pysdic.compute_triangle_6_shape_functions

compute_triangle_6_shape_functions(natural_coordinates, return_derivatives=False, *, default=0.0)

Compute the shape functions for a 6-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 \((N_{p}, 6)\), 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 \((N_{p}, 6, 2)\).

Raises:

• TypeError –

  • If the input natural_coordinates cannot be converted to a numpy array. - If the input return_derivatives is not a boolean. - If the input default is not a real number.

• ValueError –

  • If the input natural_coordinates does not have shape \((N_{p}, 2)\).

Return type:

ndarray | Tuple[ndarray, ndarray]

Notes

A 6-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	\((0, 0)\)	\(N_1(\xi, \eta) = (1 - \xi - \eta)(1 - 2\xi - 2\eta)\)	\((-3 + 4\xi + 4\eta, -3 + 4\xi + 4\eta)\)
2	\((1, 0)\)	\(N_2(\xi, \eta) = \xi(2\xi - 1)\)	\((4\xi - 1, 0)\)
3	\((0, 1)\)	\(N_3(\xi, \eta) = \eta(2\eta - 1)\)	\((0, 4\eta - 1)\)
4	\((0.5, 0)\)	\(N_4(\xi, \eta) = 4\xi(1 - \xi - \eta)\)	\((4 - 8\xi - 4\eta, -4\xi)\)
5	\((0.5, 0.5)\)	\(N_5(\xi, \eta) = 4\xi\eta\)	\((4\eta, 4\xi)\)
6	\((0, 0.5)\)	\(N_6(\xi, \eta) = 4\eta(1 - \xi - \eta)\)	\((-4\eta, 4 - 4\xi - 8\eta)\)

See also

pysdic.compute_triangle_3_shape_functions

Shape functions for 3-node triangle 2D-elements.

pysdic.get_triangle_6_gauss_points

Gauss integration points for 6-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_6_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_6_shape_functions(coords)
print("Shape function values:")
print(shape_functions)

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