3. Surface elements#

3.1. Triangles: ELREFE TR3, TR6, TR7#

_images/Object_5.svg

Node coordinates:

\(\xi\)

\(\eta\)

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

Family

Point

\(\xi\)

\(\eta\)

Weight

FPG1

1111

1/3

1/3

1/2

FPG3

1111

1/6

1/6

1/6

2222

2/3

1/6

1/6

3333

1/6

2/3

1/6

 
FPG4

1111

1/5

1/5

25/ (24*4)

2222

3/5

1/5

25/ (24*4)

3333

1/5

3/5

25/ (24*4)

4444

1/3

1/3

-27/ (24*4)

FPG6

1111

bbbb

bbbb

P2

2222

1 — 2b

bbbb

P2

3333

bbbb

1 — 2b

P2

4444

aaaa

1 — 2a

P1

5555

aaaa

aaaa

P1

6666

1 — 2a

aaaa

P1

 
COT3

1111

1/2

1/2

1/6

2222

0000

1/2

1/6

3333

1/2

0000

1/6

With

P1 = 0.11169079483905,

P2 = 0.0549758718227661,

A = 0.445948490915965,

b = 0.091576213509771

Family

Point

\(\xi\)

\(\eta\)

Weight

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

With

A = 0.470142064105115

B = 0.101286507323456

P1=0.066197076394253

P2=0.062969590272413

Family

Point

\(\xi\)

\(\eta\)

Weight

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

TR3: triangle with 3 knots

number of nodes

: 3

number of vertex nodes

: 3

shape functions and first derivatives of the triangle with 3 nodes:

\(\left\{N\right\}\)

\(\left\{\partial N/\partial \xi \right\}\)

\(\left\{\partial N/\partial \eta \right\}\)

\(1-\xi -\eta\)

\(-1\)

\(-1\)

\(\xi\)

\(1\)

\(0\)

\(\eta\)

\(0\)

\(1\)

TR6: triangle with 6 knots

number of nodes

: 6

number of vertex nodes

: 3

shape functions, first derivatives of the 6-node triangle:

\(\left\{N\right\}\)

\(\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\xi \right\}\)

\(\left\{\partial N/\partial \eta \right\}\)

\(\mathrm{-}(1\mathrm{-}\xi \mathrm{-}\eta )(1\mathrm{-}2(1\mathrm{-}\xi \mathrm{-}\eta ))\)

\(1\mathrm{-}4(1\mathrm{-}\xi \mathrm{-}\eta )\)

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

\(-\xi (1-2\xi )\)

\(-1+4\xi\)

\(0\)

\(-\eta (1-2\eta )\)

\(0\)

\(-1+4\eta\)

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

\(4(1\mathrm{-}2\xi \mathrm{-}\eta )\)

\(-4\xi\)

\(4\xi \eta\)

\(\mathrm{4\eta }\)

\(4\xi\)

\(4\eta (1\mathrm{-}\xi \mathrm{-}\eta )\)

\(\mathrm{-}4\eta\)

\(4(1\mathrm{-}\xi \mathrm{-}2\eta )\)

second derivatives of the 6-node triangle:

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\xi }^{2}\right\}\)

\(\left\{{\partial }^{2}N/\partial \xi \partial \eta \right\}\)

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}\)

4

4

4

4

0

0

0

0

4

-8

-4

0

0

4

0

0

-4

-8

TR7: triangle with 7 knots

number of nodes

: 7

number of vertex nodes

: 3

7-node triangle shape functions:

\(\left\{N\right\}\)

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

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

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

\(4\xi (1\mathrm{-}\xi \mathrm{-}4\eta +3\eta (\xi +\eta ))\)

\(4\xi \eta (\mathrm{-}2+3(\xi +\eta ))\)

\(4\eta (1\mathrm{-}4\xi \mathrm{-}\eta +3\xi (\xi +\eta ))\)

\(27\xi \eta (1\mathrm{-}\xi \mathrm{-}\eta )\)

first derivatives of the 7-node triangle:

\(\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\xi \right\}\)

\(\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\eta \right\}\)

\(\mathrm{-}3+4\xi +7\eta \mathrm{-}6\xi \eta \mathrm{-}3{\eta }^{2}\)

\(\mathrm{-}3+7\xi +4\eta \mathrm{-}6\xi \eta \mathrm{-}3{\xi }^{2}\)

\(\mathrm{-}1+4\xi +3\eta \mathrm{-}6\xi \eta \mathrm{-}3{\eta }^{2}\)

\(3\xi (1\mathrm{-}\xi \mathrm{-}2\eta )\)

\(3\xi (1\mathrm{-}2\eta \mathrm{-}\xi )\)

\(\mathrm{-}1+3\xi +4\eta \mathrm{-}6\xi \eta \mathrm{-}3{\xi }^{2}\)

\(4(1\mathrm{-}2\xi \mathrm{-}4\eta +6\xi \eta +3{\eta }^{2})\)

\(4\xi (\mathrm{-}4+3\xi +6\eta )\)

\(4\eta (\mathrm{-}2+6\xi +3\eta )\)

\(4\xi (\mathrm{-}2+3\xi +6\eta )\)

\(4\eta (\mathrm{-}4+6\xi +3\eta )\)

\(4(\mathrm{-}1\mathrm{-}4\xi \mathrm{-}2\eta +6\xi \eta +3{\xi }^{2})\)

\(27\eta (1\mathrm{-}2\xi \mathrm{-}\eta )\)

\(27\xi (1\mathrm{-}\xi \mathrm{-}2\eta )\)

second derivatives of the 7-node triangle:

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\xi }^{2}\right\}\)

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }\xi \mathrm{\partial }\eta \right\}\)

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}\)

\(4\mathrm{-}6\eta\)

\(7\mathrm{-}6\xi \mathrm{-}6\eta\)

\(4\mathrm{-}6\xi\)

\(4\mathrm{-}6\eta\)

\(3\mathrm{-}6\xi \mathrm{-}6\eta\)

\(\mathrm{-}6\xi\)

\(\mathrm{-}6\eta\)

\(3\mathrm{-}6\xi \mathrm{-}6\eta\)

\(4\mathrm{-}6\xi\)

\(4(\mathrm{-}2+6\eta )\)

\(4(\mathrm{-}4+6\xi +6\eta )\)

\(24\xi\)

\(\text{24}\eta\)

\(4(\mathrm{-}2+6\xi +6\eta )\)

\(24\xi\)

\(24\eta\)

\(4(\mathrm{-}4+6\xi +6\eta )\)

\(4(\mathrm{-}2+6\xi )\)

\(\mathrm{-}54\eta\)

\(27(1\mathrm{-}2\xi \mathrm{-}2\eta )\)

\(\mathrm{-}54\xi\)

3.2. Quadrangles: ELREFE QU4, QU8, QU9#

_images/Object_98.svg

Node coordinates:

\(\xi\)

\(\eta\)

\(\mathrm{N1}\)

-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

QU4: quadrangle with 4 knots

number of knots

: 4

number of vertex nodes

: 4

shape functions, first and second derivatives of the 4-node quadrangle:

\(\left\{N\right\}\)

\(\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\xi \right\}\)

\(\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\eta \right\}\)

\((1\mathrm{-}\xi )(1\mathrm{-}\eta )\mathrm{/}4\)

\(\mathrm{-}(1\mathrm{-}\eta )\mathrm{/}4\)

\(\mathrm{-}(1\mathrm{-}\xi )\mathrm{/}4\)

\((1+\xi )(1\mathrm{-}\eta )\mathrm{/}4\)

\((1\mathrm{-}\eta )\mathrm{/}4\)

\(\mathrm{-}(1+\xi )\mathrm{/}4\)

\((1+\xi )(1+\eta )\mathrm{/}4\)

\((1+\eta )\mathrm{/}4\)

\((1+\xi )\mathrm{/}4\)

\((1\mathrm{-}\xi )(1+\eta )\mathrm{/}4\)

\(\mathrm{-}(1+\eta )\mathrm{/}4\)

\((1-\xi )/4\)

\(\left\{{\partial }^{2}N/\partial {\xi }^{2}\right\}\)

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }\xi \mathrm{\partial }\eta \right\}\)

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}\)

0

1/4

0

0

-1/4

0

0

1/4

0

0

-1/4

0

QU8: quadrangle with 8 knots

number of knots

: 8

number of vertex nodes

: 4

shape functions and first derivatives of the 8-node quadrangle:

\(\left\{N\right\}\)

\(\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\xi \right\}\)

\(\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\eta \right\}\)

\((1\mathrm{-}\xi )(1\mathrm{-}\eta )(\mathrm{-}1\mathrm{-}\xi \mathrm{-}\eta )\mathrm{/}4\)

\((1\mathrm{-}\eta )(2\xi +\eta )\mathrm{/}4\)

\((1\mathrm{-}\xi )(\xi +2\eta )\mathrm{/}4\)

\((1+\xi )(1\mathrm{-}\eta )(\mathrm{-}1+\xi \mathrm{-}\eta )\mathrm{/}4\)

\((1\mathrm{-}\eta )(2\xi \mathrm{-}\eta )\mathrm{/}4\)

\(\mathrm{-}(1+\xi )(\xi \mathrm{-}2\eta )\mathrm{/}4\)

\((1+\xi )(1+\eta )(\mathrm{-}1+\xi +\eta )\mathrm{/}4\)

\((1+\eta )(2\xi +\eta )\mathrm{/}4\)

\((1+\xi )(\xi +2\eta )\mathrm{/}4\)

\((1\mathrm{-}\xi )(1+\eta )(\mathrm{-}1\mathrm{-}\xi +\eta )\mathrm{/}4\)

\(\mathrm{-}(1+\eta )(\mathrm{-}2\xi +\eta )\mathrm{/}4\)

\((1\mathrm{-}\xi )(\mathrm{-}\xi +2\eta )\mathrm{/}4\)

\((1\mathrm{-}{\xi }^{2})(1\mathrm{-}\eta )\mathrm{/}2\)

\(\mathrm{-}\xi (1\mathrm{-}\eta )\)

\(\mathrm{-}(1\mathrm{-}{\xi }^{2})\mathrm{/}2\)

\((1+\xi )(1\mathrm{-}{\eta }^{2})\mathrm{/}2\)

\((1\mathrm{-}{\eta }^{2})\mathrm{/}2\)

\(\mathrm{-}\eta (1+\xi )\)

\((1\mathrm{-}{\xi }^{2})(1+\eta )\mathrm{/}2\)

\(\mathrm{-}\xi (1+\eta )\)

\((1\mathrm{-}{\xi }^{2})\mathrm{/}2\)

\((1\mathrm{-}\xi )(1\mathrm{-}{\eta }^{2})\mathrm{/}2\)

\(\mathrm{-}(1\mathrm{-}{\eta }^{2})\mathrm{/}2\)

\(\mathrm{-}\eta (1\mathrm{-}\xi )\)

second derivatives of the 8-node quadrangle:

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\xi }^{2}\right\}\)

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }\xi \mathrm{\partial }\eta \right\}\)

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}\)

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

\((1-2\xi -2\eta )/4\)

\((1\mathrm{-}\xi )\mathrm{/}2\)

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

\(-(1+2\xi -2\eta )/4\)

\((1+\xi )\mathrm{/}2\)

\((1+\eta )/2\)

\((1+2\xi +2\eta )/4\)

\((1+\xi )\mathrm{/}2\)

\((1+\eta )/2\)

\(-(1-2\xi +2\eta )/4\)

\((1\mathrm{-}\xi )\mathrm{/}2\)

\(-1+\eta\)

\(\xi\)

\(0\)

\(0\)

\(-\eta\)

\(\mathrm{-}1\mathrm{-}\xi\)

\(-1-\eta\)

\(-\xi\)

\(0\)

\(0\)

\(\eta\)

\(\mathrm{-}1+\xi\)

QU9: quadrangle with 9 knots

number of nodes

: 9

number of vertex nodes

: 4

shape functions and first derivatives of the 9-node quadrangle:

\(\left\{N\right\}\)

\(\left\{\partial N/\partial \xi \right\}\)

\(\left\{\mathrm{\partial }N\mathrm{/}\mathrm{\partial }\eta \right\}\)

\(\xi \eta (\xi \mathrm{-}1)(\eta \mathrm{-}1)\mathrm{/}4\)

\((2\xi \mathrm{-}1)\eta (\eta \mathrm{-}1)\mathrm{/}4\)

\(\xi (\xi \mathrm{-}1)(2\eta \mathrm{-}1)\mathrm{/}4\)

\(\xi \eta (\xi +1)(\eta \mathrm{-}1)\mathrm{/}4\)

\((2\xi +1)\eta (\eta \mathrm{-}1)\mathrm{/}4\)

\(\xi (\xi +1)(2\eta \mathrm{-}1)\mathrm{/}4\)

\(\xi \eta (\xi +1)(\eta +1)\mathrm{/}4\)

\((2\xi +1)\eta (\eta +1)\mathrm{/}4\)

\(\xi (\xi +1)(2\eta +1)\mathrm{/}4\)

\(\xi \eta (\xi \mathrm{-}1)(\eta +1)\mathrm{/}4\)

\((2\xi \mathrm{-}1)\eta (\eta +1)\mathrm{/}4\)

\(\xi (\xi \mathrm{-}1)(2\eta +1)\mathrm{/}4\)

\((1\mathrm{-}{\xi }^{2})\eta (\eta \mathrm{-}1)\mathrm{/}2\)

\(\mathrm{-}\xi \eta (\eta \mathrm{-}1)\)

\((1\mathrm{-}{\xi }^{2})(2\eta \mathrm{-}1)\mathrm{/}2\)

\(\xi (\xi +1)(1\mathrm{-}{\eta }^{2})\mathrm{/}2\)

\((2\xi +1)(1\mathrm{-}{\eta }^{2})\mathrm{/}2\)

\(\mathrm{-}\xi \eta (\xi +1)\)

\((1\mathrm{-}{\xi }^{2})\eta (\eta +1)\mathrm{/}2\)

\(\mathrm{-}\xi \eta (\eta +1)\)

\((1\mathrm{-}{\xi }^{2})(2\eta +1)\mathrm{/}2\)

\(\xi (\xi \mathrm{-}1)(1\mathrm{-}{\eta }^{2})\mathrm{/}2\)

\((2\xi \mathrm{-}1)(1\mathrm{-}{\eta }^{2})\mathrm{/}2\)

\(\mathrm{-}\xi \eta (\xi \mathrm{-}1)\)

\((1\mathrm{-}{\xi }^{2})(1\mathrm{-}{\eta }^{2})\)

\(\mathrm{-}2\xi (1\mathrm{-}{\eta }^{2})\)

\(\mathrm{-}2\eta (1\mathrm{-}{\xi }^{2})\)

second derivatives of the 9-node quadrangle:

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\xi }^{2}\right\}\)

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }\xi \mathrm{\partial }\eta \right\}\)

\(\left\{{\mathrm{\partial }}^{2}N\mathrm{/}\mathrm{\partial }{\eta }^{2}\right\}\)

\(\eta (\eta \mathrm{-}1)\mathrm{/}2\)

\((\xi \mathrm{-}1\mathrm{/}2)(\eta \mathrm{-}1\mathrm{/}2)\)

\(\xi (\xi \mathrm{-}1)\mathrm{/}2\)

\(\eta (\eta \mathrm{-}1)\mathrm{/}2\)

\((\xi +1\mathrm{/}2)(\eta \mathrm{-}1\mathrm{/}2)\)

\(\xi (\xi +1)\mathrm{/}2\)

\(\eta (\eta +1)\mathrm{/}2\)

\((\xi +1\mathrm{/}2)(\eta +1\mathrm{/}2)\)

\(\xi (\xi +1)\mathrm{/}2\)

\(\eta (\eta +1)\mathrm{/}2\)

\((\xi \mathrm{-}1\mathrm{/}2)(\eta +1\mathrm{/}2)\)

\(\xi (\xi \mathrm{-}1)\mathrm{/}2\)

\(\mathrm{-}\eta (\eta \mathrm{-}1)\)

\(\mathrm{-}\xi (2\eta \mathrm{-}1)\)

\(1\mathrm{-}{\xi }^{2}\)

\(1\mathrm{-}{\eta }^{2}\)

\(\mathrm{-}\eta (2\xi +1)\)

\(\mathrm{-}\xi (\xi +1)\)

\(\mathrm{-}\eta (\eta +1)\)

\(\mathrm{-}\xi (\mathrm{2\eta }+1)\)

\(1\mathrm{-}{\xi }^{2}\)

\(1\mathrm{-}{\eta }^{2}\)

\(\mathrm{-}\eta (2\xi \mathrm{-}1)\)

\(\mathrm{-}\xi (\xi \mathrm{-}1)\)

\(\mathrm{-}2(1\mathrm{-}{\eta }^{2})\)

\(4\xi \eta\)

\(\mathrm{-}2(1\mathrm{-}{\xi }^{2})\)