4. Volumic elements#
4.1. Tetrahedra: ELREFE TE4, T10#
Node coordinates:
\(x\) |
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\(\mathrm{N1}\) |
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\(\mathrm{N2}\) |
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\(\mathrm{N3}\) |
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\(\mathrm{N4}\) |
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\(\mathrm{N5}\) |
0.5 |
0.5 |
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\(\mathrm{N6}\) |
0.5 |
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\(\mathrm{N7}\) |
0.5 |
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\(\mathrm{N8}\) |
0.5 |
0.5 |
0.5 |
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\(\mathrm{N9}\) |
0.5 |
0.5 |
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\(\mathrm{N10}\) |
0.5 |
Shape functions:
4-knot formula
\(\{\begin{array}{}{w}_{1}(x,y,z)=y\\ {w}_{2}(x,y,z)=z\\ {w}_{3}(x,y,z)=1-x-y-z\\ {w}_{4}(x,y,z)=x\end{array}\)
10-knot formula
Numerical integration formula:
Formula with 1 point, order 1 in \(x,y,z\): (FPG1)
Point |
\(x\) |
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Weight |
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1 |
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Formula with 4 points, order 2 in \(x,y,z\): (FPG4)
Point |
\(x\) |
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Weight |
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1 |
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2 |
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3 |
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4 |
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with: \(a=\frac{5-\sqrt{5}}{\text{20}}\), \(b=\frac{5+3\sqrt{5}}{\text{20}}\)
Formula with 5 points, order 3 in \(x,y,z\): (FPG5)
Point |
\(x\) |
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Weight |
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1 |
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2 |
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3 |
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4 |
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5 |
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With: \(a=0\text{.}\text{25}\), \(b=\frac{1}{6}\), \(c=0\text{.}5\)
Formula with 15 points, order 5 in \(x,y,z\): (FPG15)
Point |
\(x\) |
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Weight |
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1 |
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2 3 4 5 |
\({b}_{1}\) \({b}_{1}\) \({b}_{1}\) \({c}_{1}\) |
\({b}_{1}\) \({c}_{1}\) \({b}_{1}\) |
\({c}_{1}\) \({b}_{1}\) \({b}_{1}\) |
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6 7 8 9 |
\({b}_{2}\) \({b}_{2}\) \({b}_{2}\) \({c}_{2}\) |
\({b}_{2}\) \({c}_{2}\) \({b}_{2}\) |
\({c}_{2}\) \({b}_{2}\) \({b}_{2}\) |
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10 11 12 13 14 15 |
\(d\) \(d\) \(e\) \(d\) \(e\) \(e\) |
\(e\) \(d\) \(e\) \(d\) \(e\) |
\(d\) \(d\) \(e\) \(e\) \(d\) |
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with:
\(a=0\text{.}\text{25}\) |
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4.2. Pentahedra: ELREFE PE6, P15, P15, P18, P21#
N19 ROAD
N20 ROAD
N21 ROAD

Node coordinates:
\(x\) |
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\(\mathrm{N1}\) |
-1. |
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\(\mathrm{N2}\) |
-1. |
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\(\mathrm{N3}\) |
-1. |
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\(\mathrm{N4}\) |
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\(\mathrm{N5}\) |
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\(\mathrm{N6}\) |
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\(\mathrm{N7}\) |
-1. |
0.5 |
0.5. |
\(\mathrm{N8}\) |
-1. |
0.5. |
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\(\mathrm{N9}\) |
-1. |
0.5 |
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\(\mathrm{N10}\) |
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\(\mathrm{N11}\) |
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\(\mathrm{N12}\) |
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\(\mathrm{N13}\) |
0.5 |
0.5 |
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\(\mathrm{N14}\) |
0.5 |
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\(\mathrm{N15}\) |
0.5 |
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\(\mathrm{N16}\) |
0.5 |
0.5 |
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\(N17\) |
0.5 |
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\(\mathrm{N18}\) |
0.5 |
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\(N19\) |
-1. |
1/3 |
1/3 |
\(N20\) |
1/3 |
1/3 |
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\(N21\) |
1/3 |
1/3 |
Shape functions:
6-knot formula
15-knot formula
18 knot formula
6-point numerical integration formulas (order 3 in \(x\), order 2 in \(y\) and \(z\)) (FPG6)
Point |
\(x\) |
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Weight |
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1 |
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0.5 |
0.5 |
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2 |
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0.5 |
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3 |
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0.5 |
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4 |
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0.5 |
0.5 |
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5 |
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0.5 |
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6 |
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0.5 |
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8-point numerical integration formula: (FPG8)
2 Gauss points in \(x\) (order 3).
4 Hammer points in \(y\) and \(z\) (3rd order).
Point |
\(x\) |
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Weight |
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1 |
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2 |
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0.6 |
0.2 |
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3 |
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0.2 |
0.6 |
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4 |
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0.2 |
0.2 |
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5 |
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6 |
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0.6 |
0.2 |
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7 |
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0.2 |
0.6 |
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8 |
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0.2 |
0.2 |
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With \(a=0.577350269189626\)
21-point numerical integration formula: (FPG21)
3 Gauss points in \(x\) (order 5).
7 Hammer points in \(y\) and \(z\) (order 5 in \(y\) and \(z\)).
Point |
\(x\) |
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Weight |
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1 |
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2 3 4 |
\(-\alpha\) \(-\alpha\) \(-\alpha\) |
\(1-\mathrm{2a}\) \(a\) |
\(a\) \(1-\mathrm{2a}\) |
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5 6 7 |
\(-\alpha\) \(-\alpha\) \(-\alpha\) |
\(1-\mathrm{2b}\) \(b\) |
\(b\) \(1-\mathrm{2b}\) |
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8 |
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9 10 11 |
0. 0. 0. |
\(1-\mathrm{2a}\) \(a\) |
\(a\) \(1-\mathrm{2a}\) |
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12 13 14 |
0. 0. 0. |
\(1-\mathrm{2b}\) \(b\) |
\(b\) \(1-\mathrm{2b}\) |
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15 |
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16 17 18 |
\(\alpha\) \(\alpha\) \(\alpha\) |
\(1-\mathrm{2a}\) \(a\) |
\(a\) \(1-\mathrm{2a}\) |
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19 20 21 |
\(\alpha\) \(\alpha\) \(\alpha\) |
\(1-\mathrm{2b}\) \(b\) |
\(b\) \(1-\mathrm{2b}\) |
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with:
\(\alpha =\sqrt{\frac{3}{5}}\) |
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27-point numerical integration formula (FPG27): see [bib3].
Point |
\(x\) |
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Weight |
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1 |
0.0 |
0.895512822481133 |
0.052243588759434 |
0.027191062410231 |
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2 |
0.0 |
0.052243588759434 |
0.052243588759434 |
0.895512822481133 |
0.027191062410231 |
3 |
0.0 |
0.052243588759434 |
0.052243588759434 |
0.027191062410231 |
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4 |
0.0 |
0.198304865473555 |
0.270635256143164 |
0.040636041641641220 |
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5 |
0.0 |
0.198304865473555 |
0.531059878383280 |
0.040636041641641220 |
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6 |
0.0 |
0.270635256143164 |
0.531059878383280 |
0.040636041641641220 |
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7 |
0.0 |
0.531059878383280 |
0.270635256143164 |
0.040636041641641220 |
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8 |
0.0 |
0.531059878383280 |
0.198304865473555 |
0.040636041641641220 |
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9 |
0.0 |
0.270635256143164 |
0.198304865473555 |
0.040636041641641220 |
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10 |
0.936241512371697 |
0.3333333333333333333 |
0.33333333333333333 |
0.05027512371697 |
0.050275140937507 |
11 |
0.948681147283254 |
0.841699897299232 |
0.079150051350384 |
0.011774414962347 |
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12 |
0.948681147283254 |
0.079150051350384 |
0.841699897299232 |
0.011774414962347 |
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13 |
0.948681147283254 |
0.079150051350384 |
0.079150051350384 |
0.011774414962347 |
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14 |
0.600638052820557 |
0.054831294873304 |
0.308513201856883 |
0.041951149272741 |
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15 |
0.600638052820557 |
0.054831294873304 |
0.636655503269814 |
0.041951149272741 |
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16 |
0.600638052820557 |
0.308513201856883 |
0.636655503269814 |
0.041951149272741 |
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17 |
0.600638052820557 |
0.636655503269814 |
0.308513201856883 |
0.041951149272741 |
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18 |
0.600638052820557 |
0.636655503269814 |
0.054831294873304 |
0.041951149272741 |
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19 |
0.600638052820557 |
0.308513201856883 |
0.054831294873304 |
0.041951149272741 |
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20 |
-0.93624151212371697 |
0.3333333333333333333 |
0.33333333333333333 |
0.05027512371697 |
0.050275140937507 |
21 |
-0.948681147283254 |
0.841699897299232 |
0.079150051350384 |
0.011774414962347 |
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22 |
-0.948681147283254 |
0.079150051350384 |
0.841699897299232 |
0.011774414962347 |
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23 |
-0.948681147283254 |
0.079150051350384 |
0.079150051350384 |
0.011774414962347 |
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24 |
-0.600638052820557 |
0.054831294873304 |
0.308513201856883 |
0.041951149272741 |
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25 |
-0.600638052820557 |
0.054831294873304 |
0.636655503269814 |
0.0419511492820557 |
0.041951149272741 |
26 |
-0.600638052820557 |
0.308513201856883 |
0.636655503269814 |
0.0419511492820557 |
0.041951149272741 |
27 |
-0.600638052820557 |
0.636655503269814 |
0.308513201856883 |
0.041951149272741 |
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28 |
-0.600638052820557 |
0.636655503269814 |
0.054831294873304 |
0.041951149272741 |
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29 |
-0.600638052820557 |
0.308513201856883 |
0.054831294873304 |
0.041951149272741 |
4.3. Hexahedra: ELREFE HE8, H20, H27#


Node coordinates:
\(x\) |
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N1 |
-1. |
-1. |
-1. |
N2 |
-1. |
-1. |
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N3 |
-1. |
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N4 |
-1. |
-1. |
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N5 |
-1. |
-1. |
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N6 |
-1. |
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N7 |
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N8 |
-1. |
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N9 |
-1. |
-1. |
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N10 |
-1. |
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N11 |
-1. |
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N12 |
-1. |
-1. |
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N13 |
-1. |
-1. |
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N14 |
-1. |
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N15 |
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N16 |
-1. |
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N17 |
-1. |
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N18 |
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N19 |
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N20 |
-1. |
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N21 |
-1. |
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N22 |
-1. |
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N23 |
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N24 |
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N25 |
-1. |
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N26 |
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N27 |
Shape functions:
8-knot formula
20-knot formula
27-knot formula
Gauss quadrature formula with 2 points in each direction (order 3) (FPG8)
Point |
\(x\) |
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Weight |
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1 |
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2 |
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3 |
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4 |
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5 |
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6 |
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7 |
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8 |
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Gauss quadrature formula with 3 points in each direction (order 5): (FPG27)
Point |
\(x\) |
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Weight |
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1 |
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2 |
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3 |
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4 |
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5 |
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6 |
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7 |
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8 |
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9 |
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10 |
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11 |
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12 |
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13 |
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14 |
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15 |
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16 |
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17 |
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18 |
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19 |
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20 |
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21 |
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22 |
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23 |
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24 |
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25 |
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26 |
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27 |
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with:
\(\alpha =\sqrt{\frac{3}{5}}\) |
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4.4. Pyramids: ELREFE PY5, P13, P19#

The blue nodes are in the middle of the faces, the red one is in the middle of the cell.
The square base is made up of the quadrangle \({N}_{1}{N}_{2}{N}_{3}{N}_{4}\) and \({N}_{5}\) is the top of the pyramid.
\(x\) |
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\({N}_{1}\) |
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\({N}_{2}\) |
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\({N}_{3}\) |
—1. |
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\({N}_{4}\) |
—1. |
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\({N}_{5}\) |
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\({N}_{6}\) |
0.5 |
0.5 |
0.5 |
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\({N}_{7}\) |
—0.5 |
0.5 |
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\({N}_{8}\) |
—0.5 |
—0.5 |
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\({N}_{9}\) |
0.5 |
—0.5 |
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\({N}_{10}\) |
0.5 |
0.5 |
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\({N}_{11}\) |
0.5 |
0.5 |
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\({N}_{12}\) |
—0.5 |
0.5 |
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\({N}_{13}\) |
—0.5 |
0.5 |
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\({N}_{14}\) |
0 |
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\({N}_{15}\) |
1/3 |
1/3 |
1/3 |
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\({N}_{16}\) |
-1/3 |
1/3 |
1/3 |
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\({N}_{17}\) |
-1/3 |
-1/3 |
1/3 |
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\({N}_{18}\) |
1/3 |
-1/3 |
1/3 |
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\({N}_{19}\) |
0 |
0 |
0 |
0.2 |
Shape functions:
5-knot formula
\(\begin{array}{}{w}_{1}=\frac{(-x+y+z-1)(-x-y+z-1)}{4(1-z)}\\ {w}_{2}=\frac{(-x-y+z-1)(x-y+z-1)}{4(1-z)}\\ {w}_{3}=\frac{(x+y+z-1)(x-y+z-1)}{4(1-z)}\\ {w}_{4}=\frac{(x+y+z-1)(-x+y+z-1)}{4(1-z)}\\ {w}_{5}=z\end{array}\)
13-knot formula
\(\begin{array}{}{w}_{1}=\frac{(-x+y+z-1)(-x-y+z-1)(x-0\text{.}5)}{2(1-z)}\\ {w}_{2}=\frac{(-x-y+z-1)(x-y+z-1)(y-0\text{.}5)}{2(1-z)}\\ {w}_{3}=\frac{(x-y+z-1)(x+y+z-1)(-x-0\text{.}5)}{2(1-z)}\\ {w}_{4}=\frac{(x+y+z-1)(-x+y+z-1)(-y-0\text{.}5)}{2(1-z)}\\ {w}_{5}=\mathrm{2z}(z-0\text{.}5)\\ {w}_{6}=-\frac{(-x+y+z-1)(-x-y+z-1)(x-y+z-1)}{2(1-z)}\\ {w}_{7}=-\frac{(-x-y+z-1)(x-y+z-1)(x+y+z-1)}{2(1-z)}\end{array}\)
\(\begin{array}{}{w}_{8}=-\frac{(x-y+z-1)(x+y+z-1)(-x+y+z-1)}{2(1-z)}\\ {w}_{9}=-\frac{(x+y+z-1)(-x+y+z-1)(-x-y+z-1)}{2(1-z)}\\ {w}_{\text{10}}=\frac{z(-x+y+z-1)(-x-y+z-1)}{1-z}\\ {w}_{\text{11}}=\frac{z(-x-y+z-1)(x-y+z-1)}{1-z}\\ {w}_{\text{12}}=\frac{z(x-y+z-1)(x+y+z-1)}{1-z}\\ {w}_{\text{13}}=\frac{z(x+y+z-1)(-x+y+z-1)}{1-z}\end{array}\)
Numerical integration formula with 5 points of order 2 (FPG5):
Point |
\(x\) |
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Weight |
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1 |
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2 |
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3 |
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4 |
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5 |
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with:
\({h}_{1}=0.1531754163448146\)
\({h}_{2}=0.6372983346207416\)
\({p}_{1}=\frac{2}{15}\)
Numerical integration formula with 6 order 3 points (FPG6):
Point |
\(x\) |
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Weight |
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1 |
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2 |
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3 |
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4 |
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5 |
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6 |
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With:
\(a=0.5610836110587396\)
\({p}_{1}=0.1681372559485071\)
\({p}_{2}=0.07500000404404333\)
\({p}_{3}=0.1058823516685291\)
\({h}_{1}=0.1681372559485071\)
\({h}_{2}=0.00000000567585\)
\({h}_{3}=0.1058823516685291\)
10-point numerical integration formula (FPG10) of order 4, see [2]:
Point |
\(x\) |
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Weight |
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1 |
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2 |
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3 |
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4 |
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5 |
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6 |
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7 |
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8 |
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9 |
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10 |
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With:
\(a=0.3252907781991163\)
\(b=0.65796699712169\)
\({h}_{1}=0.6772327888861374\)
\({h}_{2}=0.1251369531087465\)
\({h}_{3}=0.3223841495782137\)
\({h}_{4}=0.0392482838988154\)
\({w}_{1}=0.07582792211376127\)
\({w}_{2}=0.1379222683930349\)
\({w}_{3}=0.07088305859288367\)
\({w}_{4}=0.04234606044708394\)
Numerical integration formula with 15 Gauss points (FPG15) of order 5:
Point |
\(x\) |
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Weight |
|
1 |
0.0 |
0.0 |
0.0 |
0.7298578807825067 |
0.0456235799393942674 |
2 |
0.0 |
0.0 |
0.0 |
0.300401020813769 |
0.112931409661816 |
3 |
0.0 |
0.0 |
0.0 |
0.0000000064917722 |
0.03913635721904967 |
4 |
-0.3532630157731623 |
-0.3532630157731623 |
0.125 |
0.050960862086209874681 |
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5 |
-0.3532630157731623 |
0.3532630157731623 |
0.125 |
0.050960862086209874681 |
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6 |
0.3532630157731623 |
0.3532630157731623 |
0.125 |
0.05096086209874681 |
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7 |
0.3532630157731623 |
-0.3532630157731623 |
0.125 |
0.050960862086209874681 |
|
8 |
-0.705117122727788277 |
0.531059878383280 |
0.061111907062023 |
0.0264472678827788277 |
0.02644726771976367 |
9 |
0.0 |
-0.705117121227788277 |
0.061111907062023 |
0.02644726771976367 |
|
10 |
0.0 |
0.705117121227788277 |
0.061111907062023 |
0.02644726771976367 |
|
11 |
0.705117122727788277 |
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 |
Numerical integration formula with 24 Gauss points (FPG24) of order 6:
Point |
\(x\) |
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Weight |
||||
1 |
0.0 |
0.0 |
0.0 |
0.8076457976939595 |
0.01697526244176133 |
|||
2 |
0.0 |
0.0 |
0.0 |
0.0017638088528196 |
0.0107023421167942 |
|||
3 |
0.0 |
0.0 |
0.0 |
0.1382628064637306 |
0.0797197029683492 |
|||
4 |
0.0 |
0.0 |
0.0 |
0.4214239119356371 |
0.0687071134661012 |
|||
5 |
-0.4172976755573542 |
-0.4172976755573542 |
0.097447341025462 |
0.024633755573542 |
0.02463375557353542 |
0.024633755573542 |
||
6 |
-0.4172976755573542 |
0.4172976755573542 |
0.097447341025462 |
0.024633755573542 |
0.02463375557353542 |
0.4172976755573542 |
0.097447341025462 |
0.024633755573542 |
7 |
0.4172976755573542 |
0.4172976755573542 |
0.097447341025462 |
0.024633755573542 |
0.02463375557353542 |
0.4172976755573542 |
0.097447341025462 |
0.024633755573542 |
8 |
0.4172976755573542 |
-0.4172976755573542 |
0.097447341025462 |
0.024633755573542 |
0.02463375557353542 |
0.024633725573542 |
||
9 |
-0.2169627046883496 |
-0.2169627046883496 |
0.5660745906233009 |
0.02105846883496 |
0.02105838883496 |
0.0210583863632544886 |
||
10 |
-0.2169627046883496 |
0.2169627046883496 |
0.5660745906233009 |
0.0210583846883496 |
0.02105838883496 |
0.0210583863632544886 |
||
11 |
0.2169627046883496 |
0.2169627046883496 |
0.5660745906233009 |
0.02105838883496 |
0.0210583868632544886 |
|||
12 |
0.2169627046883496 |
-0.2169627046883496 |
0.5660745906233009 |
0.02105838883496 |
0.0210583868632544886 |
|||
13 |
-0.565680854444256755 |
0.0 |
0.0294777308457207 |
0.0248000862596322 |
||||
14 |
0.0 |
-0.565680858544256755 |
0.0294777308457207 |
0.0248000862596322 |
||||
15 |
0.0 |
0.565680858544256755 |
0.0294777308457207 |
0.0248000862596322 |
||||
16 |
0.565680854444256755 |
0.0 |
0.0294777308457207 |
0.0248000862596322 |
||||
17 |
-0.498079091780705 |
0.0 |
0.2649158632121295 |
0.049254923117951295125 |
0.04925492311795127 |
|||
18 |
0.0 |
-0.498079079091780705 |
0.2649158632121295 |
0.04925492311795127 |
||||
19 |
0.0 |
0.498079099091780705 |
0.2649158632121295 |
0.04925492311795127 |
||||
20 |
0.498079091780705 |
0.0 |
0.2649158632121295 |
0.04925492311795125 |
0.04925492311795127 |
|||
21 |
-0.9508994872144825 |
0.0 |
0.0 |
0.048249070631936 |
0.0028934404244966 |
|||
22 |
0.0 |
-0.950899484872144825 |
0.048249070631936 |
0.0028934404244966 |
||||
23 |
0.0 |
0.950899484872144825 |
0.048249070631936 |
0.0028934404244966 |
||||
24 |
0.9508994872144825 |
0.0 |
0.048249070631936 |
0.0028934404244966 |