Triangular elements

numgeo provides triangular elements with either 3 or 6 nodes:

Figure 1: Triangular elements implemented in numgeo

In addition to the number of nodes, these elements differ in the selection of shape functions used for numerical interpolation, as detailed in the following section. The triangular element with 4 nodes in Figure1 has 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 (and we thus consider it a 3-node element).

Shape functions

The triangular element with three nodes uses the following linear shape functions:

\[ N_1 = 1 - \xi - \eta ~~~;~~~ N_2 = \xi ~~~;~~~ N_3 = \eta \]

For the linear element with additional center node the shape functions read:

\[ \begin{aligned} N_1 &= 1 - \xi - \eta - \frac{27 \xi \eta \left( 1.0 - \xi - \eta \right)}{3} \\ N_2 &= \xi - \frac{27 \xi \eta \left( 1 - \xi - \eta \right)}{3} \\ N_3 &= \eta - \frac{27 \xi \eta \left( 1 - \xi - \eta \right)}{3} \\ N_4 &= 27 \xi \eta \left( 1 - \xi - \eta \right) \end{aligned} \]

And finally the shape functions for the six node triangular element:

\[ \begin{aligned} N_1 &= 2.0 \left( 0.5 - \xi - \eta \right) \left( 1.0 - \xi - \eta \right) \\ N_2 &= 2.0 \xi \left( \xi - 0.5 \right) \\ N_3 &= 2.0 \eta \left( \eta - 0.5 \right) \\ N_4 &= 4.0 \xi \left( 1.0 - \xi - \eta \right) \\ N_5 &= 4.0 \xi \eta \\ N_6 &= 4.0 \eta \left( 1.0 - \xi - \eta \right) \end{aligned} \]

Numerical integration

The following integration rules are implemented in numgeo:

• 1 point integration, degree of precision = 1

  Integration point	\(\xi\)	\(\eta\)	\(w\)
  1	\(\frac{1}{3}\)	\(\frac{1}{3}\)	1.0

• 3 point integration, degree of precision = 2

  Integration point	\(\xi\)	\(\eta\)	\(w\)
  1	\(\frac{1}{6}\)	\(\frac{1}{6}\)	\(\frac{1}{3}\)
  2	\(\frac{2}{3}\)	\(\frac{1}{6}\)	\(\frac{1}{3}\)
  3	\(\frac{1}{6}\)	\(\frac{2}{3}\)	\(\frac{1}{3}\)

• 6 point integration, degree of precision = 5

  Integration point	\(\xi\)	\(\eta\)	\(w\)
  1	0.0915762135098	0.0915762135098	0.1099517436553
  2	0.8168475729805	0.0915762135098	0.1099517436553
  3	0.0915762135098	0.8168475729805	0.1099517436553
  4	0.4459484909160	0.1081030181681	0.2233815896780
  5	0.4459484909160	0.4459484909160	0.2233815896780
  6	0.1081030181681	0.4459484909160	0.2233815896780

• 6 point integration with integration points at nodes (Gauss-Lobatto integration rule), degree of precision = 2

  Integration point	\(\xi\)	\(\eta\)	\(w\)
  1	0.0	0.0	0.1099517436553
  2	1.0	0.0	0.1099517436553
  3	0.0	1.0	0.1099517436553
  4	0.5	0.0	0.2233815896780
  5	0.5	0.5	0.2233815896780
  6	0.0	0.5	0.2233815896780

General remarks

• 1-point integration is unsuitable for triangular elements with a center node because the shape functions of this element evaluate to zero at the element's center.
• The Gauss-Lobatto rule is not recommended for integrating external surface loads, as the nodal forces it produces are inconsistent with the volume integration of tetrahedral elements. However, for contact integration, the Gauss-Lobatto rule has demonstrated greater numerical robustness compared to the standard 6-point integration rule.