# 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 ($$-{\tfrac{1}{\sqrt{3}}}$$, $$\tfrac{1}{3}$$, $$\tfrac{1}{3}$$) -$$\frac{27}{96}$$ ($$-{\tfrac{1}{\sqrt{3}}}$$, $$0.6$$, $$0.2$$) $$\frac{25}{96}$$ ($$-{\tfrac{1}{\sqrt{3}}}$$, $$0.2$$, $$0.6$$) $$\frac{25}{96}$$ ($$-{\tfrac{1}{\sqrt{3}}}$$, $$0.2$$, $$0.2$$) $$\frac{25}{96}$$ ($${\tfrac{1}{\sqrt{3}}}$$, $$\tfrac{1}{3}$$, $$\tfrac{1}{3}$$) -$$\frac{27}{96}$$ ($${\tfrac{1}{\sqrt{3}}}$$, $$0.6$$, $$0.2$$) $$\frac{25}{96}$$ ($${\tfrac{1}{\sqrt{3}}}$$, $$0.2$$, $$0.6$$) $$\frac{25}{96}$$ ($${\tfrac{1}{\sqrt{3}}}$$, $$0.2$$, $$0.2$$) $$\frac{25}{96}$$