sfepy.mechanics.membranes module¶

sfepy.mechanics.membranes.create_mapping(coors, gel, order)[source]¶

Create mapping from transformed (in x-y plane) element faces to reference element faces.

Parameters:

coorsarray: The transformed coordinates of element nodes, shape (n_el, n_ep, dim). The function verifies that the all z components are zero.
gelGeometryElement instance: The geometry element corresponding to the faces.
orderint: The polynomial order of the mapping.

Returns:

mappingFEMapping instance: The reference element face mapping.

sfepy.mechanics.membranes.create_transformation_matrix(coors)[source]¶

Create a transposed coordinate transformation matrix, that transforms 3D coordinates of element face nodes so that the transformed nodes are in the x-y plane. The rotation is performed w.r.t. the first node of each face.

Parameters:

coorsarray: The coordinates of element nodes, shape (n_el, n_ep, dim).

Returns:

mtx_tarray: The transposed transformation matrix $T$ , i.e. $X_{inplane} = T^T X_{3D}$ .

Notes

$T = [t_1, t_2, n]$ , where $t_1$ , $t_2$ , are unit in-plane (column) vectors and $n$ is the unit normal vector, all mutually orthonormal.

sfepy.mechanics.membranes.describe_deformation(el_disps, bfg)[source]¶

Describe deformation of a thin incompressible 2D membrane in 3D space, composed of flat finite element faces.

The coordinate system of each element (face), i.e. the membrane mid-surface, should coincide with the x, y axes of the x-y plane.

Parameters:

el_dispsarray: The displacements of element nodes, shape (n_el, n_ep, dim).
bfgarray: The in-plane base function gradients, shape (n_el, n_qp, dim-1, n_ep).

Returns:

mtx_c ; array: The in-plane right Cauchy-Green deformation tensor $C_{ij}$ , $i, j = 1, 2$ .
c33array: The component $C_{33}$ computed from the incompressibility condition.
mtx_barray: The discrete Green strain variation operator.

sfepy.mechanics.membranes.describe_geometry(field, region, integral)[source]¶

Describe membrane geometry in a given region.

Parameters:

fieldField instance: The field defining the FE approximation.
regionRegion instance: The surface region to describe.
integralIntegral instance: The integral defining the quadrature points.

Returns:

mtx_tarray: The transposed transformation matrix $T$ , see create_transformation_matrix().
membrane_geoCMapping instance: The mapping from transformed elements to a reference elements.

sfepy.mechanics.membranes.get_green_strain_sym3d(mtx_c, c33)[source]¶

Get the 3D Green strain tensor in symmetric storage.

Parameters:

mtx_c ; array: The in-plane right Cauchy-Green deformation tensor $C_{ij}$ , $i, j = 1, 2$ , shape (n_el, n_qp, dim-1, dim-1).
c33array: The component $C_{33}$ computed from the incompressibility condition, shape (n_el, n_qp).

Returns:

mtx_earray: The membrane Green strain $E_{ij} = \frac{1}{2} (C_{ij}) - \delta_{ij}$ , symmetric storage: items (11, 22, 33, 12, 13, 23), shape (n_el, n_qp, sym, 1).

sfepy.mechanics.membranes.get_invariants(mtx_c, c33)[source]¶

Get the first and second invariants of the right Cauchy-Green deformation tensor describing deformation of an incompressible membrane.

Parameters:

mtx_c ; array: The in-plane right Cauchy-Green deformation tensor $C_{ij}$ , $i, j = 1, 2$ , shape (n_el, n_qp, dim-1, dim-1).
c33array: The component $C_{33}$ computed from the incompressibility condition, shape (n_el, n_qp).

Returns:

i1array: The first invariant of $C_{ij}$ .
i2array: The second invariant of $C_{ij}$ .

sfepy.mechanics.membranes.get_tangent_stress_matrix(stress, bfg)[source]¶

Get the tangent stress matrix of a thin incompressible 2D membrane in 3D space, given a stress.

Parameters:

stressarray: The components 11, 22, 12 of the second Piola-Kirchhoff stress tensor, shape (n_el, n_qp, 3, 1).
bfgarray: The in-plane base function gradients, shape (n_el, n_qp, dim-1, n_ep).

Returns:

mtxarray: The tangent stress matrix, shape (n_el, n_qp, dim*n_ep, dim*n_ep).

sfepy.mechanics.membranes.transform_asm_matrices(out, mtx_t)[source]¶

Transform matrix assembling contributions to global coordinate system, one node at a time.

Parameters:

outarray: The array of matrices, transformed in-place.
mtx_tarray: The transposed transformation matrix $T$ , see create_transformation_matrix().

sfepy.mechanics.membranes.transform_asm_vectors(out, mtx_t)[source]¶

Transform vector assembling contributions to global coordinate system, one node at a time.

Parameters:

outarray: The array of vectors, transformed in-place.
mtx_tarray: The transposed transformation matrix $T$ , see create_transformation_matrix().