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Nonlinear Truss Analysis#
This example requires external packages.
pip install contique
This example describes a three-dimensional system of trusses with 5 points and 6 cells (in total 5 active degrees of freedom). Given to its geometry, strong geometric nonlinearities are to be expected when the given reference load is applied. First, we create a mesh, where points are defined by their coordinates and cells by pairs of point connectivities.
import contique
import matplotlib.pyplot as plt
import numpy as np
import felupe as fem
mesh = fem.Mesh(
points=[
[2.5, 0, 0],
[-1.25, 1.25, 0],
[1, 2, 0],
[-0.5, 1.5, 1.5],
[-2.5, 4.5, 2.5],
],
cells=[[0, 3], [1, 3], [2, 3], [2, 4], [1, 4], [3, 4]],
cell_type="line",
)
region = fem.RegionTruss(mesh)
field = fem.FieldContainer([fem.Field(region, dim=3)])
Beside points and cells we have to define displacement boundary conditions, external forces and the constitutive material formulation for the trusses.
boundaries = fem.BoundaryDict(
fixed_xyz=fem.Boundary(field[0], mask=[1, 1, 1, 0, 0]),
fixed_y=fem.Boundary(field[0], mask=[0, 0, 0, 0, 1], skip=(1, 0, 1)),
)
dof0, dof1 = fem.dof.partition(field, boundaries)
solid = fem.TrussBody(
umat=fem.LinearElastic1D(E=1),
field=field,
area=[0.75, 1, 0.5, 0.75, 1, 1],
)
force_3 = np.array([1, 1, -1])
force_4 = np.array([-2, 0, -2])
load_3 = fem.PointLoad(field, [3], force_3)
load_4 = fem.PointLoad(field, [4], force_4)
The undeformed configuration is plotted in a 3d-view.
plotter = mesh.plot(
line_width=10,
render_lines_as_tubes=True,
show_edges=False,
)
plotter.add_points(
mesh.points,
color="black",
point_size=20,
render_points_as_spheres=True,
)
plotter = boundaries.plot(plotter=plotter)
plotter = load_3.plot(plotter=plotter, color="green", deformed=False)
plotter = load_4.plot(plotter=plotter, color="green", deformed=False)
plotter.show()
For the numeric continuation, the equilibrium function fun and its derivatives
w.r.t. the displacement field dfun_du and the load-proportionality-factor
dfun_dlpf have to be defined. Here, we’re only interested in the active degrees of
freedom.
def fun(x, lpf, *args):
field[0].values.ravel()[dof1] = x
load_3.update(force_3 * lpf)
load_4.update(force_4 * lpf)
return fem.tools.fun([solid, load_3, load_4], field)[dof1]
def dfun_du(x, lpf, *args):
field[0].values.ravel()[dof1] = x
K = fem.tools.jac([solid, load_3, load_4], field)
return fem.solve.partition(field, K, dof1, dof0)[2]
def dfun_dlpf(x, lpf, *args):
load_3.update(force_3)
load_4.update(force_4)
return fem.tools.fun([load_3, load_4], field)[dof1]
Now that the model is finished, some additional settings have to be chosen. Initial
allowed incremental system vector components for both the displacement vector and the
load-proportionality-factor (LPF) have to be specified. We use dlpf = 0.005 and
du = 0.05 (figured out after some trial and error). Both parameters can’t be
specified automatically, as they depend on the model configuration. The job will be
limited to a total amount of 163 increments (again, the total number has been figured
out after some job runs to get good looking plots).
|Step,C.| Control Component | Norm (Iter.#) | Message |
|-------|-------------------|---------------|-------------|
| 1,1 | 5+ => 4- | 8.8e-09 ( 6#) | => re-Cycle |
| 2 | 4- => 3- | 1.7e-14 ( 3#) |tol.Overshoot|
| 2,1 | 3- => 3- | 1.6e-14 ( 3#) | |
| 3,1 | 3- => 4- | 2.3e-14 ( 3#) |tol.Overshoot|
| 4,1 | 4- => 4- | 6.7e-14 ( 3#) | |
| 5,1 | 4- => 4- | 4.2e-13 ( 3#) | |
| 6,1 | 4- => 4- | 2.5e-12 ( 3#) | |
| 7,1 | 4- => 4- | 1.4e-11 ( 3#) | |
| 8,1 | 4- => 4- | 7.1e-11 ( 3#) | |
| 9,1 | 4- => 4- | 5.4e-11 ( 3#) | |
| 10,1 | 4- => 4- | 3.8e-11 ( 3#) | |
| 11,1 | 4- => 4- | 3.0e-11 ( 3#) | |
| 12,1 | 4- => 4- | 2.7e-11 ( 3#) | |
| 13,1 | 4- => 4- | 2.7e-11 ( 3#) | |
| 14,1 | 4- => 4- | 3.1e-11 ( 3#) | |
| 15,1 | 4- => 4- | 4.0e-11 ( 3#) | |
| 16,1 | 4- => 4- | 5.4e-11 ( 3#) | |
| 17,1 | 4- => 4- | 7.6e-11 ( 3#) | |
| 18,1 | 4- => 4- | 1.1e-10 ( 3#) | |
| 19,1 | 4- => 4- | 1.6e-10 ( 3#) | |
| 20,1 | 4- => 4- | 2.3e-10 ( 3#) | |
| 21,1 | 4- => 4- | 3.3e-10 ( 3#) | |
| 22,1 | 4- => 4- | 4.6e-10 ( 3#) | |
| 23,1 | 4- => 4- | 6.1e-10 ( 3#) | |
| 24,1 | 4- => 4- | 6.5e-10 ( 3#) | |
| 25,1 | 4- => 4- | 3.4e-10 ( 3#) | |
| 26,1 | 4- => 4- | 4.1e-10 ( 3#) | |
| 27,1 | 4- => 4- | 1.9e-15 ( 4#) | |
| 28,1 | 4- => 4- | 2.5e-13 ( 4#) | |
| 29,1 | 4- => 2- | 4.7e-11 ( 4#) |tol.Overshoot|
| 30,1 | 2- => 2- | 9.4e-13 ( 4#) | |
| 31,1 | 2- => 2- | 1.9e-12 ( 4#) | |
| 32,1 | 2- => 2- | 3.9e-09 ( 4#) | |
| 33,1 | 2- => 2- | 7.3e-03 ( 8#) |Failed |
| 34,1 | 2- => 4+ | 5.4e-02 ( 8#) |Failed |
| 35,1 | 2- => 4+ | 4.0e-11 ( 4#) |tol.Overshoot|
| 36,1 | 4+ => 4+ | 2.2e-09 ( 3#) | |
| 37,1 | 4+ => 4+ | 2.4e-10 ( 3#) | |
| 38,1 | 4+ => 4+ | 2.5e-10 ( 3#) | |
| 39,1 | 4+ => 4+ | 1.6e-10 ( 3#) | |
| 40,1 | 4+ => 4+ | 3.5e-11 ( 3#) | |
| 41,1 | 4+ => 4+ | 1.8e-10 ( 3#) | |
| 42,1 | 4+ => 4+ | 1.3e-09 ( 3#) | |
| 43,1 | 4+ => 4+ | 7.3e-10 ( 3#) | |
| 44,1 | 4+ => 4+ | 2.9e-10 ( 3#) | |
| 45,1 | 4+ => 4+ | 1.0e-10 ( 3#) | |
| 46,1 | 4+ => 4+ | 3.5e-11 ( 3#) | |
| 47,1 | 4+ => 4+ | 1.5e-11 ( 3#) | |
| 48,1 | 4+ => 4+ | 1.5e-11 ( 3#) | |
| 49,1 | 4+ => 4+ | 1.8e-11 ( 3#) | |
| 50,1 | 4+ => 4+ | 2.1e-11 ( 3#) | |
| 51,1 | 4+ => 4+ | 2.3e-11 ( 3#) | |
| 52,1 | 4+ => 4+ | 2.5e-11 ( 3#) | |
| 53,1 | 4+ => 4+ | 2.8e-11 ( 3#) | |
| 54,1 | 4+ => 4+ | 3.2e-11 ( 3#) | |
| 55,1 | 4+ => 4+ | 3.8e-11 ( 3#) | |
| 56,1 | 4+ => 4+ | 4.7e-11 ( 3#) | |
| 57,1 | 4+ => 4+ | 6.2e-11 ( 3#) | |
| 58,1 | 4+ => 4+ | 8.9e-11 ( 3#) | |
| 59,1 | 4+ => 4+ | 1.4e-10 ( 3#) | |
| 60,1 | 4+ => 4+ | 2.3e-10 ( 3#) | |
| 61,1 | 4+ => 4+ | 4.3e-10 ( 3#) | |
| 62,1 | 4+ => 4+ | 9.0e-10 ( 3#) | |
| 63,1 | 4+ => 4+ | 2.1e-09 ( 3#) | |
| 64,1 | 4+ => 4+ | 5.4e-09 ( 3#) | |
| 65,1 | 4+ => 3+ | 2.3e-15 ( 4#) |tol.Overshoot|
| 66,1 | 3+ => 3+ | 1.9e-09 ( 3#) | |
| 67,1 | 3+ => 3+ | 1.5e-09 ( 3#) | |
| 68,1 | 3+ => 3+ | 3.4e-09 ( 3#) | |
| 69,1 | 3+ => 3+ | 2.4e-15 ( 4#) | |
| 70,1 | 3+ => 3+ | 6.5e-04 ( 8#) |Failed |
| 71,1 | 3+ => 3+ | 1.4e-15 ( 4#) | |
| 72,1 | 3+ => 3+ | 7.0e-04 ( 8#) |Failed |
| 73,1 | 3+ => 3+ | 3.1e-15 ( 4#) | |
| 74,1 | 3+ => 2- | 4.2e-03 ( 8#) |Failed |
| 75,1 | 3+ => 2- | 1.5e-14 ( 4#) |tol.Overshoot|
| 76,1 | 2- => 2- | 3.6e-11 ( 3#) | |
| 77,1 | 2- => 2- | 1.2e-11 ( 3#) | |
| 78,1 | 2- => 2- | 4.1e-11 ( 3#) | |
| 79,1 | 2- => 2- | 1.4e-10 ( 3#) | |
| 80,1 | 2- => 2- | 5.4e-10 ( 3#) | |
| 81,1 | 2- => 3- | 2.5e-09 ( 3#) |tol.Overshoot|
| 82,1 | 3- => 3- | 4.2e-14 ( 4#) | |
| 83,1 | 3- => 3- | 4.3e-15 ( 4#) | |
| 84,1 | 3- => 3- | 7.9e-09 ( 3#) | |
| 85,1 | 3- => 3- | 7.5e-09 ( 3#) | |
| 86,1 | 3- => 3- | 3.7e-10 ( 3#) | |
| 87,1 | 3- => 4- | 1.7e-09 ( 3#) |tol.Overshoot|
| 88,1 | 4- => 4- | 3.0e-10 ( 3#) | |
| 89,1 | 4- => 4- | 3.0e-10 ( 3#) | |
| 90,1 | 4- => 4- | 2.5e-10 ( 3#) | |
| 91,1 | 4- => 4- | 1.9e-10 ( 3#) | |
| 92,1 | 4- => 4- | 1.3e-10 ( 3#) | |
| 93,1 | 4- => 4- | 9.3e-11 ( 3#) | |
| 94,1 | 4- => 4- | 6.5e-11 ( 3#) | |
| 95,1 | 4- => 4- | 4.8e-11 ( 3#) | |
| 96,1 | 4- => 4- | 3.9e-11 ( 3#) | |
| 97,1 | 4- => 4- | 3.7e-11 ( 3#) | |
| 98,1 | 4- => 4- | 4.3e-11 ( 3#) | |
| 99,1 | 4- => 4- | 5.9e-11 ( 3#) | |
| 100,1 | 4- => 4- | 9.7e-11 ( 3#) | |
| 101,1 | 4- => 4- | 1.8e-10 ( 3#) | |
| 102,1 | 4- => 4- | 3.9e-10 ( 3#) | |
| 103,1 | 4- => 4- | 9.8e-10 ( 3#) | |
| 104,1 | 4- => 4- | 3.0e-09 ( 3#) | |
| 105,1 | 4- => 4- | 9.0e-16 ( 4#) | |
| 106,1 | 4- => 4- | 2.9e-13 ( 4#) | |
| 107,1 | 4- => 3+ | 2.5e-11 ( 5#) | => re-Cycle |
| 2 | 3+ => 3+ | 1.5e-10 ( 4#) | |
| 108,1 | 3+ => 3+ | 1.9e-10 ( 4#) | |
| 109,1 | 3+ => 3+ | 4.3e-12 ( 4#) | |
| 110,1 | 3+ => 3+ | 6.1e-15 ( 4#) | |
| 111,1 | 3+ => 3+ | 9.7e-09 ( 3#) | |
| 112,1 | 3+ => 3+ | 5.6e-09 ( 3#) | |
| 113,1 | 3+ => 3+ | 3.2e-09 ( 3#) | |
| 114,1 | 3+ => 3+ | 2.0e-09 ( 3#) | |
| 115,1 | 3+ => 3+ | 6.7e-09 ( 3#) | |
| 116,1 | 3+ => 3+ | 3.8e-15 ( 4#) | |
| 117,1 | 3+ => 3+ | 6.1e-15 ( 4#) | |
| 118,1 | 3+ => 3+ | 8.0e-09 ( 3#) | |
| 119,1 | 3+ => 3+ | 4.2e-10 ( 3#) | |
| 120,1 | 3+ => 3+ | 7.0e-10 ( 3#) | |
| 121,1 | 3+ => 3+ | 5.9e-10 ( 3#) | |
| 122,1 | 3+ => 3+ | 3.0e-10 ( 3#) | |
| 123,1 | 3+ => 3+ | 1.3e-10 ( 3#) | |
| 124,1 | 3+ => 3+ | 1.1e-10 ( 3#) | |
| 125,1 | 3+ => 3+ | 2.6e-10 ( 3#) | |
| 126,1 | 3+ => 3+ | 4.5e-10 ( 3#) | |
| 127,1 | 3+ => 3+ | 5.5e-10 ( 3#) | |
| 128,1 | 3+ => 3+ | 5.1e-10 ( 3#) | |
| 129,1 | 3+ => 3+ | 7.0e-10 ( 3#) | |
| 130,1 | 3+ => 3+ | 2.3e-09 ( 3#) | |
| 131,1 | 3+ => 3+ | 7.3e-09 ( 3#) | |
| 132,1 | 3+ => 3+ | 5.0e-15 ( 4#) | |
| 133,1 | 3+ => 4- | 3.1e-13 ( 4#) |tol.Overshoot|
| 134,1 | 4- => 4- | 3.5e-14 ( 4#) | |
| 135,1 | 4- => 4- | 5.1e-09 ( 3#) | |
| 136,1 | 4- => 4- | 1.9e-09 ( 3#) | |
| 137,1 | 4- => 4- | 1.7e-09 ( 3#) | |
| 138,1 | 4- => 4- | 1.6e-09 ( 3#) | |
| 139,1 | 4- => 4- | 1.2e-09 ( 3#) | |
| 140,1 | 4- => 4- | 8.9e-10 ( 3#) | |
| 141,1 | 4- => 4- | 8.2e-10 ( 3#) | |
| 142,1 | 4- => 4- | 1.0e-09 ( 3#) | |
| 143,1 | 4- => 4- | 1.5e-09 ( 3#) | |
| 144,1 | 4- => 4- | 2.8e-09 ( 3#) | |
| 145,1 | 4- => 4- | 6.1e-09 ( 3#) | |
| 146,1 | 4- => 3- | 6.8e-16 ( 4#) |tol.Overshoot|
| 147,1 | 3- => 3- | 2.3e-10 ( 3#) | |
| 148,1 | 3- => 3- | 8.2e-11 ( 3#) | |
| 149,1 | 3- => 3- | 3.9e-11 ( 3#) | |
| 150,1 | 3- => 3- | 2.7e-11 ( 3#) | |
| 151,1 | 3- => 3- | 2.3e-11 ( 3#) | |
| 152,1 | 3- => 3- | 2.1e-11 ( 3#) | |
| 153,1 | 3- => 3- | 2.0e-11 ( 3#) | |
| 154,1 | 3- => 3- | 2.1e-11 ( 3#) | |
| 155,1 | 3- => 3- | 2.3e-11 ( 3#) | |
| 156,1 | 3- => 3- | 2.8e-11 ( 3#) | |
| 157,1 | 3- => 3- | 3.6e-11 ( 3#) | |
| 158,1 | 3- => 3- | 4.8e-11 ( 3#) | |
| 159,1 | 3- => 3- | 6.7e-11 ( 3#) | |
| 160,1 | 3- => 3- | 9.4e-11 ( 3#) | |
| 161,1 | 3- => 3- | 1.4e-10 ( 3#) | |
| 162,1 | 3- => 3- | 1.9e-10 ( 3#) | |
| 163,1 | 3- => 3- | 2.5e-10 ( 3#) | |
To visualize the deformed state of the model for increment 40 the deformed model plot is generated.
field[0].values.ravel()[dof1] = X[40, :-1]
force = solid.evaluate.gradient(field) * solid.area
plotter = field.view(cell_data={"Force": force}).plot(
"Force",
line_width=10,
show_undeformed=False,
view="xy",
cmap="coolwarm",
clim=[-abs(force).max(), abs(force).max()],
render_lines_as_tubes=True,
show_edges=False,
)
plotter.add_points(
mesh.points + field[0].values,
color="black",
point_size=20,
render_points_as_spheres=True,
)
plotter.show()
Path-tracing of the displacement-LPF curves#
The path-tracing of the deformation process is shown as a History Plot of Displacement-LPF curves for all active DOF. Strong geometrical nonlinearities are observed for all active DOF.
fig, ax = plt.subplots()
ax.plot(*X[:, [0, -1]].T, ".-", label="Point 3")
ax.plot(*X[:, [3, -1]].T, ".-", label="Point 4")
ax.set_xlabel("Displacement X")
ax.set_ylabel("LPF")
ax.legend()
fig, ax = plt.subplots()
ax.plot(*X[:, [1, -1]].T, ".-", label="Point 3")
ax.set_xlabel("Displacement Y")
ax.set_ylabel("LPF")
ax.legend()
fig, ax = plt.subplots()
ax.plot(*X[:, [2, -1]].T, ".-", label="Point 3")
ax.plot(*X[:, [4, -1]].T, ".-", label="Point 4")
ax.set_xlabel("Displacement Z")
ax.set_ylabel("LPF")
ax.legend()
Total running time of the script: (0 minutes 3.543 seconds)