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

Table 8 Elements properties

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

\(\frac{1}{2}\left(1+\xi\right)\)

\(\frac{1}{2}\)

Table 9 Gaussian quadrature points

Coord. (\(\xi\))

Weight

0

2

Segment 3

Table 10 Elements properties

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

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

\(-2\xi\)

Table 11 Gaussian quadrature points

Coord. (\(\xi\))

Weight

\(-1/\sqrt{3}\)

1

\(1/\sqrt{3}\)

1

2D-Shape Functions

Triangle 3

Table 12 Elements properties

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

(\(0\), \(1\))

\(\eta\)

(\(0\), \(1\))

Table 13 Gaussian quadrature points

Coord. (\(\xi\), \(\eta\))

Weight

(\(\frac{1}{3}\), \(\frac{1}{3}\))

\(\frac{1}{2}\)

Triangle 6

Table 14 Elements properties

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

(\(0\), \(\frac{1}{2}\))

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

(\(-4\eta\), \(4\left(1-\xi-2\eta\right)\))

Table 15 Gaussian quadrature points

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

Table 16 Elements properties

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

(\(-1\), \(1\))

\(\frac{1}{4}\left(1-\xi\right)\left(1+\eta\right)\)

(\(-\frac{1}{4}\left(1+\eta\right)\), \(\frac{1}{4}\left(1-\xi\right)\))

Table 17 Gaussian quadrature points

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

Table 18 Elements properties

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

(\(-1\), \(0\))

\(\frac{1}{2}\left(1-\xi\right)\left(1-\eta^{2}\right)\)

(\(-\frac{1}{2}\left(1-\eta^{2}\right)\), \(-\eta\left(1-\xi\right)\))

Table 19 Gaussian quadrature points

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

Table 20 Elements properties

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

(\(0\), \(0\), \(1\))

\(\zeta\)

(\(0\), \(0\), \(1\))

Table 21 Gaussian quadrature points

Coord. (\(\xi\), \(\eta\), \(\zeta\))

Weight

(\(\frac{1}{4}\), \(\frac{1}{4}\), \(\frac{1}{4}\))

\(\frac{1}{6}\)

Tetrahedron 10

Table 22 Elements properties

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

(\(0\), \(\frac{1}{2}\), \(\frac{1}{2}\))

\(4\eta\zeta\)

(\(0\), \(4\zeta\), \(4\eta\))

Table 23 Gaussian quadrature points

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

Table 24 Elements properties

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

(\(-1\), \(1\), \(1\))

\(\frac{1}{8}\left(1-\xi\right)\left(1+\eta\right)\left(1+\zeta\right)\)

(\(-\frac{1}{8}\left(1+\eta\right)\left(1+\zeta\right)\), \(\frac{1}{8}\left(1-\xi\right)\left(1+\zeta\right)\), \(3\))

Table 25 Gaussian quadrature points

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

Table 26 Elements properties

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

(\(1\), \(0\), \(0\))

\(\frac{1}{2}\left(1+\xi\right)\left(1-\eta-\zeta\right)\)

(\(\frac{1}{2}\left(1-\eta-\zeta\right)\), \(-\frac{1}{2}\left(1+\xi\right)\), \(3\))

Table 27 Gaussian quadrature points

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

Table 28 Elements properties

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

(\(-1\), \(0\), \(1\))

\(\frac{1}{4}\left(1-\xi\right)\left(1-\eta^{2}\right)\left(1+\zeta\right)\)

(\(\frac{1}{4}\left(\eta^{2}-1\right)\left(\zeta+1\right)\), \(\frac{1}{2}\eta\left(\xi-1\right)\left(\zeta+1\right)\), \(3\))

Table 29 Gaussian quadrature points

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

Table 30 Elements properties

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

(\(1\), \(0.5\), \(0\))

\(2\eta\left(1+\xi\right)\left(1-\eta-\zeta\right)\)

(\(-2\eta\left(\eta+\zeta-1\right)\), \(-2\left(2\eta+\zeta-1\right)\left(\xi+1\right)\), \(3\))

Table 31 Gaussian quadrature points

Coord. (\(\xi\), \(\eta\), \(\zeta\))

Weight

(\(-{\tfrac{1}{\sqrt{3}}}\), \(\tfrac{1}{3}\), \(\tfrac{1}{3}\))

-\(\frac{27}{96}\)

(\(-{\tfrac{1}{\sqrt{3}}}\), \(0.6\), \(0.2\))

\(\frac{25}{96}\)

(\(-{\tfrac{1}{\sqrt{3}}}\), \(0.2\), \(0.6\))

\(\frac{25}{96}\)

(\(-{\tfrac{1}{\sqrt{3}}}\), \(0.2\), \(0.2\))

\(\frac{25}{96}\)

(\({\tfrac{1}{\sqrt{3}}}\), \(\tfrac{1}{3}\), \(\tfrac{1}{3}\))

-\(\frac{27}{96}\)

(\({\tfrac{1}{\sqrt{3}}}\), \(0.6\), \(0.2\))

\(\frac{25}{96}\)

(\({\tfrac{1}{\sqrt{3}}}\), \(0.2\), \(0.6\))

\(\frac{25}{96}\)

(\({\tfrac{1}{\sqrt{3}}}\), \(0.2\), \(0.2\))

\(\frac{25}{96}\)

Material Parameters