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 図 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)

Φ_{1} = (2 L_{1} - 1) L_{1}

(2)

Φ_{2} = (2 L_{2} - 1) L_{2}

(3)

Φ_{3} = (2 L_{3} - 1) L_{3}

(4)

Φ_{4} = (2 L_{4} - 1) L_{4}

(5)

Φ_{5} = 4 L_{1} L_{2}

(6)

Φ_{6} = 4 L_{2} L_{3}

(7)

Φ_{7} = 4 L_{3} L_{1}

(8)

Φ_{8} = 4 L_{1} L_{4}

(9)

Φ_{9} = 4 L_{2} L_{4}

(10)

Φ_{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	$β$	$β$	$β$
b	$β$	α	$β$	$β$
c	$β$	$β$	α	$β$
d	$β$	$β$	$β$	α

With,

$α = 0 \cdot 58541020$ and $β = 0 \cdot 13819660$ .

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
No direct compatibility with contact interface and other elements.

Time Step

The time step for a regular tetrahedron is computed as:(11)

d t = \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}}; L_{c} = 0.816 a

(13)

L_{c} = a \sqrt{\frac{5 / 2}{6}}; L_{c} = 0.264 a

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

(14)

L_{c} = a \frac{\sqrt{2 / 3}}{4}; L_{c} = 0.204 a

CPU Cost and Time/Element/Cycle

The CPU cost is shown in 図 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 図 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.