pysdic.get_quadrangle_8_gauss_points#

get_quadrangle_8_gauss_points(return_weights=False)[source]#

Get the natural coordinates \((\xi, \eta)\) and weights of Gauss quadrature points for an 8-node quadrangle element.

Parameters:

return_weights (bool, optional) – If True, the function will also return the weights associated with each Gauss point. Default is False.

Returns:

  • gauss_points (numpy.ndarray) – Natural coordinates of the Gauss points. The returned array has shape (9, 2).

  • weights (numpy.ndarray, optional) – If return_weights is True, the function also returns an array of weights with shape (9,) associated with each Gauss point.

Return type:

ndarray | Tuple[ndarray, ndarray]

Notes

An 8-node quadrangle element uses the following Gauss points and weights for numerical integration:

Point No.

\((\xi, \eta)\)

Weight \(w\)

1

\((0,0)\)

\(\frac{64}{81}\)

2

\((\sqrt(\frac{3}{5}), \sqrt(\frac{3}{5}))\)

\(\frac{25}{81}\)

3

\((-\sqrt{\frac{3}{5}}, \sqrt{\frac{3}{5}})\)

\(\frac{25}{81}\)

4

\((-\sqrt{\frac{3}{5}}, -\sqrt{\frac{3}{5}})\)

\(\frac{25}{81}\)

5

\((\sqrt{\frac{3}{5}}, -\sqrt{\frac{3}{5}})\)

\(\frac{25}{81}\)

6

\((0, \sqrt{\frac{3}{5}})\)

\(\frac{40}{81}\)

7

\((-\sqrt{\frac{3}{5}}, 0)\)

\(\frac{40}{81}\)

8

\((0, -\sqrt{\frac{3}{5}})\)

\(\frac{40}{81}\)

9

\((\sqrt{\frac{3}{5}}, 0)\)

\(\frac{40}{81}\)
Gauss quadrature points for 8-node quadrangle element

See also

pysdic.compute_quadrangle_8_shape_functions

Shape functions for 8-node quadrangle 2D-elements.

Examples

Get the Gauss points for an 8-node quadrangle element:

1import numpy
2from pysdic import get_quadrangle_8_gauss_points
3
4gauss_points = get_quadrangle_8_gauss_points()
5print("Gauss points:")
6print(gauss_points)

Gauss points:
[[ 0.          0.        ]
 [ 0.77459667  0.77459667]
 [-0.77459667  0.77459667]
 [-0.77459667 -0.77459667]
 [ 0.77459667 -0.77459667]
 [ 0.          0.77459667]
 [-0.77459667  0.        ]
 [ 0.         -0.77459667]
 [ 0.77459667  0.        ]]