femlabpy.elements.solids#

Three-dimensional solid element kernels for tetrahedra and hexahedra.

Workflow role#

This module contains the 3D structural kernels used when the model carries three translational degrees of freedom per node. It covers both simplex and brick topologies and includes stiffness, internal-force, and mass-matrix construction at element and assembled levels.

Public entry points#

keT4e and qeT4e operate on one tetrahedral element.
keh8e and qeh8e operate on one hexahedral brick.
kT4e, qT4e, kh8e, and qh8e assemble those responses globally.
meT4e, mT4e, meh8e, and mh8e provide the dynamic mass terms.

Functions#

`keT4e`(Xe, Ge)	Compute the stiffness matrix for a 4-node tetrahedral solid element.
`keh8e`(Xe, Ge)	Compute the stiffness matrix for an 8-node hexahedral solid element.
`kT4e`(K, T, X, G)	Assemble T4 solid element stiffness contributions into the global matrix.
`kh8e`(K, T, X, G)	Assemble H8 solid element stiffness contributions into the global matrix.
`meT4e`(Xe, Ge, *[, lumped])	Compute the element mass matrix for a 4-node tetrahedral solid element.
`meh8e`(Xe, Ge, *[, lumped])	Compute the element mass matrix for an 8-node hexahedral solid element.
`mT4e`(M, T, X, G, *[, lumped])	Assemble T4 element mass matrices into the global mass matrix.
`mh8e`(M, T, X, G, *[, lumped])	Assemble H8 element mass matrices into the global mass matrix.
`qeT4e`(Xe, Ge, Ue)	Recover stress and strain for one tetrahedral solid element.
`qeh8e`(Xe, Ge, Ue)	Recover stress and strain at the eight H8 Gauss points.
`qT4e`(q, T, X, G, u)	Recover T4 solid stresses and assemble internal forces.
`qh8e`(q, T, X, G, u)	Recover H8 solid stresses and assemble internal forces.

Function Reference#

femlabpy.elements.solids.keT4e(Xe, Ge)[source]#

Compute the stiffness matrix for a 4-node tetrahedral solid element.

Parameters:

Xe:: Tetrahedral nodal coordinates with shape (4, 3).
Ge:: Material row [E, nu].

Returns:

ndarray: 12 x 12 element stiffness matrix.

femlabpy.elements.solids.keh8e(Xe, Ge)[source]#

Compute the stiffness matrix for an 8-node hexahedral solid element.

Parameters:

Xe:: Hexahedral nodal coordinates with shape (8, 3).
Ge:: Material row [E, nu].

Returns:

ndarray: 24 x 24 element stiffness matrix integrated with 2 x 2 x 2 Gauss quadrature.

femlabpy.elements.solids.kT4e(K, T, X, G)[source]#

Assemble T4 solid element stiffness contributions into the global matrix.

Parameters:

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

Returns:

ndarray or sparse matrix: Updated global stiffness matrix.

femlabpy.elements.solids.kh8e(K, T, X, G)[source]#

Assemble H8 solid element stiffness contributions into the global matrix.

Parameters:

K:: Global stiffness matrix.
T:: Topology table [n1, ..., n8, mat_id].
X:: Nodal coordinates.
G:: Material table.

Returns:

ndarray or sparse matrix: Updated global stiffness matrix.

femlabpy.elements.solids.meT4e(Xe, Ge, *, lumped: bool = False)[source]#

Compute the element mass matrix for a 4-node tetrahedral solid element.

Consistent (12x12) — analytical formula:

M = (rho * V / 20) * [[2I, I, I, I],: [I, 2I, I, I], [I, I, 2I, I], [I, I, I, 2I]]

Lumped (diagonal):

M = (rho * V / 4) * I_12

Parameters:

Xearray_like, shape (4, 3): Nodal coordinates.
Gearray_like: Material row [E, nu, rho]. If rho is omitted, defaults to 1.
lumpedbool: If True, return diagonally lumped mass.

Returns:

Mendarray, shape (12, 12)

femlabpy.elements.solids.meh8e(Xe, Ge, *, lumped: bool = False)[source]#

Compute the element mass matrix for an 8-node hexahedral solid element.

Consistent (24x24) via 2x2x2 Gauss quadrature:: M = sum_gp rho * N^T N * det(J) * w

Lumped via HRZ (diagonal scaling to preserve total mass).

Parameters:

Xearray_like, shape (8, 3): Nodal coordinates.
Gearray_like: Material row [E, nu, rho].
lumpedbool: If True, return diagonally lumped mass.

Returns:

Mendarray, shape (24, 24)

femlabpy.elements.solids.mT4e(M, T, X, G, *, lumped: bool = False)[source]#

Assemble T4 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 [n1, n2, n3, n4, mat_id].
Xarray_like, shape (nn, 3): Nodal coordinates.
Garray_like: Material table [E, nu, rho].
lumpedbool: If True, assemble lumped mass.

Returns:

Mndarray or sparse: Updated global mass matrix.

femlabpy.elements.solids.mh8e(M, T, X, G, *, lumped: bool = False)[source]#

Assemble H8 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, 9): Topology [n1, ..., n8, mat_id].
Xarray_like, shape (nn, 3): Nodal coordinates.
Garray_like: Material table [E, nu, rho].
lumpedbool: If True, assemble lumped mass.

Returns:

Mndarray or sparse: Updated global mass matrix.

femlabpy.elements.solids.qeT4e(Xe, Ge, Ue)[source]#

Recover stress and strain for one tetrahedral solid element.

Parameters:

Xe:: Tetrahedral nodal coordinates with shape (4, 3).
Ge:: Material row [E, nu].
Ue:: Element displacement vector.

Returns:

tuple[ndarray, ndarray, ndarray]: Element internal-force vector, stress vector, and strain vector.

femlabpy.elements.solids.qeh8e(Xe, Ge, Ue)[source]#

Recover stress and strain at the eight H8 Gauss points.

Parameters:

Xe:: Hexahedral nodal coordinates with shape (8, 3).
Ge:: Material row [E, nu].
Ue:: Element displacement vector.

Returns:

tuple[ndarray, ndarray, ndarray]: Element internal-force vector, Gauss-point stresses, and Gauss-point strains.

femlabpy.elements.solids.qT4e(q, T, X, G, u)[source]#

Recover T4 solid stresses and assemble internal forces.

Parameters:

q:: Global internal-force vector.
T:: Topology table [n1, n2, n3, n4, mat_id].
X:: Nodal coordinates.
G:: Material table.
u:: Global displacement vector.

Returns:

tuple[ndarray, ndarray, ndarray]: Updated internal-force vector, element stresses, and element strains.

femlabpy.elements.solids.qh8e(q, T, X, G, u)[source]#

Recover H8 solid stresses and assemble internal forces.

Parameters:

q:: Global internal-force vector.
T:: Topology table [n1, ..., n8, mat_id].
X:: Nodal coordinates.
G:: Material table.
u:: Global displacement vector.

Returns:

tuple[ndarray, ndarray, ndarray]: Updated internal-force vector, flattened Gauss-point stresses, and flattened Gauss-point strains.