# 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

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

$$\frac{1}{2}$$

 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

$$1-\xi^{2}$$

$$-2\xi$$

 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

($$0$$, $$1$$)

$$\eta$$

($$0$$, $$1$$)

 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

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

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

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

 Coord. ($$\xi$$, $$\eta$$) Weight ($$\frac{1}{6}$$, $$\frac{1}{6}$$) $$\frac{1}{6}$$ ($$\frac{2}{3}$$, $$\frac{1}{6}$$) $$\frac{1}{6}$$ ($$\frac{1}{6}$$, $$\frac{2}{3}$$) $$\frac{1}{6}$$

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

($$-1$$, $$1$$)

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

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

 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

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

($$-1$$, $$0$$)

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

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

 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

($$0$$, $$0$$, $$1$$)

$$\zeta$$

($$0$$, $$0$$, $$1$$)

 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

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

$$4\eta\zeta$$

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

 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

($$-1$$, $$1$$, $$1$$)

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

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

 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

($$1$$, $$0$$, $$0$$)

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

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

 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

($$-1$$, $$0$$, $$1$$)

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

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

 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

($$1$$, $$0.5$$, $$0$$)

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

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

 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¶

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