3. Les éléments surfaciques#
3.1. 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:
\(\left\lbrace {\partial}^{2}N/\partial {\xi}^{2}\right\rbrace\) |
\(\left\lbrace {\partial}^{2}N/\partial \xi \partial \eta \right\rbrace\) |
\(\left\lbrace {\partial}^{2}N/\partial {\eta}^{2}\right\rbrace\) |
\(4-6\eta\) |
\(7-6\xi -6\eta\) |
\(4-6\xi\) |
\(4-6\eta\) |
\(3-6\xi -6\eta\) |
\(-6\xi\) |
\(-6\eta\) |
\(3-6\xi -6\eta\) |
\(4-6\xi\) |
\(4(-2+6\eta )\) |
\(4(-4+6\xi +6\eta )\) |
\(24\xi\) |
\(24\eta\) |
\(4(-2+6\xi +6\eta )\) |
\(24\xi\) |
\(24\eta\) |
\(4(-4+6\xi +6\eta )\) |
\(4(-2+6\xi )\) |
\(-54\eta\) |
\(27(1-2\xi -2\eta )\) |
\(-54\xi\) |
3.2. 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:
\(\left\lbrace {\partial}^{2}N/\partial {\xi}^{2}\right\rbrace\) |
\(\left\lbrace {\partial}^{2}N/\partial \xi \partial \eta \right\rbrace\) |
\(\left\lbrace {\partial}^{2}N/\partial {\eta}^{2}\right\rbrace\) |
\(\eta (\eta -1)/2\) |
\((\xi -1/2)(\eta -1/2)\) |
\(\xi (\xi -1)/2\) |
\(\eta (\eta -1)/2\) |
\((\xi +1/2)(\eta -1/2)\) |
\(\xi (\xi +1)/2\) |
\(\eta (\eta +1)/2\) |
\((\xi +1/2)(\eta +1/2)\) |
\(\xi (\xi +1)/2\) |
\(\eta (\eta +1)/2\) |
\((\xi -1/2)(\eta +1/2)\) |
\(\xi (\xi -1)/2\) |
\(-\eta (\eta -1)\) |
\(-\xi (2\eta -1)\) |
\(1-{\xi}^{2}\) |
\(1-{\eta}^{2}\) |
\(-\eta (2\xi +1)\) |
\(-\xi (\xi +1)\) |
\(-\eta (\eta +1)\) |
\(-\xi (\mathrm{2\eta }+1)\) |
\(1-{\xi}^{2}\) |
\(1-{\eta}^{2}\) |
\(-\eta (2\xi -1)\) |
\(-\xi (\xi -1)\) |
\(-2(1-{\eta}^{2})\) |
\(4\xi \eta\) |
\(-2(1-{\xi}^{2})\) |