Tri3

class planestress.analysis.finite_element.Tri3(el_idx: int, el_tag: int, coords: npt.NDArray[np.float64], node_idxs: list[int], material: Material)

Bases: TriangularElement

Class for a three-noded linear triangular element.

Methods

b_matrix_jacobian	Calculates the B matrix and jacobian at an isoparametric point.
element_load_vector	Assembles the load vector for the element.
element_stiffness_matrix	Assembles the stiffness matrix for the element.
gauss_points	Gaussian weights and locations for n_point Gaussian integration.
get_element_results	Calculates various results for the finite element given nodal displacements.
get_triangulation	Returns a list of triangle indices for a Tri3 element.
iso_to_global	Converts a point in isoparametric coordinates to global coordinates.
nodal_isoparametric_coordinates	Returns the values of the isoparametric coordinates at the nodes.
shape_functions	Returns the shape functions at a point for a Tri3 element.
shape_functions_derivatives	Returns the derivatives of the shape functions at a point for a Tri3 element.

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

Inits the Tri3 class.

Parameters:

• el_idx (int) – Element index.

• el_tag (int) – Element mesh tag.

• coords (npt.NDArray[np.float64]) – A 2 x 3 numpy.ndarray of coordinates defining the element, i.e. [[x1, x2, x3], [y1, y2, y3]].

• node_idxs (list[int]) – A list of node indexes defining the element, i.e. [idx1, idx2, idx3].

• material (Material) – Material of the element.

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

Returns the shape functions at a point for a Tri3 element.

Parameters:

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

Returns:

The values of the shape functions [N1, N2, N3].

Return type:

ndarray[Any, dtype[float64]]

static shape_functions_derivatives(iso_coords: tuple[float, float, float]) → ndarray[Any, dtype[float64]]

Returns the derivatives of the shape functions at a point for a Tri3 element.

Parameters:

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

Returns:

The partial derivatives of the shape functions.

Return type:

ndarray[Any, dtype[float64]]

static nodal_isoparametric_coordinates() → ndarray[Any, dtype[float64]]

Returns the values of the isoparametric coordinates at the nodes.

Returns:

Values of the isoparametric coordinates at the nodes.

Return type:

ndarray[Any, dtype[float64]]

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

Returns a list of triangle indices for a Tri3 element.

Returns:

List of triangle indices.

Return type:

list[tuple[int, int, int]]

b_matrix_jacobian(iso_coords: tuple[float, float, float]) → tuple[ndarray[Any, dtype[float64]], float]

Calculates the B matrix and jacobian at an isoparametric point.

Parameters:

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

Returns:

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

Return type:

tuple[ndarray[Any, dtype[float64]], float]

element_load_vector() → ndarray[Any, dtype[float64]]

Assembles the load vector for the element.

Returns:

Element load vector.

Return type:

ndarray[Any, dtype[float64]]

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

Assembles the stiffness matrix for the element.

Returns:

Element stiffness matrix.

Return type:

ndarray[Any, dtype[float64]]

gauss_points(n_points: int) → ndarray[Any, dtype[float64]]

Gaussian weights and locations for n_point Gaussian integration.

Parameters:

n_points (int) – Number of gauss points.

Raises:

ValueError – If n_points is not 1, 3, 4 or 6.

Returns:

Gaussian weights and locations. For each gauss point - [weight, eta, xi, zeta].

Return type:

ndarray[Any, dtype[float64]]

get_element_results(u: ndarray[Any, dtype[float64]]) → ElementResults

Calculates various results for the finite element given nodal displacements.

Calculates the following:

• Stresses at nodes

• TODO

Parameters:

u (ndarray[Any, dtype[float64]]) – Displacement vector for the element.

Returns:

ElementResults object.

Return type:

ElementResults

iso_to_global(iso_coords: tuple[float, float, float]) → tuple[float, float]

Converts a point in isoparametric coordinates to global coordinates.

Parameters:

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

Returns:

Location of the point in global coordinates (x, y).

Return type:

tuple[float, float]