BENCHMARKS
Cook's membrane example
This page is based on a Jupyter notebook.
Cook’s membrane example for nearly icompressible solid
B bar method
Considering a strain decomposition: $\mathbf\epsilon = \underbrace{\mathbf\epsilon- \frac{1}{3}(\epsilon:\mathbf I)}_{\text{deviatoric}}\I + \underbrace{\frac{1}{3}(\epsilon:\mathbf I)}_{\text{dilatational}} \I$. The idea of the B bar method is to use another quadrature rule to interpolate the dilatational part, which leads to a modified B matrix [1]:
$$ \bar\B = \underbrace{\B - \B^{\text{dil}}}_{\text{original B elements}}+ \underbrace{{\bar\B}^{\text{dil}}}_{\text{by another quadrature rule} } $$There are several methods to form ${\bar\B}^{\text{dil}}$ such as selective integration, generalization of the mean-dilatation formulation. In the current OGS, we use the latter, which reads
$$ {\bar\B}^{\text{dil}} = \frac{\intD{\B^{\text{dil}}(\xi)}}{\intD{}} $$Example
To verify the implementation of the B bar method, the so called Cook’s membrane is used as a benchmark. Illustrated in the following figure, this example simulates a tapered and swept panel of unit thickness. The left edge is clamped and the right edge is applied with a distributed shearing load $F$ = 100 N/mm. The plane strain condition is considered. This numerical model is exactly the same as that is presented in the paper by T. Elguedj et al [1,2].
import os…
(click to toggle)
import os
from pathlib import Path
import matplotlib.pyplot as plt
import numpy as np
import ogstools as ot
import pyvista as pv
out_dir = Path(os.environ.get("OGS_TESTRUNNER_OUT_DIR", "_out"))
out_dir.mkdir(parents=True, exist_ok=True)
if pv.global_theme.trame.server_proxy_enabled:
pv.set_jupyter_backend("client")
else:
pv.set_jupyter_backend("static")
def run_single_test(n: int, use_bbar: bool) -> ot.MeshSeries:…
(click to toggle)
def run_single_test(n: int, use_bbar: bool) -> ot.MeshSeries:
model = ot.Project(
input_file="CooksMembrane.prj", output_file=out_dir / "modified.prj"
)
model.replace_text(f"mesh_n{n:02d}.vtu", xpath="./mesh")
model.replace_text(str(use_bbar).lower(), xpath="./processes/process/use_b_bar")
prefix = f"cooks_membrane_n_{n}_bbar_{str(use_bbar).lower()}"
model.replace_text(prefix, xpath="./time_loop/output/prefix")
model.replace_text("BiCGSTAB", xpath=".//solver_type")
model.replace_text("ILUT", xpath=".//precon_type")
model.write_input()
model.run_model(logfile=out_dir / "out.txt", args=f"-o {out_dir} -m .")
return ot.MeshSeries(out_dir / (prefix + ".pvd"))
def get_top_uy(mesh: pv.UnstructuredGrid) -> np.ndarray:
top_point = (48.0e-3, 60.0e-3, 0)
p_id = mesh.find_closest_point(top_point)
return mesh.point_data["displacement"][p_id, 1]
def compare(mesh_a: pv.UnstructuredGrid, mesh_b: pv.UnstructuredGrid) -> plt.Figure:
fig, axs = plt.subplots(2, 2, figsize=[8, 6], sharex=True, sharey=True)
u = ot.variables.displacement["y"].replace(output_unit="mm")
sig_tr = ot.variables.stress.trace
ot.plot.contourf(mesh_a, u, fig=fig, ax=axs[0, 0], show_edges=True)
ot.plot.contourf(mesh_b, u, fig=fig, ax=axs[0, 1], show_edges=True)
ot.plot.contourf(mesh_a, sig_tr, fig=fig, ax=axs[1, 0], show_edges=True)
ot.plot.contourf(mesh_b, sig_tr, fig=fig, ax=axs[1, 1], show_edges=True)
axs[0, 0].set_title("bbar = false")
axs[0, 1].set_title("bbar = true")
ot.plot.utils.update_font_sizes(fig.axes, 10)
return fig
n_range = [4, 10, 15, 20, 25, 30]
results_bbar_false = {n: run_single_test(n, use_bbar=False) for n in n_range}…
(click to toggle)
results_bbar_false = {n: run_single_test(n, use_bbar=False) for n in n_range}
uy_top_bbar_false = [get_top_uy(ms[-1]) for ms in results_bbar_false.values()]
uy_top_bbar_false_ref = np.array(
[0.002164586784123102, 0.0022603329644579383, 0.002375295856067169,
0.002519725590136146, 0.0026515294133790837, 0.002868289617025223]
) # fmt: skip
np.testing.assert_allclose(uy_top_bbar_false, uy_top_bbar_false_ref, atol=1e-10) ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
results_bbar_true = {n: run_single_test(n, use_bbar=True) for n in n_range}…
(click to toggle)
results_bbar_true = {n: run_single_test(n, use_bbar=True) for n in n_range}
uy_top_bbar_true = [get_top_uy(ms[-1]) for ms in results_bbar_true.values()]
uy_top_bbar_true_ref = np.array(
[0.006957471385697900, 0.007772616910217863, 0.007897597955618913,
0.007951479575082158, 0.007976349858390623, 0.007999718483861992]
) # fmt: skip
np.testing.assert_allclose(uy_top_bbar_true, uy_top_bbar_true_ref, atol=2e-4) ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
ogs -o /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar -m . /var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj
['ogs', '-o', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar', '-m', '.', '/var/lib/gitlab-runner/builds/K-X1vcp78/0/ogs/build/release-all/Tests/Data/Mechanics/CooksMembrane/CooksMembraneBbar/modified.prj']
Result
Vertical diplacement at the top point
The following figure shows that the convergence of the solutions obtained by using the B bar method follows the one presented in the paper by T. Elguedj et al [1]. However, the results obtained without the B bar method are quit far from the converged solution with the finest mesh.
fig, ax = plt.subplots(figsize=(4, 3))…
(click to toggle)
fig, ax = plt.subplots(figsize=(4, 3))
ax.plot(n_range, np.asarray(uy_top_bbar_true) * 1e3, "--o", label="B bar")
ax.plot(n_range, np.asarray(uy_top_bbar_false) * 1e3, "--x", label="non B bar")
ax.set_xlabel("Number of elements per side")
ax.set_ylabel("Top right corner displacement /mm")
ax.legend()
fig.tight_layout()
Comparison of results
for n in n_range:
compare(results_bbar_false[n][-1], results_bbar_true[n][-1])
The contour plots show that even with the coarsest mesh, the B bar method still gives reasonable stress results.
Reference
[1] T.J.R. Hughes (1980). Generalization of selective integration procedures to anisotropic and nonlinear media. International Journal for Numerical Methods in Engineering, 15(9), 1413-1418.
[2] T. Elguedj, Y. Bazilevs, V.M. Calo, T.J.R. Hughes (2008), $\bar\B$ and $\bar\F$ projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements, Computer Methods in Applied Mechanics and Engineering, 197(33–40), 2732-2762.
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