Tria3R - Triangular element with reduced integration (`pyfe3d.tria3r`)#

class pyfe3d.tria3r.Tria3R#

Nodal connectivity for the triangular element similar to Nastran’s CTRIA3:

3
|\
| \    positive normal in CCW
|  \
|___\
1    2

The element coordinate system is determined identically what is explained in Nastran’s quick reference guide for the CTRIA3 element, as illustrated below.

Attributes:

eid,int

Element identification number.

pid,int

Property identification number.

area,double

Element area.

alpha_shear_locking,double

Factor used to prevent shear locking, adopted from teh BFG element, affecting stiffness terms E44,E45,E45:

maxl = max(edge_12, edge_23, edge_31)
factor = alpha_shear_locking*maxl**2/thickness**2
E44 = 1 / (1 + factor) * E44
# E45 = 1 / (1 + factor) * E45
E55 = 1 / (1 + factor) * E55

The adopted default is alpha_shear_locking = 0.7, based on a linear buckling analysis of a simply supported plate, such that the result approaches the one of the Quad4R element for an equivalent mesh (see the test case test_tria3r_linear_buckling_plate.py).

alphat,double

Element drilling penalty factor for the plate drilling stiffness, defined according to Eq. 2.20 in the reference below. The default value of alphat = 1. comes from the same reference:

Adam, A.E. Mohamed, A.E. Hassaballa, Degenerated Four Nodes Shell
Element with Drilling Degree of Freedom, IOSR J. Eng. 3 (2013)
10–20. www.iosrjen.org (accessed April 20, 2020).

For those familiar with NASTRAN, alphat can be calculated based on NASTRAN’s K6ROT parameters as alphat = 1.e-6*K6ROT. The default value according to AUTODESK NASTRAN’s quick reference guide is K6ROT = 100. for static analysis and K6ROT=1.e4 for modal solutions. MSC NASTRAN’s quick reference guide states that K6ROT > 100. should not be used, but this is controversion, already being controversial to what AUTODESK NASTRAN’s manual says.

r11, r12, r13, r21, r22, r23, r31, r32, r33double

Rotation matrix from local to global coordinates.

m11, m12, m21, m22double

Rotation matrix only for the constitutive relations. Used when a material direction is used instead of the element local coordinates.

c1, c2, c3: int

Position of each node in the global stiffness matrix.

n1, n2, n3: int

Node identification number.

init_k_KC0, init_k_KG, init_k_Mint

Position in the arrays storing the sparse data for the structural matrices.

probe,Tria3RProbe object

Pointer to the probe.

Methods

`update_KC0`(self, long[, long[, double[, ...)	Update sparse vectors for linear constitutive stiffness matrix KC0
`update_KG`(self, long[, long[, double[, ...)	Update sparse vectors for geometric stiffness matrix KG
`update_KG_given_stress`(self, double Nxx, ...)	Update sparse vectors for geometric stiffness matrix KG
`update_M`(self, long[, long[, double[, ...)	Update sparse vectors for mass matrix M
`update_area`(self)	Update element area
`update_probe_ue`(self, double[)	Update the local displacement vector of the probe of the element
`update_probe_xe`(self, double[)	Update the 3D coordinates of the probe of the element
`update_rotation_matrix`(self, double[, ...)	Update the rotation matrix of the element

alpha_shear_locking#: alpha_shear_locking: ‘double’

alphat#: alphat: ‘double’

area#: area: ‘double’

c1#: c1: ‘int’

c2#: c2: ‘int’

c3#: c3: ‘int’

eid#: eid: ‘int’

init_k_KC0#: init_k_KC0: ‘int’

init_k_KG#: init_k_KG: ‘int’

init_k_M#: init_k_M: ‘int’

m11#: m11: ‘double’

m12#: m12: ‘double’

m21#: m21: ‘double’

m22#: m22: ‘double’

n1#: n1: ‘int’

n2#: n2: ‘int’

n3#: n3: ‘int’

pid#: pid: ‘int’

probe#: probe: pyfe3d.tria3r.Tria3RProbe

r11#: r11: ‘double’

r12#: r12: ‘double’

r13#: r13: ‘double’

r21#: r21: ‘double’

r22#: r22: ‘double’

r23#: r23: ‘double’

r31#: r31: ‘double’

r32#: r32: ‘double’

r33#: r33: ‘double’

update_KC0(self, long[::1] KC0r, long[::1] KC0c, double[::1] KC0v, ShellProp prop, int update_KC0v_only=0) → void#

Update sparse vectors for linear constitutive stiffness matrix KC0

Reduced integration is used with a single point in the centroid (\(N1=N2=N3=1/3\)) and weight \(weight=1\), preventing shear locking. Hourglass control is used according to Brockman 1987:

Brockman, R. A., 1987, “Dynamics of the Bilinear Mindlin Plate Element,” Int. J. Numer. Methods Eng., 24(12), pp. 2343–2356. https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.1620241208

Drilling stiffness is used according to Adam et al. 2013:

Adam, F. M., Mohamed, A. E., and Hassaballa, A. E., 2013, “Degenerated Four Nodes Shell Element with Drilling Degree of Freedom,” IOSR J. Eng., 3(8), pp. 10–20.

Parameters:

KC0rnp.array: Array to store row positions of sparse values
KC0cnp.array: Array to store column positions of sparse values
KC0vnp.array: Array to store sparse values
propShellProp object: Shell property object from where the stiffness and mass attributes are read from.
update_KC0v_onlyint: The default 0 means that the row and column indices KC0r and KC0c should also be updated. Any other value will only update the stiffness matrix values KC0v.

update_KG(self, long[::1] KGr, long[::1] KGc, double[::1] KGv, ShellProp prop, int update_KGv_only=0) → void#

Update sparse vectors for geometric stiffness matrix KG

Two-point Gauss-Legendre quadrature is used, which showed more accuracy for linear buckling load predictions.

Before this function is called, the probe Tria3RProbe attribute of the Tria3R object must be updated using update_probe_ue() with the correct pre-buckling displacements; and update_probe_xe() with the node coordinates.

Parameters:

KGrnp.array: Array to store row positions of sparse values
KGcnp.array: Array to store column positions of sparse values
KGvnp.array: Array to store sparse values
propShellProp object: Shell property object from where the stiffness and mass attributes are read from.
update_KGv_onlyint: The default \(0\) means that only \(KGv\) is updated. Any other value will lead to \(KGr\) and \(KGc\) also being updated.

update_KG_given_stress(self, double Nxx, double Nyy, double Nxy, long[::1] KGr, long[::1] KGc, double[::1] KGv, int update_KGv_only=0) → void#

Update sparse vectors for geometric stiffness matrix KG

Note

A constant stress state is assumed within the element, according to the given values of \(N_{xx}, N_{yy}, N_{xy}\).

Two-point Gauss-Legendre quadrature is used, which showed more accuracy for linear buckling load predictions.

Before this function is called, the probe Tria3RProbe attribute of the Tria3R object must be updated using update_probe_xe() with the node coordinates.

Parameters:

KGrnp.array: Array to store row positions of sparse values
KGcnp.array: Array to store column positions of sparse values
KGvnp.array: Array to store sparse values
update_KGv_onlyint: The default \(0\) means that only \(KGv\) is updated. Any other value will lead to \(KGr\) and \(KGc\) also being updated.

update_M(self, long[::1] Mr, long[::1] Mc, double[::1] Mv, ShellProp prop, int mtype=0) → void#

Update sparse vectors for mass matrix M

Different integration schemes are available by means of the mtype parameter.

Parameters:

Mrnp.array: Array to store row positions of sparse values
Mcnp.array: Array to store column positions of sparse values
Mvnp.array: Array to store sparse values
mtypeint, optional: 0 for consistent mass matrix using method from Brockman 1987 1 for reduced integration mass matrix using method from Brockman 1987 2 for lumped mass matrix using method from Brockman 1987

update_area(self) → void#: Update element area

update_probe_ue(self, double[::1] u) → void#

Update the local displacement vector of the probe of the element

Note

The probe attribute object Tria3RProbe is updated, not the element object.

Parameters:

uarray-like: Array with global displacements, for a total of \(M\) nodes in the model, this array will be arranged as: \(u_1, v_1, w_1, {r_x}_1, {r_y}_1, {r_z}_1, u_2, v_2, w_2, {r_x}_2, {r_y}_2, {r_z}_2, ..., u_M, v_M, w_M, {r_x}_M, {r_y}_M, {r_z}_M\).

update_probe_xe(self, double[::1] x) → void#

Update the 3D coordinates of the probe of the element

Note

The probe attribute object Tria3RProbe is updated, not the element object.

Parameters:

xarray-like: Array with global nodal coordinates, for a total of \(M\) nodes in the model, this array will be arranged as: \(x_1, y_1, z_1, x_2, y_2, z_2, ..., x_M, y_M, z_M\).

update_rotation_matrix(self, double[::1] x, double xmati=0., double xmatj=0., double xmatk=0.) → void#

Update the rotation matrix of the element

Attributes r11,r12,r13,r21,r22,r23,r31,r32,r33 are updated, corresponding to the rotation matrix from local to global coordinates.

The element coordinate system is determined, identifying the \(ijk\) components of each axis: \({x_e}_i, {x_e}_j, {x_e}_k\); \({y_e}_i, {y_e}_j, {y_e}_k\); \({z_e}_i, {z_e}_j, {z_e}_k\).

Parameters:

xarray-like: Array with global nodal coordinates, for a total of \(M\) nodes in the model, this array will be arranged as: \(x_1, y_1, z_1, x_2, y_2, z_2, ..., x_M, y_M, z_M\).
xmati, xmatj, xmatk: array-like: Vector in global coordinates representing the material direction. This vector is projected onto the plate element, thus becoming the material direction. The \(ABD\) matrix defining the constitutive behavior of the element is rotated from the material direction to the element \(x\) axis while calculating the stiffness matrices.

class pyfe3d.tria3r.Tria3RData#

Used to allocate memory for the sparse matrices.

Attributes:

KC0_SPARSE_SIZE,int: KC0_SPARSE_SIZE = 324
KG_SPARSE_SIZE,int: KG_SPARSE_SIZE = 81
M_SPARSE_SIZE,int: M_SPARSE_SIZE = 270

KC0_SPARSE_SIZE#: KC0_SPARSE_SIZE: ‘int’

KG_SPARSE_SIZE#: KG_SPARSE_SIZE: ‘int’

M_SPARSE_SIZE#: M_SPARSE_SIZE: ‘int’

class pyfe3d.tria3r.Tria3RProbe#

Probe used for local coordinates, local displacements, local stresses etc

Attributes:

xe,array-like: Array of size NUM_NODES*DOF//2=9 containing the nodal coordinates in the element coordinate system, in the following order \({x_e}_1, {y_e}_1, {z_e}_1, `{x_e}_2, {y_e}_2, {z_e}_2\), \({x_e}_3, {y_e}_3, {z_e}_3\).
ue,array-like: Array of size NUM_NODES*DOF=18 containing the element displacements in the following order \({u_e}_1, {v_e}_1, {w_e}_1, {{r_x}_e}_1, {{r_y}_e}_1, {{r_z}_e}_1\), \({u_e}_2, {v_e}_2, {w_e}_2, {{r_x}_e}_2, {{r_y}_e}_2, {{r_z}_e}_2\), \({u_e}_3, {v_e}_3, {w_e}_3, {{r_x}_e}_3, {{r_y}_e}_3, {{r_z}_e}_3\).

ue#: ue: ‘double[::1]’

xe#: xe: ‘double[::1]’

Tria3R - Triangular element with reduced integration (pyfe3d.tria3r)#

Tria3R - Triangular element with reduced integration (`pyfe3d.tria3r`)#