Quad4#

class planestress.analysis.finite_elements.quad4.Quad4(el_idx: int, el_tag: int, coords: npt.NDArray[np.float64], node_idxs: list[int], material: Material, orientation: bool)[source]#

Bases: FiniteElement

Class for a four-noded linear quadrilateral element.

Methods

`b_matrix_jacobian`	Calculates the B matrix and jacobian at an isoparametric point for a Quad4.
`calculate_element_stresses`	Calculates various results for a Quad4 element given nodal displacements.
`element_col_indexes`	Returns the column indexes fore the global stiffness matrix.
`element_load_vector`	Assembles the load vector for a Quad4 element.
`element_row_indexes`	Returns the row indexes fore the global stiffness matrix.
`element_stiffness_matrix`	Assembles the stiffness matrix for a Quad4 element.
`extrapolate_gauss_points_to_nodes`	Returns the extrapolation matrix for a Quad4 element.
`get_polygon_coordinates`	Returns a list of coordinates and indexes that define the element exterior.
`get_triangulation`	Returns a list of triangle indexes for a Quad4 element.
`shape_functions`	Returns the shape functions at a point for a Quad4 element.

__init__(el_idx: int, el_tag: int, coords: npt.NDArray[np.float64], node_idxs: list[int], material: Material, orientation: bool) → None[source]#

Inits the Quad4 class.

Parameters:

el_idx (int) – Element index.
el_tag (int) – Element mesh tag.
coords (npt.NDArray[np.float64]) – A 2 x 4 numpy.ndarray of coordinates defining the element, i.e. [[x1, x2, x3, x4], [y1, y2, y3, y4]].
node_idxs (list[int]) – A list of node indexes defining the element, i.e. [idx1, idx2, idx3, idx4].
material (Material) – Material of the element.
orientation (bool) – If True the element is oriented correctly, if False the element’s nodes will need reordering.

static shape_functions(iso_coords: tuple[float, float]) → ndarray[Any, dtype[float64]][source]#

Returns the shape functions at a point for a Quad4 element.

Parameters:: iso_coords (tuple[float, float]) – Location of the point in isoparametric coordinates.
Returns:: The values of the shape functions [N1, N2, N3, N4].
Return type:: ndarray[Any, dtype[float64]]

static b_matrix_jacobian(iso_coords: tuple[float, float], coords: tuple[float, ...]) → tuple[npt.NDArray[np.float64], float][source]#

Calculates the B matrix and jacobian at an isoparametric point for a Quad4.

Parameters:

iso_coords (tuple[float, float]) – Location of the point in isoparametric coordinates.
coords (tuple[float, ...]) – Flattened list of coordinates.

Raises:

RuntimeError – If the jacobian is less than zero.

Returns:

Derivatives of the shape function (B matrix) and value of the jacobian, (b_mat, j).

Return type:

tuple[npt.NDArray[np.float64], float]

element_stiffness_matrix() → ndarray[Any, dtype[float64]][source]#

Assembles the stiffness matrix for a Quad4 element.

Returns:: Element stiffness matrix.
Return type:: ndarray[Any, dtype[float64]]

element_load_vector(acceleration_field: tuple[float, float]) → ndarray[Any, dtype[float64]][source]#

Assembles the load vector for a Quad4 element.

Parameters:: acceleration_field (tuple[float, float]) – Acceleration field (a_x, a_y).
Returns:: Element load vector.
Return type:: ndarray[Any, dtype[float64]]

calculate_element_stresses(u: ndarray[Any, dtype[float64]]) → ElementResults[source]#

Calculates various results for a Quad4 element given nodal displacements.

Calculates the following:

Stress components at the nodes (:math`sigma_{xx}`, :math`sigma_{yy}`, :math`sigma_{xy}`).
TODO

Parameters:: u (ndarray[Any, dtype[float64]]) – Displacement vector for the element.
Returns:: Element results object.
Return type:: ElementResults

extrapolate_gauss_points_to_nodes() → ndarray[Any, dtype[float64]][source]#

Returns the extrapolation matrix for a Quad4 element.

Returns:: Extrapolation matrix.
Return type:: ndarray[Any, dtype[float64]]

get_polygon_coordinates() → tuple[list[int], ndarray[Any, dtype[float64]]][source]#

Returns a list of coordinates and indexes that define the element exterior.

Returns:: List of node indexes and exterior coordinates
Return type:: tuple[list[int], ndarray[Any, dtype[float64]]]

get_triangulation() → list[tuple[int, int, int]][source]#

Returns a list of triangle indexes for a Quad4 element.

Returns:: List of triangle indexes.
Return type:: list[tuple[int, int, int]]

element_col_indexes() → list[int]#

Returns the column indexes fore the global stiffness matrix.

Returns:: Column indexes.
Return type:: list[int]

element_row_indexes() → list[int]#

Returns the row indexes fore the global stiffness matrix.

Returns:: Row indexes.
Return type:: list[int]