# Solid Tetrahedron Elements

## 4-Node Solid Tetrahedron

The Radioss solid tetrahedron element is a 4 node element with one integration point and a linear shape function.

This element has no hourglass. But the drawbacks are the low convergence and the shear locking.

## 10-Node Solid Tetrahedron

The Radioss solid tetrahedron element is a 10 nodes element
with 4 integration points and a quadratic shape function as shown in Figure 1.

Introducing volume coordinates in an isoparametric frame:

${L}_{1}=r$

${L}_{2}=s$

${L}_{3}=t$

${L}_{4}=1-{L}_{1}-{L}_{2}-{L}_{3}$

The shape functions are expressed by:

$${\Phi}_{1}=\left(2{L}_{1}-1\right){L}_{1}$$

$${\Phi}_{2}=\left(2{L}_{2}-1\right){L}_{2}$$

$${\Phi}_{3}=\left(2{L}_{3}-1\right){L}_{3}$$

$${\Phi}_{4}=\left(2{L}_{4}-1\right){L}_{4}$$

$${\Phi}_{5}=4{L}_{1}{L}_{2}$$

$${\Phi}_{6}=4{L}_{2}{L}_{3}$$

$${\Phi}_{7}=4{L}_{3}{L}_{1}$$

$${\Phi}_{8}=4{L}_{1}{L}_{4}$$

$${\Phi}_{9}=4{L}_{2}{L}_{4}$$

$${\Phi}_{10}=4{L}_{3}{L}_{4}$$

Location of the 4 integration points is expressed by

^{1}.${L}_{1}$ | ${L}_{2}$ | ${L}_{3}$ | ${L}_{4}$ | |
---|---|---|---|---|

a | $\alpha $ | $\beta $ | $\beta $ | $\beta $ |

b | $\beta $ | $\alpha $ | $\beta $ | $\beta $ |

c | $\beta $ | $\beta $ | $\alpha $ | $\beta $ |

d | $\beta $ | $\beta $ | $\beta $ | $\alpha $ |

With,

$\alpha =0\cdot 58541020$ and $\beta =0\cdot 13819660$ .

a, b, c, and d are the 4 integration points.

### Advantages and Limitations

This element has various advantages:

- No hourglass
- Compatible with powerful mesh generators
- Fast convergence
- No shear locking.

But there are some drawbacks too:

- Low time step
- Not compatible with ALE formulation

### Time Step

The time step for a regular tetrahedron is computed as:

$$dt=\frac{{L}_{c}}{c}$$

Where,
${L}_{c}$
is the characteristic length of element depending on tetra
type. The different types are:

$${L}_{c}=a\sqrt{\frac{2}{3}};\text{\hspace{0.17em}}{L}_{c}=0.816a$$

$${L}_{c}=a\sqrt{\frac{5/2}{6}};\text{\hspace{0.17em}}{L}_{c}=0.264a$$

For another regular tetra obtained by the assemblage of four hexa as shown in Figure 4, the characteristic length is:

$${L}_{c}=a\frac{\sqrt{2/3}}{4};\text{\hspace{0.17em}}{L}_{c}=0.204a$$

### CPU Cost and Time/Element/Cycle

The CPU cost is shown in Figure 5:

### Example: Comparison

Below is a comparison of the 3 types of elements (8-nodes brick, 10-nodes tetra and
20-nodes brick). The results are shown in Figure 6 for a plastic strain contour.

^{1}Hammet P.C., Marlowe O.P. and Stroud A.H.,

Numerical integration over simplexes and cones, Math. Tables Aids Comp, 10, 130-7, 1956.