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Tetrahedral elements

numgeo provides tetrahedral elements with either 4, 5 or 10 nodes:

 
   
    Figure 1: Tetrahedral elements implemented in numgeo.  

These elements are defined by barycentric or volume coordinates, which are a natural choice for triangles and tetrahedra. In addition to the number of nodes, these elements differ in the selection of shape functions used for numerical interpolation.

  • 4 nodes: A linear element with nodes only at the four vertices.
  • 5 nodes: A linear element with an additional node in the center of the element. Since this node is not connected to any neighboring element, it is eliminated from the system of equations using static condensation.
  • 10 nodes: A quadratic element with nodes at the four vertices and at the midpoint of each of the six edges. This is the highest-order tetrahedral element in numgeo.

Shape functions

The interpolation within an element is defined by its shape functions, \(N_i\). For tetrahedra, it is convenient to use barycentric coordinates, which are denoted here by \(\xi\), \(\eta\), \(\zeta\), and \(\lambda = 1 - \xi - \eta - \zeta\).

4-node tetrahedron

The linear shape functions for the 4-node tetrahedron (T4) are:

\[ \begin{aligned} N_1 &= 1 - \xi - \eta - \zeta \\ N_2 &= \xi \\ N_3 &= \eta \\ N_4 &= \zeta \end{aligned} \]

5-node tetrahedron

The 5-node tetrahedron (T5) uses modified linear shape functions for the vertex nodes and an additional bubble function for the central node.

  • Vertex Nodes (1-4):

    \[ \begin{aligned} N_1 &= (1 - \xi - \eta - \zeta) - 64 (1 - \xi - \eta - \zeta)\xi\eta\zeta \\ N_2 &= \xi - 64 (1 - \xi - \eta - \zeta)\xi\eta\zeta \\ N_3 &= \eta - 64 (1 - \xi - \eta - \zeta)\xi\eta\zeta \\ N_4 &= \zeta - 64 (1 - \xi - \eta - \zeta)\xi\eta\zeta \end{aligned} \]

  • Central Node (5):

    \[ N_5 = 256 (1 - \xi - \eta - \zeta)\xi\eta\zeta \]

10-node tetrahedron

The shape functions for the 10-node quadratic tetrahedron (T10) are defined using the barycentric coordinates. Let \(\lambda = 1 - \xi - \eta - \zeta\).

  • Corner Nodes (1-4):

    \[ \begin{aligned} N_1 &= (2\lambda - 1)\lambda \\ N_2 &= (2\xi - 1)\xi \\ N_3 &= (2\eta - 1)\eta \\ N_4 &= (2\zeta - 1)\zeta \end{aligned} \]

  • Midside Nodes (5-10):

    \[ \begin{aligned} N_5 &= 4\lambda\xi \\ N_6 &= 4\xi\eta \\ N_7 &= 4\eta\lambda \\ N_8 &= 4\lambda\zeta \\ N_9 &= 4\xi\zeta \\ N_{10} &= 4\eta\zeta \end{aligned} \]

Numerical integration

Integrals over the element volume are computed numerically using Gaussian quadrature. The following schemes are implemented in numgeo for tetrahedra. The volume of the parent element is \(1/6\).

1-point integration

1-point integration, degree of precision = 1

Integration point \(\xi\) \(\eta\) \(\zeta\) \(w\)
1 0.25 0.25 0.25 1.0

4-point integration

4-point integration, degree of precision = 3

Let \(a = 0.5854101966\) and \(b = 0.1381966011\)

Integration point \(\xi\) \(\eta\) \(\zeta\) \(w\)
1 \(b\) \(b\) \(b\) 0.25
2 \(a\) \(b\) \(b\) 0.25
3 \(b\) \(a\) \(b\) 0.25
4 \(b\) \(b\) \(a\) 0.25

5-point integration

5-point integration, degree of precision = 3

Integration point \(\xi\) \(\eta\) \(\zeta\) \(w\)
1 0.0 0.0 0.0 0.125
2 1.0 0.0 0.0 0.125
3 0.0 1.0 0.0 0.125
4 0.0 0.0 1.0 0.125
5 0.25 0.25 0.25 0.5

8-point integration

8-point integration, degree of precision = 3

Integration point \(\xi\) \(\eta\) \(\zeta\) \(w\)
1 0.0158359099 0.3280546967 0.3280546967 0.138527961
2 0.3280546967 0.0158359099 0.3280546967 0.138527961
3 0.3280546967 0.3280546967 0.0158359099 0.138527961
4 0.3280546967 0.3280546967 0.3280546967 0.138527961
5 0.6791431785 0.1069522739 0.1069522739 0.111472039
6 0.1069522739 0.6791431785 0.1069522739 0.111472039
7 0.1069522739 0.1069522739 0.6791431785 0.111472039
8 0.1069522739 0.1069522739 0.1069522739 0.111472039

11-point integration

11-point Gauss-Lobatto integration rule, degree of precision = 4

Integration point \(\xi\) \(\eta\) \(\zeta\) \(w\)
1 0.0 0.0 0.0 \(1/60\)
2 1.0 0.0 0.0 \(1/60\)
3 0.0 1.0 0.0 \(1/60\)
4 0.0 0.0 1.0 \(1/60\)
5 0.5 0.0 0.0 \(1/15\)
6 0.0 0.5 0.0 \(1/15\)
7 0.0 0.0 0.5 \(1/15\)
8 0.5 0.5 0.0 \(1/15\)
9 0.5 0.0 0.5 \(1/15\)
10 0.0 0.5 0.5 \(1/15\)
11 0.25 0.25 0.25 \(8/15\)

General remarks

  • 4-node tetrahedra elements: 1-point integration is the standard scheme. Higher-order rules like the 4-point scheme can be used for improved accuracy.
  • 5-node tetrahedra elements: This element requires at least a 4-point or 5-point rule to correctly integrate the energy from the internal bubble function.
  • 10-node tetrahedra elements: The 4-point or 5-point integration rules are suitable for this element. The 11-point rule provides higher precision.