femlabpy.elements.triangles#

Triangular element kernels for constant-strain and scalar diffusion problems.

Workflow role#

This module covers the three-node triangle family in both structural and potential-flow style settings. It includes single-element stiffness and force evaluations, batch-friendly assembly wrappers, and the triangular mass matrix used in dynamic problems.

Public entry points#

ket3e and qet3e implement the structural constant-strain triangle.
ket3p and qet3p implement the scalar potential variant.
kt3e, qt3e, kt3p, and qt3p assemble those kernels globally.
met3e and mt3e provide mass-matrix support for structural triangles.

Functions#

`ket3e`(Xe, Ge)	Compute the element stiffness matrix for a 3-node triangular element (CST).
`ket3p`(Xe, Ge)	Compute the conductivity matrix for a 3-node scalar potential triangle.
`kt3e`(K, T, X, G)	Assemble T3 (CST) element stiffness matrices into global stiffness matrix.
`kt3p`(K, T, X, G)	Assemble T3 conductivity matrices into a global scalar system.
`met3e`(Xe, Ge, *[, lumped])	Compute the element mass matrix for a 3-node triangular (CST) element.
`mt3e`(M, T, X, G, *[, lumped])	Assemble T3 element mass matrices into the global mass matrix.
`qet3e`(Xe, Ge, Ue)	Compute stresses and strains for a single T3 element.
`qet3p`(Xe, Ge, Ue)	Recover gradients and fluxes for one 3-node scalar potential triangle.
`qt3e`(q, T, X, G, u)	Compute element stresses/strains for all T3 elements and assemble internal forces.
`qt3p`(q, T, X, G, u)	Recover T3 scalar gradients and assemble equivalent nodal fluxes.

Function Reference#

femlabpy.elements.triangles.ket3e(Xe, Ge)[source]#

Compute the element stiffness matrix for a 3-node triangular element (CST).

This function computes the 6x6 stiffness matrix for a constant strain triangle (T3/CST) element under plane stress or plane strain conditions.

Parameters:

Xearray_like, shape (3, 2): Nodal coordinates of the triangle. Each row contains [x, y] coordinates of a node. Node ordering: counter-clockwise.
Gearray_like: Material properties vector. Ge[0] is Young’s modulus, Ge[1] is Poisson’s ratio, and the optional Ge[2] selects the formulation: 1 or omitted for plane stress and 2 for plane strain.

Returns:

Kendarray, shape (6, 6): Element stiffness matrix. DOF ordering: [u1, v1, u2, v2, u3, v3]

Notes

The CST element assumes constant strain throughout the element. For better accuracy, use finer meshes or higher-order elements.

Examples

>>> import numpy as np
>>> from femlabpy import ket3e
>>> # Right triangle with unit sides
>>> Xe = np.array([[0, 0], [1, 0], [0, 1]])
>>> Ge = np.array([200e9, 0.3])  # Steel, plane stress
>>> Ke = ket3e(Xe, Ge)
>>> Ke.shape
(6, 6)

femlabpy.elements.triangles.ket3p(Xe, Ge)[source]#

Compute the conductivity matrix for a 3-node scalar potential triangle.

Parameters:

Xe:: Triangle coordinates with shape (3, 2).
Ge:: Material row [k] or [k, b] where k is conductivity and b is an optional reaction term.

Returns:

ndarray: 3 x 3 element conductivity matrix.

femlabpy.elements.triangles.kt3e(K, T, X, G)[source]#

Assemble T3 (CST) element stiffness matrices into global stiffness matrix.

This function loops over all triangular elements and assembles their contributions into the global stiffness matrix K.

Parameters:

Kndarray or sparse matrix, shape (ndof, ndof): Global stiffness matrix (modified in place).
Tarray_like, shape (nel, 4): Element topology matrix. Each row: [n1, n2, n3, mat_id] where n1, n2, n3 are 1-based node indices and mat_id is the material index into G.
Xarray_like, shape (nn, 2): Nodal coordinates matrix. Each row: [x, y].
Garray_like, shape (nmat, 2) or (nmat, 3): Material properties matrix. Each row: [E, nu] or [E, nu, type]. type=1: plane stress (default), type=2: plane strain.

Returns:

Kndarray or sparse matrix: Updated global stiffness matrix.

Notes

Node numbering in T is 1-based (MATLAB/Fortran convention).
Supports both dense and scipy.sparse matrices.
Uses vectorized operations for efficiency.

Examples

>>> from femlabpy import init, kt3e
>>> K, q, p, C, P, S = init(nn=10, nd=2)
>>> # T: element connectivity, X: node coords, G: materials
>>> K = kt3e(K, T, X, G)

femlabpy.elements.triangles.kt3p(K, T, X, G)[source]#

Assemble T3 conductivity matrices into a global scalar system.

Parameters:

K:: Global conductivity matrix.
T:: Topology table [n1, n2, n3, mat_id].
X:: Nodal coordinates.
G:: Material table with conductivity rows.

Returns:

ndarray or sparse matrix: Updated global conductivity matrix.

femlabpy.elements.triangles.met3e(Xe, Ge, *, lumped: bool = False)[source]#

Compute the element mass matrix for a 3-node triangular (CST) element.

Consistent (6x6):: M = (rho * t * A / 12) * [[2,1,1],[1,2,1],[1,1,2]] tensor I_2
Lumped (diagonal):: M = (rho * t * A / 3) * I_6

Parameters:

Xearray_like, shape (3, 2): Nodal coordinates.
Gearray_like: Material row. Thickness t is taken from Ge[3] if present (default 1). Density rho is taken from Ge[4] if present (default 1).

Returns:

Mendarray, shape (6, 6)

femlabpy.elements.triangles.mt3e(M, T, X, G, *, lumped: bool = False)[source]#

Assemble T3 element mass matrices into the global mass matrix.

Parameters:

Mndarray or sparse, shape (ndof, ndof): Global mass matrix (modified in place).
Tarray_like, shape (nel, 4): Topology table [n1, n2, n3, mat_id].
Xarray_like, shape (nn, 2): Nodal coordinates.
Garray_like: Material table.
lumpedbool: If True, assemble lumped mass.

Returns:

Mndarray or sparse: Updated global mass matrix.

femlabpy.elements.triangles.qet3e(Xe, Ge, Ue)[source]#

Compute stresses and strains for a single T3 element.

Parameters:

Xearray_like, shape (3, 2): Nodal coordinates [x, y] for each node.
Gearray_like: Material properties: [E, nu] or [E, nu, type]. type=1: plane stress, type=2: plane strain.
Uearray_like, shape (6,): Element nodal displacements [u1, v1, u2, v2, u3, v3].

Returns:

qendarray, shape (6, 1): Element internal force vector.
Sendarray, shape (3,): Stress components [sxx, syy, txy].
Eendarray, shape (3,): Strain components [exx, eyy, gxy].

Examples

>>> qe, stress, strain = qet3e(Xe, Ge, Ue)
>>> print(f"sxx = {stress[0]:.2f}, syy = {stress[1]:.2f}, txy = {stress[2]:.2f}")

femlabpy.elements.triangles.qet3p(Xe, Ge, Ue)[source]#

Recover gradients and fluxes for one 3-node scalar potential triangle.

Parameters:

Xe:: Triangle coordinates with shape (3, 2).
Ge:: Material row [k] or [k, b].
Ue:: Element nodal potentials.

Returns:

tuple[ndarray, ndarray, ndarray]: Element flux vector, flux components, and gradient components.

femlabpy.elements.triangles.qt3e(q, T, X, G, u)[source]#

Compute element stresses/strains for all T3 elements and assemble internal forces.

Parameters:

qndarray, shape (ndof, 1): Global internal force vector (modified in place).
Tarray_like, shape (nel, 4): Element topology: [n1, n2, n3, mat_id] per row (1-based).
Xarray_like, shape (nn, 2): Nodal coordinates.
Garray_like, shape (nmat, 2+): Material properties: [E, nu, type] per material.
uarray_like, shape (nn, 2) or (ndof,): Nodal displacement vector.

Returns:

qndarray: Updated internal force vector.
Sndarray, shape (nel, 3): Stress at each element centroid: [sxx, syy, txy].
Endarray, shape (nel, 3): Strain at each element centroid: [exx, eyy, gxy].

Examples

>>> q, stresses, strains = qt3e(q, T, X, G, u)
>>> max_stress = np.max(np.abs(stresses[:, 0]))  # max sxx

femlabpy.elements.triangles.qt3p(q, T, X, G, u)[source]#

Recover T3 scalar gradients and assemble equivalent nodal fluxes.

Parameters:

q:: Global nodal flux vector.
T:: Topology table [n1, n2, n3, mat_id].
X:: Nodal coordinates.
G:: Material table with conductivity rows.
u:: Global nodal potentials.

Returns:

tuple[ndarray, ndarray, ndarray]: Updated nodal flux vector, element fluxes, and element gradients.