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.
図 1. (a) Isoparametric 10 Node Tetrahedron; (b) Nodal Mass Distribution


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:
図 2.
Φ 1 = ( 2 L 1 − 1 ) L 1
図 3.
Φ 2 = ( 2 L 2 − 1 ) L 2
図 4.
Φ 3 = ( 2 L 3 − 1 ) L 3
図 5.
Φ 4 = ( 2 L 4 − 1 ) L 4
図 6.
Φ 5 = 4 L 1 L 2
図 7.
Φ 6 = 4 L 2 L 3
図 8.
Φ 7 = 4 L 3 L 1
図 9.
Φ 8 = 4 L 1 L 4
図 10.
Φ 9 = 4 L 2 L 4
図 11.
Φ 10 = 4 L 3 L 4
Location of the 4 integration points is expressed by 1.
L 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaaIXaaabeaaaaa@37AE@ L 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaaIXaaabeaaaaa@37AE@ L 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaaIXaaabeaaaaa@37AE@ L 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaaBa aaleaacaaIXaaabeaaaaa@37AE@
a α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3792@ β β β
b β α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3792@ β β
c β β α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3792@ β
d β β β α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3792@

With,

α = 0 ⋅ 58541020 and β = 0 ⋅ 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:
図 12.
d t = L c c
Where, L c is the characteristic length of element depending on tetra type. The different types are:
図 13. Regular 4 Nodes Tetra


図 14.
L c = a 2 3 ;   L c = 0.816 a
図 15. Regular 10 Nodes Tetra


図 16.
L c = a 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:
図 17. Other Regular Tetra


図 18.
L c = a 2 / 3 4 ;   L c = 0.204 a

CPU Cost and Time/Element/Cycle

The CPU cost is shown in 図 5:
図 19. CPU Cost in TEC


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.
図 20. Comparison (plastic strain max = 60%)


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.