pysdic.compute_quadrangle_4_shape_functions

compute_quadrangle_4_shape_functions(natural_coordinates, return_derivatives=False, *, default=0.0)

Compute the shape functions for a 4-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 \((N_{p}, 4)\), 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 \((N_{p}, 4, 2)\).

Return type:

ndarray | Tuple[ndarray, ndarray]

Notes

A 4-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	\((-1, -1)\)	\(N_1(\xi, \eta) = \frac{1}{4}(1 - \xi)(1 - \eta)\)	\((\frac{1}{4}(\eta - 1), \frac{1}{4}(\xi - 1))\)
2	\((1, -1)\)	\(N_2(\xi, \eta) = \frac{1}{4}(1 + \xi)(1 - \eta)\)	\((\frac{1}{4}(1 - \eta), -\frac{1}{4}(1 + \xi))\)
3	\((1, 1)\)	\(N_3(\xi, \eta) = \frac{1}{4}(1 + \xi)(1 + \eta)\)	\((\frac{1}{4}(1 + \eta), \frac{1}{4}(1 + \xi))\)
4	\((-1, 1)\)	\(N_4(\xi, \eta) = \frac{1}{4}(1 - \xi)(1 + \eta)\)	\((-\frac{1}{4}(1 + \eta), \frac{1}{4}(1 - \xi))\)

See also

pysdic.compute_quadrangle_8_shape_functions

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

pysdic.get_quadrangle_4_gauss_points

Gauss integration points for 4-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_4_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_4_shape_functions(coords)
print("Shape function values:")
print(shape_functions)

Shape function values:
[[1.   0.   0.   0.  ]
 [0.5  0.5  0.   0.  ]
 [0.   0.5  0.5  0.  ]
 [0.   0.   0.   0.  ]]