pysdic.compute_segment_2_shape_functions

compute_segment_2_shape_functions(natural_coordinates, return_derivatives=False, *, default=0.0)

Compute the shape functions for a 2-node segment for given natural_coordinates \(\xi\).

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 \((N_{p},)\) or \((N_{p}, 1)\), 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 \([-1, 1]\)). By default, 0.0.

Returns:

• shape_functions (numpy.ndarray) – Shape functions evaluated at the given natural coordinates. The returned array has shape \((N_{p}, 2)\), 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 \((N_{p}, 2, 1)\).

Return type:

ndarray | Tuple[ndarray, ndarray]

Notes

A 2-node segment represented in the figure below has the following shape functions:

Node No.	\((\xi)\)	Shape Function \(N\)	First Derivative \((\frac{dN}{d\xi})\)
1	\(-1\)	\(N_1(\xi) = \frac{1}{2}(1 - \xi)\)	\(-\frac{1}{2}\)
2	\(1\)	\(N_2(\xi) = \frac{1}{2}(1 + \xi)\)	\(\frac{1}{2}\)

See also

pysdic.compute_segment_3_shape_functions

Shape functions for 3-node segment 1D-elements.

pysdic.get_segment_2_gauss_points

Gauss integration points for 2-node segment 1D-elements.

Examples

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

import numpy
from pysdic import compute_segment_2_shape_functions

coords = numpy.array([[-1.0], [0.0], [1.0], [1.5]])
shape_functions = compute_segment_2_shape_functions(coords)
print("Shape function values:")
print(shape_functions)

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