pysdic.compute_shape_functions

compute_shape_functions(natural_coordinates, element_type, return_derivatives=False, *, default=0.0)

Compute the shape functions for a given element_type at specified natural_coordinates.

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 \((N_{p}, K)\), where \(N_{p}\) is the number of points to evaluate and \(K\) is the topological dimension of the element.

• element_type (str) –

  Type of the finite element. Supported types are:

  • 'segment_2': 2-node segment element.

  • 'segment_3': 3-node segment element.

  • 'triangle_3': 3-node triangle element.

  • 'triangle_6': 6-node triangle element.

  • 'quadrangle_4': 4-node quadrangle element.

  • 'quadrangle_8': 8-node quadrangle element.

• 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 the valid range for the specified element type). By default, 0.0.

Returns:

• shape_functions (numpy.ndarray) – Shape functions evaluated at the given natural coordinates. The returned array has shape \((N_{p}, N_{vpe})\), where each row corresponds to a point and each column to a node, and \(N_{vpe}\) is the number of vertices per element for the specified element type.

• 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 \((N_{p}, N_{vpe}, K)\).

Return type:

ndarray | Tuple[ndarray, ndarray]

See also

pysdic.segment_2_shape_functions

Shape functions for 2-node segment 1D-elements.

pysdic.compute_segment_3_shape_functions

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

pysdic.compute_triangle_3_shape_functions

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

pysdic.compute_triangle_6_shape_functions

Shape functions for 6-node triangle 2D-elements.

pysdic.compute_quadrangle_4_shape_functions

Shape functions for 4-node quadrangle 2D-elements.

pysdic.compute_quadrangle_8_shape_functions

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

Examples

Compute shape functions for a 6-node triangle element for 3 valid points and 1 invalid point:

import numpy
from pysdic import compute_shape_functions

coords = numpy.array([[0.0, 0.0], [0.5, 0.0], [0.0, 0.5], [0.7, 0.5]])
shape_funcs = compute_shape_functions(coords, element_type='triangle_6')
print("Shape function values for triangle_6:")
print(shape_funcs)

Shape function values for triangle_6:
[[ 1.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  1.  0.  0.]
 [ 0.  0.  0.  0.  1.  0.]
 [ 0.  0.  0.  0.  0.  0.]]