femlabpy.elements.quads#

Quadrilateral element kernels for elastic, plastic, scalar, and dynamic cases.

Workflow role#

This is the largest element module in the project. It contains the bilinear Q4 family used in most two-dimensional examples, including elastic plane-stress and plane-strain kernels, scalar diffusion kernels, elastoplastic updates, and consistent or lumped mass matrices.

Public entry points#

keq4e and qeq4e are the core elastic structural kernels.
keq4p and qeq4p cover the scalar potential form.
keq4eps and keq4epe with their matching q functions implement plane-stress and plane-strain plasticity.
kq4*, qq4*, and mq4e are the assembled global wrappers.

Functions#

`keq4e`(Xe, Ge)	Compute element stiffness matrix for a 4-node quadrilateral (Q4) element.
`keq4epe`(Xe, Ge, Se, Ee[, mtype])	Compute the consistent tangent of a plane-strain elastoplastic Q4 element.
`keq4eps`(Xe, Ge, Se, Ee[, mtype])	Compute the consistent tangent of a plane-stress elastoplastic Q4 element.
`keq4p`(Xe, Ge)	Compute the scalar conductivity matrix for a 4-node quadrilateral.
`kq4e`(K, T, X, G)	Assemble Q4 element stiffness matrices into global stiffness matrix.
`kq4epe`(K, T, X, G, S, E[, mtype])	Assemble plane-strain plastic Q4 tangent matrices into the global matrix.
`kq4eps`(K, T, X, G, S, E[, mtype])	Assemble plane-stress plastic Q4 tangent matrices into the global matrix.
`kq4p`(K, T, X, G)	Assemble Q4 scalar conductivity matrices into the global system.
`meq4e`(Xe, Ge, *[, lumped])	Compute the element mass matrix for a 4-node quadrilateral (Q4) element.
`mq4e`(M, T, X, G, *[, lumped])	Assemble Q4 element mass matrices into the global mass matrix.
`qeq4e`(Xe, Ge, Ue)	Compute stresses and strains at Gauss points for a single Q4 element.
`qeq4epe`(Xe, Ge, Ue, Se, Ee[, mtype])	Update plane-strain elastoplastic Q4 response at the four Gauss points.
`qeq4eps`(Xe, Ge, Ue, Se, Ee[, mtype])	Update plane-stress elastoplastic Q4 response at the four Gauss points.
`qeq4p`(Xe, Ge, Ue)	Recover gradients and fluxes for one Q4 scalar potential element.
`qq4e`(q, T, X, G, u)	Compute stresses/strains for all Q4 elements and assemble internal forces.
`qq4epe`(q, T, X, G, u, S, E[, mtype])	Compute plane-strain plastic Q4 internal forces and updated history fields.
`qq4eps`(q, T, X, G, u, S, E[, mtype])	Compute plane-stress plastic Q4 internal forces and updated history fields.
`qq4p`(q, T, X, G, u)	Recover Q4 scalar gradients and assemble equivalent nodal fluxes.

Function Reference#

femlabpy.elements.quads.keq4e(Xe, Ge)[source]#

Compute element stiffness matrix for a 4-node quadrilateral (Q4) element.

Uses 2x2 Gauss quadrature for numerical integration.

Parameters:

Xearray_like, shape (4, 2): Nodal coordinates [x, y] for each node. Node ordering: counter-clockwise starting from bottom-left.
Gearray_like: Material properties: [E, nu] or [E, nu, type]. - E: Young’s modulus - nu: Poisson’s ratio - type: 1=plane stress (default), 2=plane strain

Returns:

Kendarray, shape (8, 8): Element stiffness matrix. DOF ordering: [u1, v1, u2, v2, u3, v3, u4, v4]

Notes

The Q4 element uses bilinear shape functions: Ni = 0.25 * (1 + xi_i * xi) * (1 + eta_i * eta)

Examples

>>> from femlabpy import keq4e
>>> Xe = np.array([[0,0], [1,0], [1,1], [0,1]])
>>> Ge = np.array([200e9, 0.3])  # E, nu for plane stress
>>> Ke = keq4e(Xe, Ge)
>>> Ke.shape
(8, 8)

femlabpy.elements.quads.keq4epe(Xe, Ge, Se, Ee, mtype: int = 1)[source]#

Compute the consistent tangent of a plane-strain elastoplastic Q4 element.

Parameters:

Xe:: Element coordinates with shape (4, 2).
Ge:: Material row [E, nu, Sy0, H] or [E, nu, Sy0, H, phi].
Se, Ee:: Previous Gauss-point stress and history tables with shape (4, 5).
mtype:: 1 for von Mises and 2 for Drucker-Prager.

Returns:

ndarray: 8 x 8 consistent tangent stiffness matrix.

femlabpy.elements.quads.keq4eps(Xe, Ge, Se, Ee, mtype: int = 1)[source]#

Compute the consistent tangent of a plane-stress elastoplastic Q4 element.

Parameters:

Xe:: Element coordinates with shape (4, 2).
Ge:: Material row [E, nu, Sy0, H] or [E, nu, Sy0, H, phi].
Se, Ee:: Previous Gauss-point stress and history tables with shape (4, 4).
mtype:: 1 for von Mises and 2 for Drucker-Prager.

Returns:

ndarray: 8 x 8 consistent tangent stiffness matrix.

femlabpy.elements.quads.keq4p(Xe, Ge)[source]#

Compute the scalar conductivity matrix for a 4-node quadrilateral.

Parameters:

Xe:: Element coordinates with shape (4, 2).
Ge:: Material row [k] or [k, b] where b is an optional reaction coefficient.

Returns:

ndarray: 4 x 4 conductivity matrix integrated with 2 x 2 Gauss points.

femlabpy.elements.quads.kq4e(K, T, X, G)[source]#

Assemble Q4 element stiffness matrices into global stiffness matrix.

Parameters:

Kndarray, shape (ndof, ndof): Global stiffness matrix (modified in place).
Tarray_like, shape (nel, 5): Element topology: [n1, n2, n3, n4, 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.

Returns:

Kndarray: Updated global stiffness matrix.

Examples

>>> from femlabpy import init, kq4e
>>> K, q, p, C, P, S = init(nn=27, nd=2)
>>> K = kq4e(K, T, X, G)

femlabpy.elements.quads.kq4epe(K, T, X, G, S, E, mtype: int = 1)[source]#: Assemble plane-strain plastic Q4 tangent matrices into the global matrix.

femlabpy.elements.quads.kq4eps(K, T, X, G, S, E, mtype: int = 1)[source]#: Assemble plane-stress plastic Q4 tangent matrices into the global matrix.

femlabpy.elements.quads.kq4p(K, T, X, G)[source]#

Assemble Q4 scalar conductivity matrices into the global system.

Parameters:

K:: Global conductivity matrix.
T:: Topology table [n1, n2, n3, n4, mat_id].
X:: Nodal coordinates.
G:: Material table with conductivity rows.

Returns:

ndarray or sparse matrix: Updated global conductivity matrix.

femlabpy.elements.quads.meq4e(Xe, Ge, *, lumped: bool = False)[source]#

Compute the element mass matrix for a 4-node quadrilateral (Q4) element.

Parameters:

Xearray_like, shape (4, 2): Nodal coordinates.
Gearray_like: Material row. Thickness t from Ge[3] (default 1). Density rho from Ge[4] (default 1).
lumpedbool: If True, return the diagonally lumped mass matrix.

Returns:

Mendarray, shape (8, 8)

femlabpy.elements.quads.mq4e(M, T, X, G, *, lumped: bool = False)[source]#

Assemble Q4 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, 5): Topology table [n1, n2, n3, n4, 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.quads.qeq4e(Xe, Ge, Ue)[source]#

Compute stresses and strains at Gauss points for a single Q4 element.

Parameters:

Xearray_like, shape (4, 2): Nodal coordinates.
Gearray_like: Material properties [E, nu] or [E, nu, type].
Uearray_like, shape (8,): Element displacements [u1, v1, u2, v2, u3, v3, u4, v4].

Returns:

qendarray, shape (8, 1): Element internal force vector.
Sendarray, shape (4, 3): Stresses at 4 Gauss points: [sxx, syy, txy] per row.
Eendarray, shape (4, 3): Strains at 4 Gauss points: [exx, eyy, gxy] per row.

Examples

>>> qe, stresses, strains = qeq4e(Xe, Ge, Ue)
>>> avg_stress = np.mean(stresses, axis=0)  # element average

femlabpy.elements.quads.qeq4epe(Xe, Ge, Ue, Se, Ee, mtype: int = 1)[source]#

Update plane-strain elastoplastic Q4 response at the four Gauss points.

Parameters:

Xe, Ge:: Element coordinates and material row.
Ue:: Element displacement vector.
Se, Ee:: Previous Gauss-point stress and history tables with shape (4, 5).
mtype:: 1 for von Mises and 2 for Drucker-Prager.

Returns:

tuple[ndarray, ndarray, ndarray]: Element internal-force vector together with updated stress and history tables.

femlabpy.elements.quads.qeq4eps(Xe, Ge, Ue, Se, Ee, mtype: int = 1)[source]#

Update plane-stress elastoplastic Q4 response at the four Gauss points.

Parameters:

Xe, Ge:: Element coordinates and material row.
Ue:: Element displacement vector.
Se, Ee:: Previous Gauss-point stress and history tables with shape (4, 4).
mtype:: 1 for von Mises and 2 for Drucker-Prager.

Returns:

tuple[ndarray, ndarray, ndarray]: Element internal-force vector together with updated stress and history tables.

femlabpy.elements.quads.qeq4p(Xe, Ge, Ue)[source]#

Recover gradients and fluxes for one Q4 scalar potential element.

Parameters:

Xe:: Element coordinates with shape (4, 2).
Ge:: Material row [k] or [k, b].
Ue:: Element nodal potentials.

Returns:

tuple[ndarray, ndarray, ndarray]: Element flux vector, Gauss-point fluxes, and Gauss-point gradients.

femlabpy.elements.quads.qq4e(q, T, X, G, u)[source]#

Compute stresses/strains for all Q4 elements and assemble internal forces.

Parameters:

qndarray, shape (ndof, 1): Global internal force vector (modified in place).
Tarray_like, shape (nel, 5): Element topology: [n1, n2, n3, n4, mat_id] per row (1-based).
Xarray_like, shape (nn, 2): Nodal coordinates.
Garray_like, shape (nmat, 2+): Material properties.
uarray_like, shape (nn, 2) or (ndof,): Nodal displacement vector.

Returns:

qndarray: Updated internal force vector.
Sndarray, shape (nel, 12): Stresses at 4 Gauss points per element. Each row: [sxx_1, syy_1, txy_1, sxx_2, …, txy_4]
Endarray, shape (nel, 12): Strains at 4 Gauss points per element.

Examples

>>> q, S, E = qq4e(q, T, X, G, u)
>>> # Get element-averaged stresses
>>> S_avg = S.reshape(-1, 4, 3).mean(axis=1)

femlabpy.elements.quads.qq4epe(q, T, X, G, u, S, E, mtype: int = 1)[source]#: Compute plane-strain plastic Q4 internal forces and updated history fields.

femlabpy.elements.quads.qq4eps(q, T, X, G, u, S, E, mtype: int = 1)[source]#: Compute plane-stress plastic Q4 internal forces and updated history fields.

femlabpy.elements.quads.qq4p(q, T, X, G, u)[source]#

Recover Q4 scalar gradients and assemble equivalent nodal fluxes.

Parameters:

q:: Global nodal flux vector.
T:: Topology table [n1, n2, n3, n4, mat_id].
X:: Nodal coordinates.
G:: Material table with conductivity rows.
u:: Global nodal potentials.

Returns:

tuple[ndarray, ndarray, ndarray]: Updated nodal flux vector, Gauss-point fluxes, and Gauss-point gradients.