pysdic.compute_quadrangle_8_shape_functions#

compute_quadrangle_8_shape_functions(natural_coordinates, return_derivatives=False, *, default=0.0)[source]#

Compute the shape functions for a 8-node quadrangle for given natural_coordinates \((\xi, \eta)\).

Note

Input natural_coordinates will be converted to numpy.float64.
Output arrays will be numpy.float64.

Parameters:

natural_coordinates (ArrayLike) – Natural coordinates where to evaluate the shape functions. The array must have shape \((N_{p}, 2)\), where \(N_{p}\) is the number of points to evaluate.
return_derivatives (bool, optional) – If True, the function will also return the first derivatives of the shape functions with respect to the natural coordinates. By default, False.
default (Real, optional) – Default value to assign to shape functions when the input natural coordinates are out of the valid range (i.e., not in \(\xi, \eta \in [-1, 1]\)). By default, 0.0.

Returns:

shape_functions (numpy.ndarray) – Shape functions evaluated at the given natural coordinates. The returned array has shape \((N_{p}, 8)\), where each row corresponds to a point and each column to a node.
shape_function_derivatives (numpy.ndarray, optional) – If return_derivatives is True, the function also returns an array of the first derivatives of the shape functions with respect to the natural coordinates. The returned array has shape \((N_{p}, 8, 2)\).

Return type:

ndarray | Tuple[ndarray, ndarray]

Notes

An 8-node quadrangle represented in the figure below has the following shape functions:

Node No.	\((\xi, \eta)\)	Shape Function \(N\)	First Derivative \((\frac{dN}{d\xi}, \frac{dN}{d\eta})\)
1	\((-1, -1)\)	\(N_1(\xi, \eta) = \frac{1}{4}(1 - \xi)(1 - \eta)(- \xi - \eta - 1)\)	\((\frac{1}{4}(1 - \eta)(2\xi + \eta), \frac{1}{4}(1 - \xi)(\xi + 2\eta))\)
2	\((1, -1)\)	\(N_2(\xi, \eta) = \frac{1}{4}(1 + \xi)(1 - \eta)(\xi - \eta - 1)\)	\((\frac{1}{4}(1 - \eta)(2\xi - \eta), -\frac{1}{4}(1 + \xi)(\xi - 2\eta))\)
3	\((1, 1)\)	\(N_3(\xi, \eta) = \frac{1}{4}(1 + \xi)(1 + \eta)(\xi + \eta - 1)\)	\((\frac{1}{4}(1 + \eta)(2\xi + \eta), \frac{1}{4}(1 + \xi)(\xi + 2\eta))\)
4	\((-1, 1)\)	\(N_4(\xi, \eta) = \frac{1}{4}(1 - \xi)(1 + \eta)(- \xi + \eta - 1)\)	\((\frac{1}{4}(1 + \eta)(2\xi - \eta), \frac{1}{4}(1 - \xi)(- \xi + 2\eta))\)
5	\((0, -1)\)	\(N_5(\xi, \eta) = \frac{1}{2}(1 - \xi^2)(1 - \eta)\)	\((-\xi(1 - \eta), -\frac{1}{2}(1 - \xi^2))\)
6	\((1, 0)\)	\(N_6(\xi, \eta) = \frac{1}{2}(1 + \xi)(1 - \eta^2)\)	\((\frac{1}{2}(1 - \eta^2), -\eta(1 + \xi))\)
7	\((0, 1)\)	\(N_7(\xi, \eta) = \frac{1}{2}(1 - \xi^2)(1 + \eta)\)	\((-\xi(1 + \eta), \frac{1}{2}(1 - \xi^2))\)
8	\((-1, 0)\)	\(N_8(\xi, \eta) = \frac{1}{2}(1 - \xi)(1 - \eta^2)\)	\((-\frac{1}{2}(1 - \eta^2), -\eta(1 - \xi))\)

See also

pysdic.compute_quadrangle_4_shape_functions: Shape functions for 4-node quadrangle 2D-elements.
pysdic.get_quadrangle_8_gauss_points: Gauss integration points for 8-node quadrangle 2D-elements.

Examples

Compute shape functions without derivatives for 3 valid points and 1 invalid point:

import numpy
from pysdic import compute_quadrangle_8_shape_functions

coords = numpy.array([[-1.0, -1.0], [0.0, -1.0], [1.0, 0.0], [1.5, 0.5]])
shape_functions = compute_quadrangle_8_shape_functions(coords)
print("Shape function values:")
print(shape_functions)

Shape function values:
[[ 1. -0. -0. -0.  0.  0.  0.  0.]
 [ 0.  0. -0. -0.  1.  0.  0.  0.]
 [-0.  0.  0. -0.  0.  1.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.]]

pysdic.compute_quadrangle_8_shape_functions#

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