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

Linear elastic isotropic

Keyword: elastic

Parameters:

  • rho: (Real) Density

  • E: (Real) Young’s modulus

  • nu: (Real) Poisson’s ratio

  • Plane_stress: (bool) Plane stress simplification (only 2D problems)

Energies:

  • potential: elastic potential energy

Linear elastic anisotropic

Keyword: elastic_anisotropic

Parameters:

  • rho: (Real) Density

  • n1: (Vector<Real>) Direction of main material axis

  • n2: (Vector<Real>) Direction of second material axis

  • n3: (Vector<Real>) Direction of third material axis

  • C..: (Real) Coefficient ij of material tensor C (all the 36 values in Voigt notation can be entered)

  • alpha: (Real) Viscous propertion (default is 0)

Linear elastic orthotropic

Keyword: elastic_orthotropic

Parameters:

  • rho: (Real) Density

  • n1: (Vector<Real>) Direction of main material axis

  • n2: (Vector<Real>) Direction of second material axis (if applicable)

  • n3: (Vector<Real>) Direction of third material axis (if applicable)

  • E1: (Real) Young’s modulus (n1)

  • E2: (Real) Young’s modulus (n2)

  • E3: (Real) Young’s modulus (n3)

  • nu1: (Real) Poisson’s ratio (n1)

  • nu2: (Real) Poisson’s ratio (n2)

  • nu3: (Real) Poisson’s ratio (n3)

  • G12: (Real) Shear modulus (12)

  • G13: (Real) Shear modulus (13)

  • G23: (Real) Shear modulus (23)

Neohookean (finite strains)

Keyword: neohookean

Parameters:

  • rho: (Real) Density

  • E: (Real) Young’s modulus

  • nu: (Real) Poisson’s ratio

  • Plane_stress: (bool) Plane stress simplification (only 2D problems)

Standard linear solid

Keyword: sls_deviatoric

Parameters:

  • rho: (Real) Density

  • E: (Real) Young’s modulus

  • nu: (Real) Poisson’s ratio

  • Plane_stress: (bool) Plane stress simplification (only 2D problems)

  • Eta: (Real) Viscosity

  • Ev: (Real) Stiffness of viscous element

Energies:

  • dissipated: energy dissipated with viscosity

Elasto-plastic linear isotropic hardening

Keyword: plastic_linear_isotropic_hardening

Parameters:

  • rho: (Real) Density

  • E: (Real) Young’s modulus

  • nu: (Real) Poisson’s ratio

  • h: (Real) Hardening modulus

  • sigma_y: (Real) Yield stress

Energies:

  • potential: elastic part of the potential energy

  • plastic: dissipated plastic energy (integrated over time)

Marigo

Keyword: marigo

Parameters:

  • rho: (Real) Density

  • E: (Real) Young’s modulus

  • nu: (Real) Poisson’s ratio

  • Plane_stress: (bool) Plane stress simplification (only 2D problems)

  • Yd: (Random) Hardening modulus

  • Sd: (Real) Damage energy

Energies:

  • dissipated: energy dissipated in damage

Mazars

Keyword: mazars

Parameters:

  • rho: (Real) Density

  • E: (Real) Young’s modulus

  • nu: (Real) Poisson’s ratio

  • At: (Real) Traction post-peak asymptotic value

  • Bt: (Real) Traction decay shape

  • Ac: (Real) Compression post-peak asymptotic value

  • Bc: (Real) Compression decay shape

  • K0: (Real) Damage threshold

  • beta: (Real) Shear parameter

Energies:

  • dissipated: energy dissipated in damage