pysdic.compute_triangle_3_shape_functions#

compute_triangle_3_shape_functions(natural_coordinates, return_derivatives=False, *, default=0.0)[source]#

Compute the shape functions for a 3-node triangle 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 \(\xi, \eta \in [0, 1]\) with \(\xi + \eta \leq 1\)). By default, 0.0.

Returns:

shape_functions (numpy.ndarray) – Shape functions evaluated at the given natural coordinates. The returned array has shape \((N_{p}, 3)\), 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 \((N_{p}, 3, 2)\).

Return type:

ndarray | Tuple[ndarray, ndarray]

Notes

A 3-node triangle 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	\((0, 0)\)	\(N_1(\xi, \eta) = 1 - \xi - \eta\)	\((-1, -1)\)
2	\((1, 0)\)	\(N_2(\xi, \eta) = \xi\)	\((1, 0)\)
3	\((0, 1)\)	\(N_3(\xi, \eta) = \eta\)	\((0, 1)\)

See also

pysdic.compute_triangle_6_shape_functions: Shape functions for 6-node triangle 2D-elements.
pysdic.get_triangle_3_gauss_points: Gauss integration points for 3-node triangle 2D-elements.

Examples

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

import numpy
from pysdic import compute_triangle_3_shape_functions

coords = numpy.array([[0.0, 0.0], [0.5, 0.0], [0.0, 0.5], [0.7, 0.5]])
shape_functions = compute_triangle_3_shape_functions(coords)
print("Shape function values:")
print(shape_functions)

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

pysdic.compute_triangle_3_shape_functions#

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