r3.01.01 Fonctions de forme et points d’intégration des éléments finis#

Résumé:

On décrit la géométrie et la topologie des éléments finis implantés dans Code_Aster ; pour chaque élément de référence, l’expression des fonctions de forme et les différentes familles de points d’intégration ainsi que les poids associés sont détaillés.

Les éléments linéiques : SE2, SE3 et SE4#

SE2 : segment à 2 nœuds

nombre de nœuds	: 2
nombre de nœuds sommets	: 2

SE3 : segment à 3 nœuds

nombre de nœuds	: 3
nombre de nœuds sommets	: 2

	\(x\)
\(\mathrm{N1}\)	-1.0
\(\mathrm{N2}\)	1.0
\(\mathrm{N3}\)	0.0

../../../../_images/1000037600000D200000082A48C4D0285F94A1B5.svg

SE4 : segment à 4 nœuds

nombre de nœuds	: 4
nombre de nœuds sommets	: 2

	\(x\)
\(\mathrm{N1}\)	-1.0
\(\mathrm{N2}\)	1.0
\(\mathrm{N3}\)	-1/3
\(\mathrm{N4}\)	+1/3

fonctions de forme du segment à 2 nœuds:

\({w}_{1}(x)=0.5(1-x)\phantom{\rule{4em}{0ex}}{w}_{2}(x)=0.5(1+x)\)

fonctions de forme du segment à 3 nœuds:

\({w}_{1}(x)=-0.5(1-x)x\phantom{\rule{4em}{0ex}}{w}_{2}(x)=0.5(1+x)x\phantom{\rule{4em}{0ex}}{w}_{3}(x)=(1+x)(1-x)\)

fonctions de forme du segment à 4 nœuds:

\(\begin{array}{c}{w}_{1}(x)=\frac{9}{16}\left(1-x\right)\left(x+\frac{1}{3}\right)\left(x-\frac{1}{3}\right)\\ {w}_{2}(x)=-\frac{9}{16}\left(1+x\right)\left(\frac{1}{3}-x\right)\left(x+\frac{1}{3}\right)\\ {w}_{3}(x)=\frac{27}{16}\left(x-1\right)\left(x+1\right)\left(x-\frac{1}{3}\right)\\ {w}_{4}(x)=-\frac{27}{16}\left(x-1\right)\left(x+1\right)\left(x+\frac{1}{3}\right)\end{array}\)

Nombre de points d’intégration	Point	\(x\)	Poids
1	1	0.000000000000000	2.000000000000000
2	1	0.577350269189626	1.000000000000000
	2	-0.577350269189626	1.000000000000000
3	1	-0.774596669241000	0.000000000000000
	2	0.000000000000000	0.000000000000000
	3	0.770000000000000	0.000000000000000
4	1	0.339981043584856	0.652145154862546
	2	-0.339981043584856	0.652145154862546
	3	0.861136311594053	0.347854845137454
	4	-0.861136311594053	0.347854845137454

Les éléments surfaciques#

Triangles : ELREFE TR3, TR6, TR7#

Coordonnées des nœuds:

	\(\xi\)	\(\eta\)
N1	0.0	0.0
N2	1.0	0.0
N3	0.0	1.0
N4	0.5	0.0
N5	0.5	0.5
N6	0.0	0.5
N7	1/3	1/3

Famille	Point	\(\xi\)	\(\eta\)	Poids

FPG1	1	1/3	1/3	1/2

FPG3	1	1/6	1/6	1/6
	2	2/3	1/6	1/6
	3	1/6	2/3	1/6

FPG4	1	1/5	1/5	25/(24*4)
	2	3/5	1/5	25/(24*4)
	3	1/5	3/5	25/(24*4)
	4	1/3	1/3	-27/(24*4)

FPG6	1	b	b	P2
	2	1 – 2b	b	P2
	3	b	1 – 2b	P2
	4	a	1 – 2a	P1
	5	a	a	P1
	6	1 – 2a	a	P1

COT3	1	1/2	1/2	1/6
	2	0	1/2	1/6
	3	1/2	0	1/6

Avec	P1 = 0.11169079483905,	P2 = 0.0549758718227661,
	A = 0.445948490915965,	b = 0.091576213509771

Famille	Point	\(\xi\)	\(\eta\)	Poids

FPG7	1	1/3	1/3	9/80
	2	A	A	P1
	3	1-2A	A	P1
	4	A	1-2A	P1
	5	B	B	P2
	6	1-2B	B	P2
	7	B	1-2B	P2

Avec	A = 0.470142064105115
	B = 0.101286507323456
	P1 = 0.066197076394253
	P2 = 0.062969590272413

Famille	Point	\(\xi\)	\(\eta\)	Poids

FPG12	1	A	A	P1
	2	1-2A	A	P1
	3	A	1-2A	P1
	4	B	B	P2
	5	1-2B	B	P2
	6	B	1-2B	P2
	7	C	D	P3
	8	D	C	P3
	9	1-C-D	C	P3
	10	1-C-D	D	P3
	11	C	1-C-D	P3
	12	D	1-C-D	P3

Avec	A = 0.063089014491502
	B = 0.249286745170910
	C = 0.310352451033785
	D = 0.053145049844816
	P1 = 0.025422453185103
	P2 = 0.058393137863189
	P3 = 0.041425537809187

TR3 : triangle à 3 nœuds

nombre de nœuds	: 3
nombre de nœuds sommets	: 3

fonctions de forme et dérivées premières du triangle à 3 nœuds:

\(\left\lbrace N\right\rbrace\)	\(\left\lbrace \partial N/\partial \xi \right\rbrace\)	\(\left\lbrace \partial N/\partial \eta \right\rbrace\)
\(1-\xi -\eta\)	\(-1\)	\(-1\)
\(\xi\)	\(1\)	\(0\)
\(\eta\)	\(0\)	\(1\)

TR6 : triangle à 6 nœuds

nombre de nœuds	: 6
nombre de nœuds sommets	: 3

fonctions de forme, dérivées premières du triangle à 6 nœuds:

\(\left\lbrace N\right\rbrace\)	\(\left\lbrace \partial N/\partial \xi \right\rbrace\)	\(\left\lbrace \partial N/\partial \eta \right\rbrace\)
\(-(1-\xi -\eta )(1-2(1-\xi -\eta ))\)	\(1-4(1-\xi -\eta )\)	\(1-4(1-\xi -\eta )\)
\(-\xi (1-2\xi )\)	\(-1+4\xi\)	\(0\)
\(-\eta (1-2\eta )\)	\(0\)	\(-1+4\eta\)
\(4\xi (1-\xi -\eta )\)	\(4(1-2\xi -\eta )\)	\(-4\xi\)
\(4\xi \eta\)	\(\mathrm{4\eta }\)	\(4\xi\)
\(4\eta (1-\xi -\eta )\)	\(-4\eta\)	\(4(1-\xi -2\eta )\)

dérivées secondes du triangle à 6 nœuds:

\(\left\lbrace {\partial}^{2}N/\partial {\xi}^{2}\right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial \xi \partial \eta \right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial {\eta}^{2}\right\rbrace\)
4	4	4
4	0	0
0	0	4
-8	-4	0
0	4	0
0	-4	-8

TR7 : triangle à 7 nœuds

nombre de nœuds	: 7
nombre de nœuds sommets	: 3

fonctions de forme du triangle à 7 nœuds:

\(\left\lbrace N\right\rbrace\)

\(1-3(\xi +\eta )+2({\xi}^{2}+{\eta}^{2})+7\xi \eta -3\xi \eta (\xi +\eta )\)

\(\xi (-1+2\xi +3\eta -3\eta (\xi +\eta ))\)

\(\eta (-1+2\xi +3\eta -3\xi (\xi +\eta ))\)

\(4\xi (1-\xi -4\eta +3\eta (\xi +\eta ))\)

\(4\xi \eta (-2+3(\xi +\eta ))\)

\(4\eta (1-4\xi -\eta +3\xi (\xi +\eta ))\)

\(27\xi \eta (1-\xi -\eta )\)

dérivées premières du triangle à 7 nœuds:

\(\left\lbrace \partial N/\partial \xi \right\rbrace\)	\(\left\lbrace \partial N/\partial \eta \right\rbrace\)
\(-3+4\xi +7\eta -6\xi \eta -3{\eta}^{2}\)	\(-3+7\xi +4\eta -6\xi \eta -3{\xi}^{2}\)
\(-1+4\xi +3\eta -6\xi \eta -3{\eta}^{2}\)	\(3\xi (1-\xi -2\eta )\)
\(3\xi (1-2\eta -\xi )\)	\(-1+3\xi +4\eta -6\xi \eta -3{\xi}^{2}\)
\(4(1-2\xi -4\eta +6\xi \eta +3{\eta}^{2})\)	\(4\xi (-4+3\xi +6\eta )\)
\(4\eta (-2+6\xi +3\eta )\)	\(4\xi (-2+3\xi +6\eta )\)
\(4\eta (-4+6\xi +3\eta )\)	\(4(-1-4\xi -2\eta +6\xi \eta +3{\xi}^{2})\)
\(27\eta (1-2\xi -\eta )\)	\(27\xi (1-\xi -2\eta )\)

dérivées secondes du triangle à 7 nœuds:

\(\left\lbrace {\partial}^{2}N/\partial {\xi}^{2}\right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial \xi \partial \eta \right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial {\eta}^{2}\right\rbrace\)
\(4-6\eta\)	\(7-6\xi -6\eta\)	\(4-6\xi\)
\(4-6\eta\)	\(3-6\xi -6\eta\)	\(-6\xi\)
\(-6\eta\)	\(3-6\xi -6\eta\)	\(4-6\xi\)
\(4(-2+6\eta )\)	\(4(-4+6\xi +6\eta )\)	\(24\xi\)
\(24\eta\)	\(4(-2+6\xi +6\eta )\)	\(24\xi\)
\(24\eta\)	\(4(-4+6\xi +6\eta )\)	\(4(-2+6\xi )\)
\(-54\eta\)	\(27(1-2\xi -2\eta )\)	\(-54\xi\)

Quadrangles : ELREFE QU4, QU8, QU9#

Coordonnées des nœuds:

	\(\xi\)	\(\eta\)
\(\mathrm{N1}\)	-1.0	-1.0
\(\mathrm{N2}\)	1.0	-1.0
\(\mathrm{N3}\)	1.0	1.0
\(\mathrm{N4}\)	-1.0	1.0
\(\mathrm{N5}\)	0.0	-1.0
\(\mathrm{N6}\)	1.0	0.0
\(\mathrm{N7}\)	0.0	1.0
\(\mathrm{N8}\)	-1.0	0.0
\(\mathrm{N9}\)	0.0	0.0

Famille	Point	\(\xi\)	\(\eta\)	Poids
FPG1	1	0	0	4
FPG4	1	\(-a\)	\(-a\)	1.0
	2	\(a\)	\(-a\)	1.0
	3	\(a\)	\(a\)	1.0
	4	\(-a\)	\(a\)	1.0
		\(a=1/\sqrt{3}\)
FPG9	1	\(-a\)	\(-a\)	25/81
	2	\(a\)	\(-a\)	25/81
	3	\(a\)	\(a\)	25/81
	4	\(-a\)	\(a\)	25/81
	5	0.0	\(-a\)	40/81
	6	\(a\)	0.0	40/81
	7	0.0	a	40/81
	8	\(-a\)	0.0	40/81
	9	0.0	0.0	64/81
		\(a=0.774596669241483\)

QU4 : quadrangle à 4 nœuds

nombre de nœuds	: 4
nombre de nœuds sommets	: 4

fonctions de forme, dérivées premières et secondes du quadrangle à 4 nœuds:

\(\left\lbrace N\right\rbrace\)	\(\left\lbrace \partial N/\partial \xi \right\rbrace\)	\(\left\lbrace \partial N/\partial \eta \right\rbrace\)
\((1-\xi )(1-\eta )/4\)	\(-(1-\eta )/4\)	\(-(1-\xi )/4\)
\((1+\xi )(1-\eta )/4\)	\((1-\eta )/4\)	\(-(1+\xi )/4\)
\((1+\xi )(1+\eta )/4\)	\((1+\eta )/4\)	\((1+\xi )/4\)
\((1-\xi )(1+\eta )/4\)	\(-(1+\eta )/4\)	\((1-\xi )/4\)

\(\left\lbrace {\partial}^{2}N/\partial {\xi}^{2}\right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial \xi \partial \eta \right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial {\eta}^{2}\right\rbrace\)
0	1/4	0
0	-1/4	0
0	1/4	0
0	-1/4	0

QU8 : quadrangle à 8 nœuds

nombre de nœuds	: 8
nombre de nœuds sommets	: 4

fonctions de forme et dérivées premières du quadrangle à 8 nœuds:

\(\left\lbrace N\right\rbrace\)	\(\left\lbrace \partial N/\partial \xi \right\rbrace\)	\(\left\lbrace \partial N/\partial \eta \right\rbrace\)
\((1-\xi )(1-\eta )(-1-\xi -\eta )/4\)	\((1-\eta )(2\xi +\eta )/4\)	\((1-\xi )(\xi +2\eta )/4\)
\((1+\xi )(1-\eta )(-1+\xi -\eta )/4\)	\((1-\eta )(2\xi -\eta )/4\)	\(-(1+\xi )(\xi -2\eta )/4\)
\((1+\xi )(1+\eta )(-1+\xi +\eta )/4\)	\((1+\eta )(2\xi +\eta )/4\)	\((1+\xi )(\xi +2\eta )/4\)
\((1-\xi )(1+\eta )(-1-\xi +\eta )/4\)	\(-(1+\eta )(-2\xi +\eta )/4\)	\((1-\xi )(-\xi +2\eta )/4\)
\((1-{\xi}^{2})(1-\eta )/2\)	\(-\xi (1-\eta )\)	\(-(1-{\xi}^{2})/2\)
\((1+\xi )(1-{\eta}^{2})/2\)	\((1-{\eta}^{2})/2\)	\(-\eta (1+\xi )\)
\((1-{\xi}^{2})(1+\eta )/2\)	\(-\xi (1+\eta )\)	\((1-{\xi}^{2})/2\)
\((1-\xi )(1-{\eta}^{2})/2\)	\(-(1-{\eta}^{2})/2\)	\(-\eta (1-\xi )\)

dérivées secondes du quadrangle à 8 nœuds:

\(\left\lbrace {\partial}^{2}N/\partial {\xi}^{2}\right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial \xi \partial \eta \right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial {\eta}^{2}\right\rbrace\)
\((1-\eta )/2\)	\((1-2\xi -2\eta )/4\)	\((1-\xi )/2\)
\((1-\eta )/2\)	\(-(1+2\xi -2\eta )/4\)	\((1+\xi )/2\)
\((1+\eta )/2\)	\((1+2\xi +2\eta )/4\)	\((1+\xi )/2\)
\((1+\eta )/2\)	\(-(1-2\xi +2\eta )/4\)	\((1-\xi )/2\)
\(-1+\eta\)	\(\xi\)	\(0\)
\(0\)	\(-\eta\)	\(-1-\xi\)
\(-1-\eta\)	\(-\xi\)	\(0\)
\(0\)	\(\eta\)	\(-1+\xi\)

QU9 : quadrangle à 9 nœuds

nombre de nœuds	: 9
nombre de nœuds sommets	: 4

fonctions de forme et dérivées premières du quadrangle à 9 nœuds:

\(\left\lbrace N\right\rbrace\)	\(\left\lbrace \partial N/\partial \xi \right\rbrace\)	\(\left\lbrace \partial N/\partial \eta \right\rbrace\)
\(\xi \eta (\xi -1)(\eta -1)/4\)	\((2\xi -1)\eta (\eta -1)/4\)	\(\xi (\xi -1)(2\eta -1)/4\)
\(\xi \eta (\xi +1)(\eta -1)/4\)	\((2\xi +1)\eta (\eta -1)/4\)	\(\xi (\xi +1)(2\eta -1)/4\)
\(\xi \eta (\xi +1)(\eta +1)/4\)	\((2\xi +1)\eta (\eta +1)/4\)	\(\xi (\xi +1)(2\eta +1)/4\)
\(\xi \eta (\xi -1)(\eta +1)/4\)	\((2\xi -1)\eta (\eta +1)/4\)	\(\xi (\xi -1)(2\eta +1)/4\)
\((1-{\xi}^{2})\eta (\eta -1)/2\)	\(-\xi \eta (\eta -1)\)	\((1-{\xi}^{2})(2\eta -1)/2\)
\(\xi (\xi +1)(1-{\eta}^{2})/2\)	\((2\xi +1)(1-{\eta}^{2})/2\)	\(-\xi \eta (\xi +1)\)
\((1-{\xi}^{2})\eta (\eta +1)/2\)	\(-\xi \eta (\eta +1)\)	\((1-{\xi}^{2})(2\eta +1)/2\)
\(\xi (\xi -1)(1-{\eta}^{2})/2\)	\((2\xi -1)(1-{\eta}^{2})/2\)	\(-\xi \eta (\xi -1)\)
\((1-{\xi}^{2})(1-{\eta}^{2})\)	\(-2\xi (1-{\eta}^{2})\)	\(-2\eta (1-{\xi}^{2})\)

dérivées secondes du quadrangle à 9 nœuds:

\(\left\lbrace {\partial}^{2}N/\partial {\xi}^{2}\right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial \xi \partial \eta \right\rbrace\)	\(\left\lbrace {\partial}^{2}N/\partial {\eta}^{2}\right\rbrace\)
\(\eta (\eta -1)/2\)	\((\xi -1/2)(\eta -1/2)\)	\(\xi (\xi -1)/2\)
\(\eta (\eta -1)/2\)	\((\xi +1/2)(\eta -1/2)\)	\(\xi (\xi +1)/2\)
\(\eta (\eta +1)/2\)	\((\xi +1/2)(\eta +1/2)\)	\(\xi (\xi +1)/2\)
\(\eta (\eta +1)/2\)	\((\xi -1/2)(\eta +1/2)\)	\(\xi (\xi -1)/2\)
\(-\eta (\eta -1)\)	\(-\xi (2\eta -1)\)	\(1-{\xi}^{2}\)
\(1-{\eta}^{2}\)	\(-\eta (2\xi +1)\)	\(-\xi (\xi +1)\)
\(-\eta (\eta +1)\)	\(-\xi (\mathrm{2\eta }+1)\)	\(1-{\xi}^{2}\)
\(1-{\eta}^{2}\)	\(-\eta (2\xi -1)\)	\(-\xi (\xi -1)\)
\(-2(1-{\eta}^{2})\)	\(4\xi \eta\)	\(-2(1-{\xi}^{2})\)

Les éléments volumiques#

Tétraèdres : ELREFE TE4, T10#

../../../../_images/100012360000182F0000170C3B52F571AC5ECFD5.svg

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

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

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:

\(a=0.25\)

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

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

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

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

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

Formule à 15 nœuds

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

Formule à 18 nœuds

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

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

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

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

Formule à 20 nœuds

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

Formule à 27 nœuds

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

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:

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

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

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

Pyramides : ELREFE PY5, P13,P19#

../../../../_images/10000201000003E8000003215D64B9F61CD54C22.png

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

Bibliographie#

[bib1]

DHATT G., TOUZOT G.: Une présentation de la méthode des éléments finis 2ème édition. Editeur: MALOINE S.A. Année 1984

[bib2]

Freddie Witherden, Peter Vincent: On the identification of symmetric quadrature rules for finite element methods, Computers and Mathematics with Applications, Volume 69, pages 1232-1241, 2015.

[bib3]

KUBATKO, YAEGER, MAGGI: New computationally efficient quadrature formulas for triangular prism elements. Computers & Fluids 73 (2013) 187-201