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Quadrilateral elements

numgeo provides quadrilateral elements with either 4, 8 or 9 nodes:

Figure 1: Quadrilateral 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 quadrilateral element with 5 nodes in Figure1 has one 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 4-node element).

Shape functions

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

\[ \begin{aligned} N_1 &= 0.25 \left(1.0 - \xi \right) \left(1.0 - \eta \right) \\ N_2 &= 0.25 \left(1.0 + \xi \right) \left(1.0 - \eta \right) \\ N_3 &= 0.25 \left(1.0 + \xi \right) \left(1.0 + \eta \right) \\ N_4 &= 0.25 \left(1.0 - \xi \right) \left(1.0 + \eta \right) \end{aligned} \]

For the linear element four nodes plus additional center node the shape functions read:

\[ \begin{aligned} N_1 &= 0.25 \left( 1.0 - \xi \right) \left( 1.0 - \eta \right) - 0.25 \left( 1.0 - \xi^2 \right) \left( 1.0 - \eta^2 \right) \\ N_2 &= 0.25 \left( 1.0 + \xi \right) \left( 1.0 - \eta \right) - 0.25 \left( 1.0 - \xi^2 \right) \left( 1.0 - \eta^2 \right) \\ N_3 &= 0.25 \left( 1.0 + \xi \right) \left( 1.0 + \eta \right) - 0.25 \left( 1.0 - \xi^2 \right) \left( 1.0 - \eta^2 \right) \\ N_4 &= 0.25 \left( 1.0 - \xi \right) \left( 1.0 + \eta \right) - 0.25 \left( 1.0 - \xi^2 \right) \left( 1.0 - \eta^2 \right) \\ N_5 &= \left( 1.0 - \xi^2 \right) \left( 1.0 - \eta^2 \right) \end{aligned} \]

The shape functions for the eight node quadrilateral element read:

\[ \begin{aligned} N_1 &= \frac{(1.0 - \xi)(1.0 - \eta)}{4.0} - 0.5 \left( N_8 + N_5 \right) \\ N_2 &= \frac{(1.0 + \xi)(1.0 - \eta)}{4.0} - 0.5 \left( N_5 + N_6 \right) \\ N_3 &= \frac{(1.0 + \xi)(1.0 + \eta)}{4.0} - 0.5 \left( N_6 + N_7 \right) \\ N_4 &= \frac{(1.0 - \xi)(1.0 + \eta)}{4.0} - 0.5 \left( N_7 + N_8 \right)\\ N_5 &= 0.5 \left( 1.0 - \xi^2 \right) \left( 1.0 - \eta \right) \\ N_6 &= 0.5 \left( 1.0 + \xi \right) \left( 1.0 - \eta^2 \right) \\ N_7 &= 0.5 \left( 1.0 - \xi^2 \right) \left( 1.0 + \eta \right) \\ N_8 &= 0.5 \left( 1.0 - \xi \right) \left( 1.0 - \eta^2 \right) \end{aligned} \]

And finally the shape functions for the nine node quadrilateral element:

\[ \begin{aligned} N_1 &= \frac{\xi \eta (1.0 - \xi)(1.0 - \eta)}{4.0} \\ N_2 &= -\frac{\xi \eta (1.0 + \xi)(1.0 - \eta)}{4.0} \\ N_3 &= \frac{\xi \eta (1.0 + \xi)(1.0 + \eta)}{4.0} \\ N_4 &= -\frac{\xi \eta (1.0 - \xi)(1.0 + \eta)}{4.0} \\ N_5 &= -0.5 \left( 1.0 - \xi^2 \right) \left( 1.0 - \eta \right) \eta \\ N_6 &= 0.5 \left( 1.0 - \eta^2 \right) \left( \xi + 1.0 \right) \xi \\ N_7 &= 0.5 \left( 1.0 - \xi^2 \right) \left( \eta + 1.0 \right) \eta \\ N_8 &= -0.5 \left( 1.0 - \eta^2 \right) \left( 1.0 - \xi \right) \xi \\ N_9 &= \left( 1.0 - \xi^2 \right) \left( 1.0 - \eta^2 \right) \end{aligned} \]

Numerical integration

The following integration rules are implemented in numgeo:

  • 1 point integration, degree of precision = 1

    Integration point \(\xi\) \(\eta\) \(w\)
    1 0.0 0.0 4.0

  • 4 point integration, degree of precision = 2

    Integration point \(\xi\) \(\eta\) \(w\)
    1 \(-\frac{1}{\sqrt{3}}\) \(-\frac{1}{\sqrt{3}}\) 1.0
    2 \(\frac{1}{\sqrt{3}}\) \(-\frac{1}{\sqrt{3}}\) 1.0
    3 \(\frac{1}{\sqrt{3}}\) \(\frac{1}{\sqrt{3}}\) 1.0
    4 \(-\frac{1}{\sqrt{3}}\) \(\frac{1}{\sqrt{3}}\) 1.0

  • 8 point "lumped" integration with integration points at nodes

    Integration point \(\xi\) \(\eta\) \(w\)
    1 -1.0 -1.0 0.5
    2 1.0 -1.0 0.5
    3 1.0 1.0 0.5
    4 -1.0 1.0 0.5
    5 0.0 -1.0 0.5
    6 1.0 0.0 0.5
    7 0.0 1.0 0.5
    8 -1.0 0.0 0.5

  • 9 point integration, degree of precision = 5

    Integration point \(\xi\) \(\eta\) \(w\)
    1 \(-\sqrt{0.6}\) \(-\sqrt{0.6}\) 0.308641975308642
    2 \(\sqrt{0.6}\) \(-\sqrt{0.6}\) 0.308641975308642
    3 \(\sqrt{0.6}\) \(\sqrt{0.6}\) 0.308641975308642
    4 \(-\sqrt{0.6}\) \(\sqrt{0.6}\) 0.308641975308642
    5 \(0.0\) \(-\sqrt{0.6}\) 0.493827160493827
    6 \(\sqrt{0.6}\) \(0.0\) 0.493827160493827
    7 \(0.0\) \(\sqrt{0.6}\) 0.493827160493827
    8 \(-\sqrt{0.6}\) \(0.0\) 0.493827160493827
    9 \(0.0\) \(0.0\) 0.790123456790123

  • 9 point integration with 8 integration points at nodes (Gauss-Lobatto integration rule), degree of precision = 3

    Integration point \(\xi\) \(\eta\) \(w\)
    1 -1.0 -1.0 0.308641975308642
    2 1.0 -1.0 0.308641975308642
    3 1.0 1.0 0.308641975308642
    4 -1.0 1.0 0.308641975308642
    5 0.0 -1.0 0.493827160493827
    6 1.0 0.0 0.493827160493827
    7 0.0 1.0 0.493827160493827
    8 -1.0 0.0 0.493827160493827
    9 0.0 0.0 0.790123456790123

General remarks

  • 1-point integration is unsuitable for quadrilateral elements with a center node because the shape functions of this element evaluate to zero at the element's center.
  • The 8-point "lumped" integration scheme is only used for the integration of contact stresses along element edges of 20-node hexahedral elements.