0.636655503269814	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. Hexaèdres : ELREFE HE8, H20, H27

Coordonnées des nœuds:

	\(x\)	\(y\)	\(z\)
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

Fonctions de forme:

Formule à 8 nœuds

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

Formule à 20 nœuds

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

Formule à 27 nœuds

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

Formule de quadrature de Gauss à 2points dans chaque direction (ordre3) (FPG8)

Point	\(x\)	\(y\)	\(z\)	Poids
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	\(1/\sqrt{3}\)	\(1/\sqrt{3}\)	\(1/\sqrt{3}\)

Formule de quadrature de Gauss à 3points dans chaque direction (ordre5): (FPG27)

Point	\(x\)	\(y\)	\(z\)	Poids
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	\(\alpha\)	\(\alpha\)	\(\alpha\)	\({c}_{1}^{3}\)

avec:

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

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

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

4.4. Pyramides : ELREFE PY5, P13,P19

Les nœuds en bleu sont au milieu des faces, celui en rouge au milieu de la cellule.

La base carrée est constituée par le quadrangle \({N}_{1}{N}_{2}{N}_{3}{N}_{4}\) et \({N}_{5}\) est le sommet de la pyramide.

	\(x\)	\(y\)	\(z\)
\({N}_{1}\)
\({N}_{2}\)
\({N}_{3}\)	–1.
\({N}_{4}\)		–1.
\({N}_{5}\)
\({N}_{6}\)	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	0.2

Fonctions de forme:

Formule à 5 nœuds

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

Formule à 13 nœuds

\(\begin{array}{}{w}_{1}=\frac{(-x+y+z-1)(-x-y+z-1)(x-0.5)}{2(1-z)}\\ {w}_{2}=\frac{(-x-y+z-1)(x-y+z-1)(y-0.5)}{2(1-z)}\\ {w}_{3}=\frac{(x-y+z-1)(x+y+z-1)(-x-0.5)}{2(1-z)}\\ {w}_{4}=\frac{(x+y+z-1)(-x+y+z-1)(-y-0.5)}{2(1-z)}\\ {w}_{5}=\mathrm{2z}(z-0.5)\\ {w}_{6}=-\frac{(-x+y+z-1)(-x-y+z-1)(x-y+z-1)}{2(1-z)}\\ {w}_{7}=-\frac{(-x-y+z-1)(x-y+z-1)(x+y+z-1)}{2(1-z)}\end{array}\)

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

Formule d’intégration numérique à 5 points d’ordre 2 (FPG5):

Point	\(x\)	\(y\)	\(z\)	Poids
1	\(0,5\)	\(0\)	\({h}_{1}\)	\({p}_{1}\)
2	\(0\)	\(0,5\)	\({h}_{1}\)	\({p}_{1}\)
3	\(–0,5\)	\(0\)	\({h}_{1}\)	\({p}_{1}\)
4	\(0\)	\(–0,5\)	\({h}_{1}\)	\({p}_{1}\)
5	\(0\)	\(0\)	\({h}_{1}\)	\({p}_{1}\)

avec:

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

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

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

Formule d’intégration numérique à 6 points d’ordre 3 (FPG6):

Point	\(x\)	\(y\)	\(z\)	Poids
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	\(a\)	\(0\)	\({h}_{3}\)	\({p}_{3}\)

Avec:

\(a=0.5610836110587396\)

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

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

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

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

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

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

Formule d’intégration numérique à 10 points (FPG10) d’ordre 4, voir [ 2 ]:

Point	\(x\)	\(y\)	\(z\)	Poids
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	\(b\)	\(0\)	\({h}_{4}\)	\({w}_{4}\)

Avec:

\(a=0.3252907781991163\)

\(b=0.65796699712169\)

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

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

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

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

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

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

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

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

Formule d’intégration numérique à 15 points de Gauss (FPG15) d’ordre 5:

Point	\(x\)	\(y\)	\(z\)	Poids
1	0.0	0.0	0.7298578807825067	0.04562357993942674
2	0.0	0.0	0.300401020813769	0.112931409661816
3	0.0	0.0	0.0000000064917722	0.03913635721904967
4	-0.3532630157731623	-0.3532630157731623	0.125	0.05096086209874681
5	-0.3532630157731623	0.3532630157731623	0.125	0.05096086209874681
6	0.3532630157731623	0.3532630157731623	0.125	0.05096086209874681
7	0.3532630157731623	-0.3532630157731623	0.125	0.05096086209874681
8	-0.7051171227788277	0.531059878383280	0.061111907062023	0.02644726771976367
9	0.0	-0.7051171227788277	0.061111907062023	0.02644726771976367
10	0.0	0.7051171227788277	0.061111907062023	0.02644726771976367
11	0.7051171227788277	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

Formule d’intégration numérique à 24 points de Gauss (FPG24) d’ordre 6:

Point	\(x\)	\(y\)	\(z\)	Poids
1	0.0	0.0	0.8076457976939595	0.01697526244176133
2	0.0	0.0	0.0017638088528196	0.0107023421167942
3	0.0	0.0	0.1382628064637306	0.0797197029683492
4	0.0	0.0	0.4214239119356371	0.0687071134661012
5	-0.4172976755573542	-0.4172976755573542	0.097447341025462	0.02463372529088633
6	-0.4172976755573542	0.4172976755573542	0.097447341025462	0.02463372529088633
7	0.4172976755573542	0.4172976755573542	0.097447341025462	0.02463372529088633
8	0.4172976755573542	-0.4172976755573542	0.097447341025462	0.02463372529088633
9	-0.2169627046883496	-0.2169627046883496	0.5660745906233009	0.02105838632544886
10	-0.2169627046883496	0.2169627046883496	0.5660745906233009	0.02105838632544886
11	0.2169627046883496	0.2169627046883496	0.5660745906233009	0.02105838632544886
12	0.2169627046883496	-0.2169627046883496	0.5660745906233009	0.02105838632544886
13	-0.5656808544256755	0.0	0.0294777308457207	0.0248000862596322
14	0.0	-0.5656808544256755	0.0294777308457207	0.0248000862596322
15	0.0	0.5656808544256755	0.0294777308457207	0.0248000862596322
16	0.5656808544256755	0.0	0.0294777308457207	0.0248000862596322
17	-0.498079091780705	0.0	0.2649158632121295	0.04925492311795127
18	0.0	-0.498079091780705	0.2649158632121295	0.04925492311795127
19	0.0	0.498079091780705	0.2649158632121295	0.04925492311795127
20	0.498079091780705	0.0	0.2649158632121295	0.04925492311795127
21	-0.9508994872144825	0.0	0.048249070631936	0.0028934404244966
22	0.0	-0.9508994872144825	0.048249070631936	0.0028934404244966
23	0.0	0.9508994872144825	0.048249070631936	0.0028934404244966
24	0.9508994872144825	0.0	0.048249070631936	0.0028934404244966