# 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:(1) ${\Phi }_{1}=\left(2{L}_{1}-1\right){L}_{1}$ (2) ${\Phi }_{2}=\left(2{L}_{2}-1\right){L}_{2}$ (3) ${\Phi }_{3}=\left(2{L}_{3}-1\right){L}_{3}$ (4) ${\Phi }_{4}=\left(2{L}_{4}-1\right){L}_{4}$ (5) ${\Phi }_{5}=4{L}_{1}{L}_{2}$ (6) ${\Phi }_{6}=4{L}_{2}{L}_{3}$ (7) ${\Phi }_{7}=4{L}_{3}{L}_{1}$ (8) ${\Phi }_{8}=4{L}_{1}{L}_{4}$ (9) ${\Phi }_{9}=4{L}_{2}{L}_{4}$ (10) ${\Phi }_{10}=4{L}_{3}{L}_{4}$
Location of the 4 integration points is expressed by 1.
${L}_{1}$ ${L}_{2}$ ${L}_{3}$ ${L}_{4}$
a α $\beta$ $\beta$ $\beta$
b $\beta$ α $\beta$ $\beta$
c $\beta$ $\beta$ α $\beta$
d $\beta$ $\beta$ $\beta$ α

With,

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

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

• 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:(11) $dt=\frac{{L}_{c}}{c}$
Where, ${L}_{c}$ is the characteristic length of element depending on tetra type. The different types are: (12) ${L}_{c}=a\sqrt{\frac{2}{3}};\text{\hspace{0.17em}}{L}_{c}=0.816a$ (13) ${L}_{c}=a\sqrt{\frac{5/2}{6}};\text{\hspace{0.17em}}{L}_{c}=0.264a$
For another regular tetra obtained by the assemblage of four hexa as shown in Figure 4, the characteristic length is: (14) ${L}_{c}=a\frac{\sqrt{2/3}}{4};\text{\hspace{0.17em}}{L}_{c}=0.204a$

### 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.