Shape Functions
Schematic overview of all the element types defined in Akantu is described in Section Elements. In this appendix, more detailed information (shape function, location of Gaussian quadrature points, and so on) of each of these types is listed. For each element type, the coordinates of the nodes are given in the iso-parametric frame of reference, together with the shape functions (and their derivatives) on these respective nodes. Also all the Gaussian quadrature points within each element are assigned (together with the weight that is applied on these points). The graphical representations of all the element types can be found in Section Elements.
Iso-parametric Elements
1D-Shape Functions
Segment 2
Node (\(i\)) |
Coord. (\(\xi\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\)) |
|---|---|---|---|
1 |
-1 |
\(\frac{1}{2}\left(1-\xi\right)\) |
\(-\frac{1}{2}\) |
2 |
1 |
\(\frac{1}{2}\left(1+\xi\right)\) |
\(\frac{1}{2}\) |
Coord. (\(\xi\)) |
Weight |
0 |
2 |
Segment 3
Node (\(i\)) |
Coord. (\(\xi\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\)) |
|---|---|---|---|
1 |
-1 |
\(\frac{1}{2}\xi\left(\xi-1\right)\) |
\(\xi-\frac{1}{2}\) |
2 |
1 |
\(\frac{1}{2}\xi\left(\xi+1\right)\) |
\(\xi+\frac{1}{2}\) |
3 |
0 |
\(1-\xi^{2}\) |
\(-2\xi\) |
Coord. (\(\xi\)) |
Weight |
\(-1/\sqrt{3}\) |
1 |
\(1/\sqrt{3}\) |
1 |
2D-Shape Functions
Triangle 3
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\)) |
|---|---|---|---|
1 |
(\(0\), \(0\)) |
\(1-\xi-\eta\) |
(\(-1\), \(-1\)) |
2 |
(\(1\), \(0\)) |
\(\xi\) |
(\(1\), \(0\)) |
3 |
(\(0\), \(1\)) |
\(\eta\) |
(\(0\), \(1\)) |
Coord. (\(\xi\), \(\eta\)) |
Weight |
(\(\frac{1}{3}\), \(\frac{1}{3}\)) |
\(\frac{1}{2}\) |
Triangle 6
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\)) |
|---|---|---|---|
1 |
(\(0\), \(0\)) |
\(-\left(1-\xi-\eta\right)\left(1-2\left(1-\xi-\eta\right)\right)\) |
(\(1-4\left(1-\xi-\eta\right)\), \(1-4\left(1-\xi-\eta\right)\)) |
2 |
(\(1\), \(0\)) |
\(-\xi\left(1-2\xi\right)\) |
(\(4\xi-1\), \(0\)) |
3 |
(\(0\), \(1\)) |
\(-\eta\left(1-2\eta\right)\) |
(\(0\), \(4\eta-1\)) |
4 |
(\(\frac{1}{2}\), \(0\)) |
\(4\xi\left(1-\xi-\eta\right)\) |
(\(4\left(1-2\xi-\eta\right)\), \(-4\xi\)) |
5 |
(\(\frac{1}{2}\), \(\frac{1}{2}\)) |
\(4\xi\eta\) |
(\(4\eta\), \(4\xi\)) |
6 |
(\(0\), \(\frac{1}{2}\)) |
\(4\eta\left(1-\xi-\eta\right)\) |
(\(-4\eta\), \(4\left(1-\xi-2\eta\right)\)) |
Coord. (\(\xi\), \(\eta\)) |
Weight |
(\(\frac{1}{6}\), \(\frac{1}{6}\)) |
\(\frac{1}{6}\) |
(\(\frac{2}{3}\), \(\frac{1}{6}\)) |
\(\frac{1}{6}\) |
(\(\frac{1}{6}\), \(\frac{2}{3}\)) |
\(\frac{1}{6}\) |
Quadrangle 4
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\)) |
|---|---|---|---|
1 |
(\(-1\), \(-1\)) |
\(\frac{1}{4}\left(1-\xi\right)\left(1-\eta\right)\) |
(\(-\frac{1}{4}\left(1-\eta\right)\), \(-\frac{1}{4}\left(1-\xi\right)\)) |
2 |
(\(1\), \(-1\)) |
\(\frac{1}{4}\left(1+\xi\right)\left(1-\eta\right)\) |
(\(\frac{1}{4}\left(1-\eta\right)\), \(-\frac{1}{4}\left(1+\xi\right)\)) |
3 |
(\(1\), \(1\)) |
\(\frac{1}{4}\left(1+\xi\right)\left(1+\eta\right)\) |
(\(\frac{1}{4}\left(1+\eta\right)\), \(\frac{1}{4}\left(1+\xi\right)\)) |
4 |
(\(-1\), \(1\)) |
\(\frac{1}{4}\left(1-\xi\right)\left(1+\eta\right)\) |
(\(-\frac{1}{4}\left(1+\eta\right)\), \(\frac{1}{4}\left(1-\xi\right)\)) |
Coord. (\(\xi\), \(\eta\)) |
Weight |
(\(-\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\)) |
1 |
(\(\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\)) |
1 |
(\(\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\)) |
1 |
(\(-\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\)) |
1 |
Quadrangle 8
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\)) |
|---|---|---|---|
1 |
(\(-1\), \(-1\)) |
\(\frac{1}{4}\left(1-\xi\right)\left(1-\eta\right)\left(-1-\xi-\eta\right)\) |
(\(\frac{1}{4}\left(1-\eta\right)\left(2\xi+\eta\right)\), \(\frac{1}{4}\left(1-\xi\right)\left(\xi+2\eta\right)\)) |
2 |
(\(1\), \(-1\)) |
\(\frac{1}{4}\left(1+\xi\right)\left(1-\eta\right)\left(-1+\xi-\eta\right)\) |
(\(\frac{1}{4}\left(1-\eta\right)\left(2\xi-\eta\right)\), \(-\frac{1}{4}\left(1+\xi\right)\left(\xi-2\eta\right)\)) |
3 |
(\(1\), \(1\)) |
\(\frac{1}{4}\left(1+\xi\right)\left(1+\eta\right)\left(-1+\xi+\eta\right)\) |
(\(\frac{1}{4}\left(1+\eta\right)\left(2\xi+\eta\right)\), \(\frac{1}{4}\left(1+\xi\right)\left(\xi+2\eta\right)\)) |
4 |
(\(-1\), \(1\)) |
\(\frac{1}{4}\left(1-\xi\right)\left(1+\eta\right)\left(-1-\xi+\eta\right)\) |
(\(\frac{1}{4}\left(1+\eta\right)\left(2\xi-\eta\right)\), \(-\frac{1}{4}\left(1-\xi\right)\left(\xi-2\eta\right)\)) |
5 |
(\(0\), \(-1\)) |
\(\frac{1}{2}\left(1-\xi^{2}\right)\left(1-\eta\right)\) |
(\(-\xi\left(1-\eta\right)\), \(-\frac{1}{2}\left(1-\xi^{2}\right)\)) |
6 |
(\(1\), \(0\)) |
\(\frac{1}{2}\left(1+\xi\right)\left(1-\eta^{2}\right)\) |
(\(\frac{1}{2}\left(1-\eta^{2}\right)\), \(-\eta\left(1+\xi\right)\)) |
7 |
(\(0\), \(1\)) |
\(\frac{1}{2}\left(1-\xi^{2}\right)\left(1+\eta\right)\) |
(\(-\xi\left(1+\eta\right)\), \(\frac{1}{2}\left(1-\xi^{2}\right)\)) |
8 |
(\(-1\), \(0\)) |
\(\frac{1}{2}\left(1-\xi\right)\left(1-\eta^{2}\right)\) |
(\(-\frac{1}{2}\left(1-\eta^{2}\right)\), \(-\eta\left(1-\xi\right)\)) |
Coord. (\(\xi\), \(\eta\)) |
Weight |
(\(0\), \(0\)) |
\(\frac{64}{81}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{25}{81}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{25}{81}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{25}{81}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{25}{81}\) |
(\(0\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{40}{81}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(0\)) |
\(\frac{40}{81}\) |
(\(0\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{40}{81}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(0\)) |
\(\frac{40}{81}\) |
3D-Shape Functions
Tetrahedron 4
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\), \(\frac{\partial N_i}{\partial \zeta}\)) |
|---|---|---|---|
1 |
(\(0\), \(0\), \(0\)) |
\(1-\xi-\eta-\zeta\) |
(\(-1\), \(-1\), \(-1\)) |
2 |
(\(1\), \(0\), \(0\)) |
\(\xi\) |
(\(1\), \(0\), \(0\)) |
3 |
(\(0\), \(1\), \(0\)) |
\(\eta\) |
(\(0\), \(1\), \(0\)) |
4 |
(\(0\), \(0\), \(1\)) |
\(\zeta\) |
(\(0\), \(0\), \(1\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Weight |
(\(\frac{1}{4}\), \(\frac{1}{4}\), \(\frac{1}{4}\)) |
\(\frac{1}{6}\) |
Tetrahedron 10
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\), \(\frac{\partial N_i}{\partial \zeta}\)) |
|---|---|---|---|
1 |
(\(0\), \(0\), \(0\)) |
\(\left(1-\xi-\eta-\zeta\right)\left(1-2\xi-2\eta-2\zeta\right)\) |
\(4\xi+4\eta+4\zeta-3\), \(4\xi+4\eta+4\zeta-3\), \(4\xi+4\eta+4\zeta-3\) |
2 |
(\(1\), \(0\), \(0\)) |
\(\xi\left(2\xi-1\right)\) |
(\(4\xi-1\), \(0\), \(0\)) |
3 |
(\(0\), \(1\), \(0\)) |
\(\eta\left(2\eta-1\right)\) |
(\(0\), \(4\eta-1\), \(0\)) |
4 |
(\(0\), \(0\), \(1\)) |
\(\zeta\left(2\zeta-1\right)\) |
(\(0\), \(0\), \(4\zeta-1\)) |
5 |
(\(\frac{1}{2}\), \(0\), \(0\)) |
\(4\xi\left(1-\xi-\eta-\zeta\right)\) |
(\(4-8\xi-4\eta-4\zeta\), \(-4\xi\), \(-4\xi\)) |
6 |
(\(\frac{1}{2}\), \(\frac{1}{2}\), \(0\)) |
\(4\xi\eta\) |
(\(4\eta\), \(4\xi\), \(0\)) |
7 |
(\(0\), \(\frac{1}{2}\), \(0\)) |
\(4\eta\left(1-\xi-\eta-\zeta\right)\) |
(\(-4\eta\), \(4-4\xi-8\eta-4\zeta\), \(-4\eta\)) |
8 |
(\(0\), \(0\), \(\frac{1}{2}\)) |
\(4\zeta\left(1-\xi-\eta-\zeta\right)\) |
(\(-4\zeta\), \(-4\zeta\), \(4-4\xi-4\eta-8\zeta\)) |
9 |
(\(\frac{1}{2}\), \(0\), \(\frac{1}{2}\)) |
\(4\xi\zeta\) |
(\(4\zeta\), \(0\), \(4\xi\)) |
10 |
(\(0\), \(\frac{1}{2}\), \(\frac{1}{2}\)) |
\(4\eta\zeta\) |
(\(0\), \(4\zeta\), \(4\eta\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Weight |
(\(\frac{5-\sqrt{5}}{20}\), \(\frac{5-\sqrt{5}}{20}\), \(\frac{5-\sqrt{5}}{20}\)) |
\(\frac{1}{24}\) |
(\(\frac{5+3\sqrt{5}}{20}\), \(\frac{5-\sqrt{5}}{20}\), \(\frac{5-\sqrt{5}}{20}\)) |
\(\frac{1}{24}\) |
(\(\frac{5-\sqrt{5}}{20}\), \(\frac{5+3\sqrt{5}}{20}\), \(\frac{5-\sqrt{5}}{20}\)) |
\(\frac{1}{24}\) |
(\(\frac{5-\sqrt{5}}{20}\), \(\frac{5-\sqrt{5}}{20}\), \(\frac{5+3\sqrt{5}}{20}\)) |
\(\frac{1}{24}\) |
Hexahedron 8
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\), \(\frac{\partial N_i}{\partial \zeta}\)) |
|---|---|---|---|
1 |
(\(-1\), \(-1\), \(-1\)) |
\(\frac{1}{8}\left(1-\xi\right)\left(1-\eta\right)\left(1-\zeta\right)\) |
(\(-\frac{1}{8}\left(1-\eta\right)\left(1-\zeta\right)\), \(-\frac{1}{8}\left(1-\xi\right)\left(1-\zeta\right)\), \(3\)) |
2 |
(\(1\), \(-1\), \(-1\)) |
\(\frac{1}{8}\left(1+\xi\right)\left(1-\eta\right)\left(1-\zeta\right)\) |
(\(\frac{1}{8}\left(1-\eta\right)\left(1-\zeta\right)\), \(-\frac{1}{8}\left(1+\xi\right)\left(1-\zeta\right)\), \(3\)) |
3 |
(\(1\), \(1\), \(-1\)) |
\(\frac{1}{8}\left(1+\xi\right)\left(1+\eta\right)\left(1-\zeta\right)\) |
(\(\frac{1}{8}\left(1+\eta\right)\left(1-\zeta\right)\), \(\frac{1}{8}\left(1+\xi\right)\left(1-\zeta\right)\), \(3\)) |
4 |
(\(-1\), \(1\), \(-1\)) |
\(\frac{1}{8}\left(1-\xi\right)\left(1+\eta\right)\left(1-\zeta\right)\) |
(\(-\frac{1}{8}\left(1+\eta\right)\left(1-\zeta\right)\), \(\frac{1}{8}\left(1-\xi\right)\left(1-\zeta\right)\), \(3\)) |
5 |
(\(-1\), \(-1\), \(1\)) |
\(\frac{1}{8}\left(1-\xi\right)\left(1-\eta\right)\left(1+\zeta\right)\) |
(\(-\frac{1}{8}\left(1-\eta\right)\left(1+\zeta\right)\), \(-\frac{1}{8}\left(1-\xi\right)\left(1+\zeta\right)\), \(3\)) |
6 |
(\(1\), \(-1\), \(1\)) |
\(\frac{1}{8}\left(1+\xi\right)\left(1-\eta\right)\left(1+\zeta\right)\) |
(\(\frac{1}{8}\left(1-\eta\right)\left(1+\zeta\right)\), \(-\frac{1}{8}\left(1+\xi\right)\left(1+\zeta\right)\), \(3\)) |
7 |
(\(1\), \(1\), \(1\)) |
\(\frac{1}{8}\left(1+\xi\right)\left(1+\eta\right)\left(1+\zeta\right)\) |
(\(\frac{1}{8}\left(1+\eta\right)\left(1+\zeta\right)\), \(\frac{1}{8}\left(1+\xi\right)\left(1+\zeta\right)\), \(3\)) |
8 |
(\(-1\), \(1\), \(1\)) |
\(\frac{1}{8}\left(1-\xi\right)\left(1+\eta\right)\left(1+\zeta\right)\) |
(\(-\frac{1}{8}\left(1+\eta\right)\left(1+\zeta\right)\), \(\frac{1}{8}\left(1-\xi\right)\left(1+\zeta\right)\), \(3\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Weight |
(\(-\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\)) |
1 |
(\(\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\)) |
1 |
(\(\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\)) |
1 |
(\(-\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\)) |
1 |
(\(-\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\)) |
1 |
(\(\frac{1}{\sqrt{3}}\), \(-\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\)) |
1 |
(\(\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\)) |
1 |
(\(-\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\), \(\frac{1}{\sqrt{3}}\)) |
1 |
Pentahedron 6
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\), \(\frac{\partial N_i}{\partial \zeta}\)) |
|---|---|---|---|
1 |
(\(-1\), \(1\), \(0\)) |
\(\frac{1}{2}\left(1-\xi\right)\eta\) |
(\(-\frac{1}{2}\eta\), \(\frac{1}{2}\left(1-\xi\right)\), \(3\)) |
2 |
(\(-1\), \(0\), \(1\)) |
\(\frac{1}{2}\left(1-\xi\right)\zeta\) |
(\(-\frac{1}{2}\zeta\), \(0.0\), \(3\)) |
3 |
(\(-1\), \(0\), \(0\)) |
\(\frac{1}{2}\left(1-\xi\right)\left(1-\eta-\zeta\right)\) |
(\(-\frac{1}{2}\left(1-\eta-\zeta\right)\), \(-\frac{1}{2}\left(1-\xi\right)\), \(3\)) |
4 |
(\(1\), \(1\), \(0\)) |
\(\frac{1}{2}\left(1+\xi\right)\eta\) |
(\(\frac{1}{2}\eta\), \(\frac{1}{2}\left(1+\xi\right)\), \(3\)) |
5 |
(\(1\), \(0\), \(1\)) |
\(\frac{1}{2}\left(1+\xi\right)\zeta\) |
(\(\frac{1}{2}\zeta\), \(0.0\), \(3\)) |
6 |
(\(1\), \(0\), \(0\)) |
\(\frac{1}{2}\left(1+\xi\right)\left(1-\eta-\zeta\right)\) |
(\(\frac{1}{2}\left(1-\eta-\zeta\right)\), \(-\frac{1}{2}\left(1+\xi\right)\), \(3\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Weight |
(\(-\frac{1}{\sqrt{3}}\), \(0.5\), \(0.5\)) |
\(\frac{1}{6}\) |
(\(-\frac{1}{\sqrt{3}}\), \(0.0\), \(0.5\)) |
\(\frac{1}{6}\) |
(\(-\frac{1}{\sqrt{3}}\), \(0.5\), \(0.0\)) |
\(\frac{1}{6}\) |
(\(\frac{1}{\sqrt{3}}\), \(0.5\), \(0.5\)) |
\(\frac{1}{6}\) |
(\(\frac{1}{\sqrt{3}}\), \(0.0\), \(0.5\)) |
\(\frac{1}{6}\) |
(\(\frac{1}{\sqrt{3}}\), \(0.5\), \(0.0\)) |
\(\frac{1}{6}\) |
Hexahedron 20
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\), \(\frac{\partial N_i}{\partial \zeta}\)) |
|---|---|---|---|
1 |
(\(-1\), \(-1\), \(-1\)) |
\(\frac{1}{8}\left(1-\xi\right)\left(1-\eta\right)\left(1-\zeta\right)\left(-2-\xi-\eta-\zeta\right)\) |
(\(\frac{1}{4}\left(\xi+\frac{1}{2}\left(\eta+\zeta+1\right)\right)\left(\eta-1\right)\left(\zeta-1\right)\), \(\frac{1}{4}\left(\eta+\frac{1}{2}\left(\xi+\zeta+1\right)\right)\left(\xi-1\right)\left(\zeta-1\right)\), \(3\)) |
2 |
(\(1\), \(-1\), \(-1\)) |
\(\frac{1}{8}\left(1+\xi\right)\left(1-\eta\right)\left(1-\zeta\right)\left(-2+\xi-\eta-\zeta\right)\) |
(\(\frac{1}{4}\left(\xi-\frac{1}{2}\left(\eta+\zeta+1\right)\right)\left(\eta-1\right)\left(\zeta-1\right)\), \(-\frac{1}{4}\left(\eta-\frac{1}{2}\left(\xi-\zeta-1\right)\right)\left(\xi+1\right)\left(\zeta-1\right)\), \(3\)) |
3 |
(\(1\), \(1\), \(-1\)) |
\(\frac{1}{8}\left(1+\xi\right)\left(1+\eta\right)\left(1-\zeta\right)\left(-2+\xi+\eta-\zeta\right)\) |
(\(-\frac{1}{4}\left(\xi+\frac{1}{2}\left(\eta-\zeta-1\right)\right)\left(\eta+1\right)\left(\zeta-1\right)\), \(-\frac{1}{4}\left(\eta+\frac{1}{2}\left(\xi-\zeta-1\right)\right)\left(\xi+1\right)\left(\zeta-1\right)\), \(3\)) |
4 |
(\(-1\), \(1\), \(-1\)) |
\(\frac{1}{8}\left(1-\xi\right)\left(1+\eta\right)\left(1-\zeta\right)\left(-2-\xi+\eta-\zeta\right)\) |
(\(-\frac{1}{4}\left(\xi-\frac{1}{2}\left(\eta-\zeta-1\right)\right)\left(\eta+1\right)\left(\zeta-1\right)\), \(\frac{1}{4}\left(\eta-\frac{1}{2}\left(\xi+\zeta+1\right)\right)\left(\xi-1\right)\left(\zeta-1\right)\), \(3\)) |
5 |
(\(-1\), \(-1\), \(1\)) |
\(\frac{1}{8}\left(1-\xi\right)\left(1-\eta\right)\left(1+\zeta\right)\left(-2-\xi-\eta+\zeta\right)\) |
(\(-\frac{1}{4}\left(\xi+\frac{1}{2}\left(\eta-\zeta+1\right)\right)\left(\eta-1\right)\left(\zeta+1\right)\), \(-\frac{1}{4}\left(\eta+\frac{1}{2}\left(\xi-\zeta+1\right)\right)\left(\xi-1\right)\left(\zeta+1\right)\), \(3\)) |
6 |
(\(1\), \(-1\), \(1\)) |
\(\frac{1}{8}\left(1+\xi\right)\left(1-\eta\right)\left(1+\zeta\right)\left(-2+\xi-\eta+\zeta\right)\) |
(\(-\frac{1}{4}\left(\xi-\frac{1}{2}\left(\eta-\zeta+1\right)\right)\left(\eta-1\right)\left(\zeta+1\right)\), \(\frac{1}{4}\left(\eta-\frac{1}{2}\left(\xi+\zeta-1\right)\right)\left(\xi+1\right)\left(\zeta+1\right)\), \(3\)) |
7 |
(\(1\), \(1\), \(1\)) |
\(\frac{1}{8}\left(1+\xi\right)\left(1+\eta\right)\left(1+\zeta\right)\left(-2+\xi+\eta+\zeta\right)\) |
(\(\frac{1}{4}\left(\xi+\frac{1}{2}\left(\eta+\zeta-1\right)\right)\left(\eta+1\right)\left(\zeta+1\right)\), \(\frac{1}{4}\left(\eta+\frac{1}{2}\left(\xi+\zeta-1\right)\right)\left(\xi+1\right)\left(\zeta+1\right)\), \(3\)) |
8 |
(\(-1\), \(1\), \(1\)) |
\(\frac{1}{8}\left(1-\xi\right)\left(1+\eta\right)\left(1+\zeta\right)\left(-2-\xi+\eta+\zeta\right)\) |
(\(\frac{1}{4}\left(\xi-\frac{1}{2}\left(\eta+\zeta-1\right)\right)\left(\eta+1\right)\left(\zeta+1\right)\), \(-\frac{1}{4}\left(\eta-\frac{1}{2}\left(\xi-\zeta+1\right)\right)\left(\xi-1\right)\left(\zeta+1\right)\), \(3\)) |
9 |
(\(0\), \(-1\), \(-1\)) |
\(\frac{1}{4}\left(1-\xi^{2}\right)\left(1-\eta\right)\left(1-\zeta\right)\) |
(\(-\frac{1}{2}\xi\left(\eta-1\right)\left(\zeta-1\right)\), \(-\frac{1}{4}\left(\xi^{2}-1\right)\left(\zeta-1\right)\), \(3\)) |
10 |
(\(1\), \(0\), \(-1\)) |
\(\frac{1}{4}\left(1+\xi\right)\left(1-\eta^{2}\right)\left(1-\zeta\right)\) |
(\(\frac{1}{4}\left(\eta^{2}-1\right)\left(\zeta-1\right)\), \(\frac{1}{2}\eta\left(\xi+1\right)\left(\zeta-1\right)\), \(3\)) |
11 |
(\(0\), \(1\), \(-1\)) |
\(\frac{1}{4}\left(1-\xi^{2}\right)\left(1+\eta\right)\left(1-\zeta\right)\) |
(\(\frac{1}{2}\xi\left(\eta+1\right)\left(\zeta-1\right)\), \(\frac{1}{4}\left(\xi^{2}-1\right)\left(\zeta-1\right)\), \(3\)) |
12 |
(\(-1\), \(0\), \(-1\)) |
\(\frac{1}{4}\left(1-\xi\right)\left(1-\eta^{2}\right)\left(1-\zeta\right)\) |
(\(-\frac{1}{4}\left(\eta^{2}-1\right)\left(\zeta-1\right)\), \(-\frac{1}{2}\eta\left(\xi-1\right)\left(\zeta-1\right)\), \(3\)) |
13 |
(\(-1\), \(-1\), \(0\)) |
\(\frac{1}{4}\left(1-\xi\right)\left(1-\eta\right)\left(1-\zeta^{2}\right)\) |
(\(-\frac{1}{4}\left(\eta-1\right)\left(\zeta^{2}-1\right)\), \(-\frac{1}{4}\left(\xi-1\right)\left(\zeta^{2}-1\right)\), \(3\)) |
14 |
(\(1\), \(-1\), \(0\)) |
\(\frac{1}{4}\left(1+\xi\right)\left(1-\eta\right)\left(1-\zeta^{2}\right)\) |
(\(\frac{1}{4}\left(\eta-1\right)\left(\zeta^{2}-1\right)\), \(\frac{1}{4}\left(\xi+1\right)\left(\zeta^{2}-1\right)\), \(3\)) |
15 |
(\(1\), \(1\), \(0\)) |
\(\frac{1}{4}\left(1+\xi\right)\left(1+\eta\right)\left(1-\zeta^{2}\right)\) |
(\(-\frac{1}{4}\left(\eta+1\right)\left(\zeta^{2}-1\right)\), \(-\frac{1}{4}\left(\xi+1\right)\left(\zeta^{2}-1\right)\), \(3\)) |
16 |
(\(-1\), \(1\), \(0\)) |
\(\frac{1}{4}\left(1-\xi\right)\left(1+\eta\right)\left(1-\zeta^{2}\right)\) |
(\(\frac{1}{4}\left(\eta+1\right)\left(\zeta^{2}-1\right)\), \(\frac{1}{4}\left(\xi-1\right)\left(\zeta^{2}-1\right)\), \(3\)) |
17 |
(\(0\), \(-1\), \(1\)) |
\(\frac{1}{4}\left(1-\xi^{2}\right)\left(1-\eta\right)\left(1+\zeta\right)\) |
(\(\frac{1}{2}\xi\left(\eta-1\right)\left(\zeta+1\right)\), \(\frac{1}{4}\left(\xi^{2}-1\right)\left(\zeta+1\right)\), \(3\)) |
18 |
(\(1\), \(0\), \(1\)) |
\(\frac{1}{4}\left(1+\xi\right)\left(1-\eta^{2}\right)\left(1+\zeta\right)\) |
(\(-\frac{1}{4}\left(\eta^{2}-1\right)\left(\zeta+1\right)\), \(-\frac{1}{2}\eta\left(\xi+1\right)\left(\zeta+1\right)\), \(3\)) |
19 |
(\(0\), \(1\), \(1\)) |
\(\frac{1}{4}\left(1-\xi^{2}\right)\left(1+\eta\right)\left(1+\zeta\right)\) |
(\(-\frac{1}{2}\xi\left(\eta+1\right)\left(\zeta+1\right)\), \(-\frac{1}{4}\left(\xi^{2}-1\right)\left(\zeta+1\right)\), \(3\)) |
20 |
(\(-1\), \(0\), \(1\)) |
\(\frac{1}{4}\left(1-\xi\right)\left(1-\eta^{2}\right)\left(1+\zeta\right)\) |
(\(\frac{1}{4}\left(\eta^{2}-1\right)\left(\zeta+1\right)\), \(\frac{1}{2}\eta\left(\xi-1\right)\left(\zeta+1\right)\), \(3\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Weight |
(\(-\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{125}{729}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\), \(0\)) |
\(\frac{200}{729}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{125}{729}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(0\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{200}{729}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(0\), \(0\)) |
\(\frac{320}{729}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(0\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{200}{729}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{125}{729}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\), \(0\)) |
\(\frac{200}{729}\) |
(\(-\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{125}{729}\) |
(\(0\), \(-\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{200}{729}\) |
(\(0\), \(-\sqrt{\tfrac{3}{5}}\), \(0\)) |
\(\frac{320}{729}\) |
(\(0\), \(-\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{200}{729}\) |
(\(0\), \(0\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{320}{729}\) |
(\(0\), \(0\), \(0\)) |
\(\frac{512}{729}\) |
(\(0\), \(0\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{320}{729}\) |
(\(0\), \(\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{200}{729}\) |
(\(0\), \(\sqrt{\tfrac{3}{5}}\), \(0\)) |
\(\frac{320}{729}\) |
(\(0\), \(\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{200}{729}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{125}{729}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\), \(0\)) |
\(\frac{200}{729}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{125}{729}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(0\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{200}{729}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(0\), \(0\)) |
\(\frac{320}{729}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(0\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{200}{729}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\), \(-\sqrt{\tfrac{3}{5}}\)) |
\(\frac{125}{729}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\), \(0\)) |
\(\frac{200}{729}\) |
(\(\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\), \(\sqrt{\tfrac{3}{5}}\)) |
\(\frac{125}{729}\) |
Pentahedron 15
Node (\(i\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Shape function (\(N_i\)) |
Derivative (\(\frac{\partial N_i}{\partial \xi}\), \(\frac{\partial N_i}{\partial \eta}\), \(\frac{\partial N_i}{\partial \zeta}\)) |
|---|---|---|---|
1 |
(\(-1\), \(1\), \(0\)) |
\(\frac{1}{2}\eta\left(1-\xi\right)\left(2\eta-2-\xi\right)\) |
(\(\frac{1}{2}\eta\left(2\xi-2\eta+1\right)\), \(-\frac{1}{2}\left(\xi-1\right)\left(4\eta-\xi-2\right)\), \(3\)) |
2 |
(\(-1\), \(0\), \(1\)) |
\(\frac{1}{2}\zeta\left(1-\xi\right)\left(2\zeta-2-\xi\right)\) |
(\(\frac{1}{2}\zeta\left(2\xi-2\zeta+1\right)\), \(0.0\), \(3\)) |
3 |
(\(-1\), \(0\), \(0\)) |
\(\frac{1}{2}\left(\xi-1\right)\left(1-\eta-\zeta\right)\left(\xi+2\eta+2\zeta\right)\) |
(\(-\frac{1}{2}\left(2\xi+2\eta+2\zeta-1\right)\left(\eta+\zeta-1\right)\), \(-\frac{1}{2}\left(\xi-1\right)\left(4\eta+\xi+2\left(2\zeta-1\right)\right)\), \(3\)) |
4 |
(\(1\), \(1\), \(0\)) |
\(\frac{1}{2}\eta\left(1+\xi\right)\left(2\eta-2+\xi\right)\) |
(\(\frac{1}{2}\eta\left(2\xi+2\eta-1\right)\), \(\frac{1}{2}\left(\xi+1\right)\left(4\eta+\xi-2\right)\), \(3\)) |
5 |
(\(1\), \(0\), \(1\)) |
\(\frac{1}{2}\zeta\left(1+\xi\right)\left(2\zeta-2+\xi\right)\) |
(\(\frac{1}{2}\zeta\left(2\xi+2\zeta-1\right)\), \(0.0\), \(3\)) |
6 |
(\(1\), \(0\), \(0\)) |
\(\frac{1}{2}\left(-\xi-1\right)\left(1-\eta-\zeta\right)\left(-\xi+2\eta+2\zeta\right)\) |
(\(-\frac{1}{2}\left(\eta+\zeta-1\right)\left(2\xi-2\eta-2\zeta+1\right)\), \(\frac{1}{2}\left(\xi+1\right)\left(4\eta-\xi+2\left(2\zeta-1\right)\right)\), \(3\)) |
7 |
(\(-1\), \(0.5\), \(0.5\)) |
\(2\eta\zeta\left(1-\xi\right)\) |
(\(-2\eta\zeta\), \(-2\left(\xi-1\right)\zeta\), \(3\)) |
8 |
(\(-1\), \(0\), \(0.5\)) |
\(2\zeta\left(1-\eta-\zeta\right)\left(1-\xi\right)\) |
(\(2\zeta\left(\eta+\zeta-1\right)\), \(2\zeta-\left(\xi-1\right)\), \(3\)) |
9 |
(\(-1\), \(0.5\), \(0\)) |
\(2\eta\left(1-\xi\right)\left(1-\eta-\zeta\right)\) |
(\(2\eta\left(\eta+\zeta-1\right)\), \(2\left(2\eta+\zeta-1\right)\left(\xi-1\right)\), \(3\)) |
10 |
(\(0\), \(1\), \(0\)) |
\(\eta\left(1-\xi^{2}\right)\) |
(\(-2\xi\eta\), \(-\left(\xi^{2}-1\right)\), \(3\)) |
11 |
(\(0\), \(0\), \(1\)) |
\(\zeta\left(1-\xi^{2}\right)\) |
(\(-2\xi\zeta\), \(0.0\), \(3\)) |
12 |
(\(0\), \(0\), \(0\)) |
\(\left(1-\xi^{2}\right)\left(1-\eta-\zeta\right)\) |
(\(2\xi\left(\eta+\zeta-1\right)\), \(\left(\xi^{2}-1\right)\), \(3\)) |
13 |
(\(1\), \(0.5\), \(0.5\)) |
\(2\eta\zeta\left(1+\xi\right)\) |
(\(2\eta\zeta\), \(2\zeta\left(\xi+1\right)\), \(3\)) |
14 |
(\(1\), \(0\), \(0.5\)) |
\(2\zeta\left(1+\xi\right)\left(1-\eta-\zeta\right)\) |
(\(-2\zeta\left(\eta+\zeta-1\right)\), \(-2\zeta\left(\xi+1\right)\), \(3\)) |
15 |
(\(1\), \(0.5\), \(0\)) |
\(2\eta\left(1+\xi\right)\left(1-\eta-\zeta\right)\) |
(\(-2\eta\left(\eta+\zeta-1\right)\), \(-2\left(2\eta+\zeta-1\right)\left(\xi+1\right)\), \(3\)) |
Coord. (\(\xi\), \(\eta\), \(\zeta\)) |
Weight |
(\(-{\tfrac{1}{\sqrt{3}}}\), \(\tfrac{1}{3}\), \(\tfrac{1}{3}\)) |
-\(\frac{27}{96}\) |
(\(-{\tfrac{1}{\sqrt{3}}}\), \(0.6\), \(0.2\)) |
\(\frac{25}{96}\) |
(\(-{\tfrac{1}{\sqrt{3}}}\), \(0.2\), \(0.6\)) |
\(\frac{25}{96}\) |
(\(-{\tfrac{1}{\sqrt{3}}}\), \(0.2\), \(0.2\)) |
\(\frac{25}{96}\) |
(\({\tfrac{1}{\sqrt{3}}}\), \(\tfrac{1}{3}\), \(\tfrac{1}{3}\)) |
-\(\frac{27}{96}\) |
(\({\tfrac{1}{\sqrt{3}}}\), \(0.6\), \(0.2\)) |
\(\frac{25}{96}\) |
(\({\tfrac{1}{\sqrt{3}}}\), \(0.2\), \(0.6\)) |
\(\frac{25}{96}\) |
(\({\tfrac{1}{\sqrt{3}}}\), \(0.2\), \(0.2\)) |
\(\frac{25}{96}\) |