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

Résumé:

On décrit la géométrie et la topologie des éléments finis; 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

Fig. 128 Support géométrique linéique quadratique

Segments à deux noeuds SE2

nombre de nœuds	: 2
nombre de nœuds sommets	: 2

fonctions de forme du segment à 2 nœuds:

\[\begin{split}\begin{array}{l} {w}_{1}(x)=0.5(1-x)\\ {w}_{2}(x)=0.5(1+x) \end{array}\end{split}\]

Segments à trois noeuds SE3

nombre de nœuds	: 3
nombre de nœuds sommets	: 2

	\(x\)
N1	-1.0
N2	1.0
N3	0.0

fonctions de forme du segment à 3 nœuds:

\[\begin{split}\begin{array}{l} {w}_{1}(x)=-0.5(1-x)x\\ {w}_{2}(x)=0.5(1+x)x\\ {w}_{3}(x)=(1+x)(1-x) \end{array}\end{split}\]

Segments à quatre noeuds SE4

Fig. 129 Support géométrique linéique cubique

nombre de nœuds	: 4
nombre de nœuds sommets	: 2

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

fonctions de forme du segment à 4 nœuds:

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

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

Fig. 130 Support géométrique triangulaire quadratique

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

Familles d’intégration:

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 \(P1 = 0,11169079483905\), \(P2 = 0,0549758718227661\),:math:A = 0,445948490915965 et \(b = 0,091576213509771\)

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 \(A = 0.470142064105115\), \(B = 0.101286507323456\), \(P1 = 0.066197076394253\) et \(P2 = 0.062969590272413\)

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 \(A = 0.063089014491502\), \(B = 0.249286745170910\), \(C = 0.310352451033785\), \(D = 0.053145049844816\), \(P1 = 0.025422453185103\), \(P2 = 0.058393137863189\) et \(P3 = 0.041425537809187\).

Triangle à 3 nœuds TR3

nombre de nœuds	: 3
nombre de nœuds sommets	: 3

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

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

Triangle à 6 nœuds TR6

nombre de nœuds	: 6
nombre de nœuds sommets	: 3

Fonctions de forme et dérivées premières du triangle à 6 nœuds:

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

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

Triangle à 7 nœuds TR7

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:

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

Triangle à 10 nœuds TR1

Fig. 131 Support géométrique triangulaire cubique

nombre de nœuds	10
nombre de nœuds sommets	3

Coordonnées des nœuds:

	\(\xi\)	\(\eta\)
N1	0.0	0.0
N2	1.0	0.0
N3	0.0	1.0
N4	1/3	0.0
N5	2/3	0.0
N6	2/3	1/3
N7	1/3	2/3
N8	0.0	2/3
N9	0.0	1/3
N10	1/3	1/3

Fonctions de forme du triangle à 10 nœuds:

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

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

\(\xi(3\xi - 2)(3\xi - 1)/2\)

\(\eta(3\eta - 2)(3\eta - 1)/2\)

\(9\xi(\eta + \xi - 1)(3\eta + 3\xi - 2)/2\)

\(-9\xi(3\xi - 1)(\eta + \xi - 1)/2\)

\(9\eta\xi(3\xi - 1)/2\)

\(9\eta\xi(3\eta - 1)/2\)

\(-9\eta(3\eta - 1)(\eta + \xi - 1)/2\)

\(9\eta(\eta + \xi - 1)(3\eta + 3\xi - 2)/2\)

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

Dérivées premières du triangle à 10 nœuds:

\(\left\lbrace \partial N/\partial \xi \right\rbrace\)	\(\left\lbrace \partial N/\partial \eta \right\rbrace\)
\(-(27\eta^2 + 54\eta\xi - 36\eta + 27\xi^2 - 36\xi + 11)/2\)	\(-(27\eta^2 + 54\eta\xi - 36\eta + 27\xi^2 - 36\xi + 11)/2\)
\((27\xi^2 - 18\xi + 2)/2\)	\(0\)
\(0\)	\((27\eta^2 - 18\eta + 2)/2\)
\(9(3\eta^2 + 12\eta\xi - 5\eta + 9\xi^2 - 10\xi + 2)/2\)	\(9\xi(6\eta + 6\xi - 5)/2\)
\(-9(6\eta\xi - \eta + 9\xi^2 - 8\xi + 1)/2\)	\(-9\xi(3\xi - 1)/2\)
\(9\eta(6\xi - 1)/2\)	\(9\xi(3\xi - 1)/2\)
\(9\eta(3\eta - 1)/2\)	\(9\xi(6\eta - 1)/2\)
\(-9\eta(3\eta - 1)/2\)	\(-9(9\eta^2 + 6\eta\xi - 8\eta - \xi + 1)/2\)
\(9\eta(6\eta + 6\xi - 5)/2\)	\(9(9\eta^2 + 12\eta\xi - 10\eta + 3\xi^2 - 5\xi + 2)/2\)
\(-27\eta(\eta + 2\xi - 1)\)	\(-27\xi(2\eta + \xi - 1)\)

Quadrangles

Fig. 132 Support géométrique quadrangulaire quadratique

Coordonnées des nœuds:

	\(\xi\)	\(\eta\)
N1	-1.0	-1.0
N2	1.0	-1.0
N3	1.0	1.0
N4	-1.0	1.0
N5	0.0	-1.0
N6	1.0	0.0
N7	0.0	1.0
N8	-1.0	0.0
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
FPG9	1	\(-b\)	\(-b\)	25/81
	2	\(b\)	\(-b\)	25/81
	3	\(b\)	\(b\)	25/81
	4	\(-b\)	\(b\)	25/81
	5	0.0	\(-b\)	40/81
	6	\(b\)	0.0	40/81
	7	0.0	b	40/81
	8	\(-b\)	0.0	40/81
	9	0.0	0.0	64/81

avec \(a=1/\sqrt{3}\) et \(b=0.774596669241483\)

Quadrangle à 4 nœuds QU4

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

Quadrangle à 8 nœuds QU8

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:

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

Quadrangle à 9 nœuds QU9

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:

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

Quadrangle à 12 nœuds Q12

Fig. 133 Support géométrique quadrangulaire cubique

nombre de nœuds	12
nombre de nœuds sommets	4

Coordonnées des nœuds:

	\(\xi\)	\(\eta\)
N1	-1.0	-1.0
N2	1.0	-1.0
N3	1.0	1.0
N4	-1.0	1.0
N5	-1/3	-1.0
N6	1/3	-1.0
N7	1.0	-1/3
N8	1.0	1/3
N9	1/3	1.0
N10	-1/3	1.0
N11	-1.0	1/3
N12	-1.0	-1/3

fonctions de forme du quadrangle à 12 nœuds:

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

\((\eta - 1)(\xi - 1)(9\eta^2 + 9\xi^2 - 10)/32\)

\(-(\eta - 1)(\xi + 1)(9\eta^2 + 9\xi^2 - 10)/32\)

\((\eta + 1)(\xi + 1)(9\eta^2 + 9\xi^2 - 10)/32\)

\(-(\eta + 1)(\xi - 1)(9\eta^2 + 9\xi^2 - 10)/32\)

\(-9(\eta - 1)(\xi - 1)(\xi + 1)(3\xi - 1)/32\)

\(9(\eta - 1)(\xi - 1)(\xi + 1)(3\xi + 1)/32\)

\(9(\eta - 1)(\eta + 1)(3\eta - 1)(\xi + 1)/32\)

\(-9(\eta - 1)(\eta + 1)(3\eta + 1)(\xi + 1)/32\)

\(-9(\eta + 1)(\xi - 1)(\xi + 1)(3\xi + 1)/32\)

\(9(\eta + 1)(\xi - 1)(\xi + 1)(3\xi - 1)/32\)

\(9(\eta - 1)(\eta + 1)(3\eta + 1)(\xi - 1)/32\)

\(-9(\eta - 1)(\eta + 1)(3\eta - 1)(\xi - 1)/32\)

dérivées premières du quadrangle à 12 nœuds:

\(\left\lbrace \partial N/\partial \xi \right\rbrace\)	\(\left\lbrace \partial N/\partial \eta \right\rbrace\)
\((\eta - 1)(9\eta^2 + 27\xi^2 - 18\xi - 10)/32\)	\((\xi - 1)(27\eta^2 - 18\eta + 9\xi^2 - 10)/32\)
\(-(\eta - 1)(9\eta^2 + 27\xi^2 + 18\xi - 10)/32\)	\(-(\xi + 1)(27\eta^2 - 18\eta + 9\xi^2 - 10)/32\)
\((\eta + 1)(9\eta^2 + 27\xi^2 + 18\xi - 10)/32\)	\((\xi + 1)(27\eta^2 + 18\eta + 9\xi^2 - 10)/32\)
\(-(\eta + 1)(9\eta^2 + 27\xi^2 - 18\xi - 10)/32\)	\(-(\xi - 1)(27\eta^2 + 18\eta + 9\xi^2 - 10)/32\)
\(-9(\eta - 1)(9\xi^2 - 2\xi - 3)/32\)	\(-9(\xi - 1)(\xi + 1)(3\xi - 1)/32\)
\(9(\eta - 1)(9\xi^2 + 2\xi - 3)/32\)	\(9(\xi - 1)(\xi + 1)(3\xi + 1)/32\)
\(9(\eta - 1)(\eta + 1)(3\eta - 1)/32\)	\(9(\xi + 1)(9\eta^2 - 2\eta - 3)/32\)
\(-9(\eta - 1)(\eta + 1)(3\eta + 1)/32\)	\(-9(\xi + 1)(9\eta^2 + 2\eta - 3)/32\)
\(-9(\eta + 1)(9\xi^2 + 2\xi - 3)/32\)	\(-9(\xi - 1)(\xi + 1)(3\xi + 1)/32\)
\(9(\eta + 1)(9\xi^2 - 2\xi - 3)/32\)	\(9(\xi - 1)(\xi + 1)(3\xi - 1)/32\)
\(9(\eta - 1)(\eta + 1)(3\eta + 1)/32\)	\(9(\xi - 1)(9\eta^2 + 2\eta - 3)/32\)
\(-9(\eta - 1)(\eta + 1)(3\eta - 1)/32\)	\(-9(\xi - 1)(9\eta^2 - 2\eta - 3)/32\)

Les éléments volumiques

Les tétraèdres linéaires et quadratiques

Fig. 134 Support géométrique tétraédrique quadratique

Coordonnées des nœuds:

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

Fonctions de forme pour 4 nœuds

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

Fonctions de forme pour 10 nœuds

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

Formule d’intégration à 1 point, du premier ordre en \(x,y,z\) : FPG1

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

Formule d’intégration à 4 points, du deuxième ordre 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 d’intégration à 5 points, du troisième ordre 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 d’intégration à 15 points, du cinquième ordre 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 \(a=0.25\), \({b}_{1}=\frac{7+\sqrt{15}}{34}\), \({b}_{2}=\frac{7-\sqrt{15}}{34}\), \({c}_{1}=\frac{13-3\sqrt{15}}{34}\), \({c}_{2}=\frac{13+3\sqrt{15}}{34}\), \(d=\frac{5-\sqrt{15}}{20}\) et \(e=\frac{5+\sqrt{15}}{20}\)

Les tétraèdres cubiques

Fig. 135 Support géométrique tétraédrique cubique

nombre de nœuds	20
nombre de nœuds sommets	4

Coordonnées des nœuds:

	\(\xi\)	\(\eta\)	\(\zeta\)
N1	0.0	1.0	0.0
N2	0.0	0.0	1.0
N3	0.0	0.0	0.0
N4	1.0	0.0	0.0
N5	0.0	2/3	1/3
N6	0.0	1/3	2/3
N7	0.0	0.0	2/3
N8	0.0	0.0	1/3
N9	0.0	2/3	0.0
N10	0.0	1/3	0.0
N11	1/3	2/3	0.0
N12	2/3	1/3	0.0
N13	1/3	0.0	2/3
N14	2/3	0.0	1/3
N15	1/3	0.0	0.0
N16	2/3	0.0	0.0
N17	0.0	1/3	1/3
N18	1/3	1/3	1/3
N19	1/3	1/3	0.0
N20	1/3	0.0	1/3

fonctions de forme du tétraèdres à 20 nœuds:

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

\(\eta(3\eta - 2)(3\eta - 1)/2\)

\(\zeta(3\zeta - 2)(3\zeta - 1)/2\)

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

\(\xi(3\xi - 2)(3\xi - 1)/2\)

\(9\eta\zeta(3\eta - 1)/2\)

\(9\eta\zeta(3\zeta - 1)/2\)

\(-9\zeta(3\zeta - 1)(\eta + \xi + \zeta - 1)/2\)

\(9\zeta(\eta + \xi + \zeta - 1)(3\eta + 3\xi + 3\zeta - 2)/2\)

\(-9\eta(3\eta - 1)(\eta + \xi + \zeta - 1)/2\)

\(9\eta(\eta + \xi + \zeta - 1)(3\eta + 3\xi + 3\zeta - 2)/2\)

\(9\eta\xi(3\eta - 1)/2\)

\(9\eta\xi(3\xi - 1)/2\)

\(9\xi\zeta(3\zeta - 1)/2\)

\(9\xi\zeta(3\xi - 1)/2\)

\(9\xi(\eta + \xi + \zeta - 1)(3\eta + 3\xi + 3\zeta - 2)/2\)

\(-9\xi(3\xi - 1)(\eta + \xi + \zeta - 1)/2\)

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

\(27\eta\xi\zeta\)

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

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

Les pentaèdres

Fig. 136 Support géométrique pentaédrique

Coordonnées des nœuds:

	\(x\)	\(y\)	\(z\)
N1	-1.
N2	-1.
N3	-1.
N4
N5
N6
N7	-1.	0.5	0.5.
N8	-1.		0.5.
N9	-1.	0.5
N10
N11
N12
N13		0.5	0.5
N14			0.5
N15		0.5
N16		0.5	0.5
N17			0.5
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

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

Fonctions de forme, formule à 15 nœuds

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

Fonctions de forme, formule à 18 nœuds

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

Formules d’intégration numérique à 6 points (ordre 3 en \(x\) , ordre 2 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 \(z\) (ordre 3).

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 \(z\) (ordre 5 en \(y\) et \(z\) ).

Point	\(x\)	\(y\)	\(z\)	Poids
1	\(-\alpha\)	\(1/3\)	\(1/3\)	\({c}_{1}\frac{9}{80}\)
2;3;4	\(-\alpha\) ; \(-\alpha\) ; \(-\alpha\)	\(a\) ; \(1-2a\) ; \(a\)	\(a\) ; \(a\) ; \(1-2a\)	\({c}_{1}(\frac{155+\sqrt{15}}{2400})\)
5;6;7	\(-\alpha\) ; \(-\alpha\) ; \(-\alpha\)	\(b\) ; \(1-2b\) ; \(b\)	\(b\) ; \(b\) ; \(1-2b\)	\({c}_{1}(\frac{155-\sqrt{15}}{2400})\)
8		\(1/3\)	\(1/3\)	\({c}_{2}\frac{9}{80}\)
9;10;11	0.;0.;0.	\(a\) ; \(1-2a\) ; \(a\)	\(a\) ; \(a\) ; \(1-2a\)	\({c}_{2}(\frac{155+\sqrt{15}}{2400})\)
12;13;14	0.;0.;0.	\(b\) ; \(1-2b\) ; \(b\)	\(b\) ; \(b\) ; \(1-2b\)	\({c}_{2}(\frac{155-\sqrt{15}}{2400})\)
15	\(\alpha\)	\(1/3\)	\(1/3\)	\({c}_{1}\frac{9}{80}\)
16;17;18	\(\alpha\) ; \(\alpha\) ; \(\alpha\)	\(b\) ; \(1-2a\) ; \(a\)	\(a\) ; \(a\) ; \(1-2a\)	\({c}_{1}(\frac{155+\sqrt{15}}{2400})\)
19;20;21	\(\alpha\) ; \(\alpha\) ; \(\alpha\)	\(b\) ; \(1-2b\) ; \(b\)	\(b\) ; \(b\) ; \(1-2b\)	\({c}_{1}(\frac{155-\sqrt{15}}{2400})\)

avec \(\alpha =\sqrt{\frac{3}{5}}\), \({c}_{1}=\frac{5}{9}\), \({c}_{2}=\frac{8}{9}\), \(a=\frac{6+\sqrt{15}}{21}\) et \(b=\frac{6-\sqrt{15}}{21}\).

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

Les hexaèdres linéaires et quadratiques

Fig. 137 Support géométrique hexaédrique linéaire

Fig. 138 Support géométrique hexaédrique quadratique

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

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

Fonctions de forme, formule à 20 nœuds

\[\begin{split}\begin{array}{l} {w}_{1}=\frac{1}{8}(1-x)(1-y)(1-z)(-2-x-y-z)\\ {w}_{2}=\frac{1}{8}(1+x)(1-y)(1-z)(-2+x-y-z)\\ {w}_{3}=\frac{1}{8}(1+x)(1+y)(1-z)(-2+x+y-z)\\ {w}_{4}=\frac{1}{8}(1-x)(1+y)(1-z)(-2-x+y-z)\\ {w}_{5}=\frac{1}{8}(1-x)(1-y)(1+z)(-2-x-y+z)\\ {w}_{6}=\frac{1}{8}(1+x)(1-y)(1+z)(-2+x-y+z)\\ {w}_{7}=\frac{1}{8}(1+x)(1+y)(1+z)(-2+x+y+z)\\ {w}_{8}=\frac{1}{8}(1-x)(1+y)(1+z)(-2-x+y+z)\\ {w}_{9}=\frac{1}{4}(1-{x}^{2})(1-y)(1-z)\\ {w}_{10}=\frac{1}{4}(1-{y}^{2})(1+x)(1-z)\\ {w}_{11}=\frac{1}{4}(1-{x}^{2})(1+y)(1-z)\\ {w}_{12}=\frac{1}{4}(1-{y}^{2})(1-x)(1-z)\\ {w}_{13}=\frac{1}{4}(1-{z}^{2})(1-y)(1-x)\\ {w}_{14}=\frac{1}{4}(1-{z}^{2})(1-y)(1+x)\\ {w}_{15}=\frac{1}{4}(1-{z}^{2})(1+y)(1+x)\\ {w}_{16}=\frac{1}{4}(1-{z}^{2})(1-x)(1+y)\\ {w}_{17}=\frac{1}{4}(1-{x}^{2})(1+z)(1-y)\\ {w}_{18}=\frac{1}{4}(1-{y}^{2})(1+z)(1+x)\\ {w}_{19}=\frac{1}{4}(1-{x}^{2})(1+z)(1+y)\\ {w}_{20}=\frac{1}{4}(1-{y}^{2})(1+z)(1-x)\\ \end{array}\end{split}\]

Fonctions de forme, formule à 27 nœuds

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

Formule de quadrature de Gauss à 2 points dans chaque direction (ordre 3): 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 3 points dans chaque direction (ordre 5): 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 \(\alpha =\sqrt{\frac{3}{5}}\), \({c}_{1}=\frac{5}{9}\) et \({c}_{2}=\frac{8}{9}\).

Les pyramides linéaires et quadratiques

Fig. 139 Support géométrique pyramide quadratique

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.

	\(x\)	\(y\)	\(z\)
N1
N2
N3	–1.
N4		–1.
N5
N6	0.5	0.5
N7	–0.5	0.5
N8	–0.5	–0.5
N9	0.5	–0.5
N10	0.5		0.5
N11		0.5	0.5
N12	–0.5		0.5
N13		–0.5	0.5
N14			0
N15	1/3	1/3	1/3
N16	-1/3	1/3	1/3
N17	-1/3	-1/3	1/3
N18	1/3	-1/3	1/3
N19	0	0	0.2

Fonctions de forme, formule à 5 nœuds

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

Formule à 13 nœuds

\[\begin{split}\begin{array}{l} {w}_{1}=\frac{(-x+y+z-1)(-x-y+z-1)(x-0.5)}{2(1-z)}\\ {w}_{2}=\frac{(-x-y+z-1)(x-y+z-1)(y-0.5)}{2(1-z)}\\ {w}_{3}=\frac{(x-y+z-1)(x+y+z-1)(-x-0.5)}{2(1-z)}\\ {w}_{4}=\frac{(x+y+z-1)(-x+y+z-1)(-y-0.5)}{2(1-z)}\\ {w}_{5}={2z}(z-0.5)\\ {w}_{6}=-\frac{(-x+y+z-1)(-x-y+z-1)(x-y+z-1)}{2(1-z)}\\ {w}_{7}=-\frac{(-x-y+z-1)(x-y+z-1)(x+y+z-1)}{2(1-z)} {w}_{8}=-\frac{(x-y+z-1)(x+y+z-1)(-x+y+z-1)}{2(1-z)}\\ {w}_{9}=-\frac{(x+y+z-1)(-x+y+z-1)(-x-y+z-1)}{2(1-z)}\\ {w}_{10}=\frac{z(-x+y+z-1)(-x-y+z-1)}{1-z}\\ {w}_{11}=\frac{z(-x-y+z-1)(x-y+z-1)}{1-z}\\ {w}_{12}=\frac{z(x-y+z-1)(x+y+z-1)}{1-z}\\ {w}_{13}=\frac{z(x+y+z-1)(-x+y+z-1)}{1-z} \end{array}\end{split}\]

Formule d’intégration numérique à 5 points du deuxième ordre (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 \({h}_{1}=0.1531754163448146\), \({h}_{2}=0.6372983346207416\) et \({p}_{1}=\frac{2}{15}\).

Formule d’intégration numérique à 6 points du troisième ordre (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 \(a=0.5610836110587396\), \({p}_{1}=0.1681372559485071\), \({p}_{2}=0.07500000404404333\), \({p}_{3}=0.1058823516685291\), \({h}_{1}=0.1681372559485071\), \({h}_{2}=0.00000000567585\) et \({h}_{3}=0.1058823516685291\).

Formule d’intégration numérique à 10 points (FPG10) d’ordre 4, voir [bib2] :

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 \(a=0.3252907781991163\), \(b=0.65796699712169\), \({h}_{1}=0.6772327888861374\), \({h}_{2}=0.1251369531087465\), \({h}_{3}=0.3223841495782137\), \({h}_{4}=0.0392482838988154\), \({w}_{1}=0.07582792211376127\), \({w}_{2}=0.1379222683930349\), \({w}_{3}=0.07088305859288367\) et \({w}_{4}=0.04234606044708394\).

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

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