Cook's membrane example
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Cook’s membrane example for nearly incompressible solid under large deformation
The example has been analyzed in many references, for example with the F-bar method in [1] and [2]. This example is also used as benchmark for SmallDeformation. Hereby we analyze it again under finite strain assumption using the F-bar method for the total Lagrange formulation (see the attached PDF for its theory). For the finite strain assumption, the constitutive law is replaced with a hyperelasticity one, the Neo-Hookean. The Neo-Hookean model defines an energy function as the sum of volumetric component $U_{\text{dil}}(\F)$ and deviatoric component $U_{\text{dev}}(\F)$ as
$$ \begin{align} W(\F) = U_{\text{dil}}(\F) + U_{\text{dev}}(\F), \end{align} $$where
$$ \begin{align} & U_{\text{dil}}(\F) = \dfrac{1}{2} K (\det(\F)-1)^2\\ & U_{\text{dev}}(\F) = \dfrac{1}{2} G \left(\text{tr} (\det(\F)^{-\frac{2}{3}}\F\F^{\text{T}})-3\right) \end{align} $$with $K$ the bulk modulus, and $G$ the shear modulus. The values of $K$ and $G$ are taken from references [1] and [2], which are $40.0942\cdot 10^{4}$ MPa and $80.1938$ MPa, respectively. For OGS input, the corresponding Young’s modulus and Poisson ratio are 240.565 MPa, and 0.499, respectively.
import os…
(click to toggle)
import os
from pathlib import Path
import matplotlib.pyplot as plt
import numpy as np
import ogstools as ot
out_dir = Path(os.environ.get("OGS_TESTRUNNER_OUT_DIR", "_out"))
out_dir.mkdir(parents=True, exist_ok=True)
def run_single_test(n: int, fbar: bool) -> ot.MeshSeries:…
(click to toggle)
def run_single_test(n: int, fbar: 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")
if not fbar:
model.replace_text("none", xpath="./processes/process/f_bar")
prefix = f"cooks_membrane_n_{n}_bbar_{str(fbar).lower()}"
model.replace_text(prefix, xpath="./time_loop/output/prefix")
if 20 <= n <= 25:
model.replace_text("0.5", xpath=".//time_stepping/timesteps/pair/delta_t")
model.replace_text("FRamp", xpath=".//boundary_condition[3]/parameter")
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: ot.Mesh) -> 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: ot.Mesh, mesh_b: ot.Mesh) -> 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("fbar = false")
axs[0, 1].set_title("fbar = true")
ot.plot.utils.update_font_sizes(fig.axes, 10)
return fig
n_range = [4, 10, 15, 20, 25, 30]
results_fbar_false = {n: run_single_test(n, fbar=False) for n in n_range}…
(click to toggle)
results_fbar_false = {n: run_single_test(n, fbar=False) for n in n_range}
uy_top_fbar_false = [get_top_uy(ms[-1]) for ms in results_fbar_false.values()]
uy_top_fbar_false_ref = np.array(
[0.0022867221436878916, 0.0028406921658587160, 0.0033745807476269606,
0.0038811997172937490, 0.0042429289476765735, 0.004702757454266369]
) # fmt: skip
np.testing.assert_allclose(uy_top_fbar_false, uy_top_fbar_false_ref, atol=1e-10)Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.1587212085723877 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.16601085662841797 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.18331336975097656 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.23560142517089844 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.31732773780822754 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.3555943965911865 s
results_fbar_true = {n: run_single_test(n, fbar=True) for n in n_range}…
(click to toggle)
results_fbar_true = {n: run_single_test(n, fbar=True) for n in n_range}
uy_top_fbar_true = [get_top_uy(ms[-1]) for ms in results_fbar_true.values()]
uy_top_fbar_true_ref = np.array(
[0.0061413833572504320, 0.006746283955378773, 0.0068241828268382505,
0.0068589364838052315, 0.006873125862208623, 0.006891409184641067]
) # fmt: skip
np.testing.assert_allclose(uy_top_fbar_true, uy_top_fbar_true_ref, atol=2e-4)Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.18892884254455566 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.27666425704956055 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.4601013660430908 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 0.6773874759674072 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 1.4099366664886475 s
Project file written to output.
Simulation: /var/lib/gitlab-runner/builds/t1_R7hwLX/1/ogs/build/release-all/Tests/Data/LargeDeformation/CooksMembrane/CooksMembraneFbar/modified.prj
Status: finished successfully.
Execution took 1.520535945892334 s
Result
Vertical diplacement at the top point
The following figure shows that the convergence of the solutions obtained by using the F bar method follows the one presented in the paper by T. Elguedj et al [1]. However, the results obtained without the F 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_fbar_true) * 1e3, "--o", label="B bar")
ax.plot(n_range, np.asarray(uy_top_fbar_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
The contour plots show that even with the coarsest mesh, the F bar method still gives reasonable stress result.
for n in n_range:
compare(results_fbar_false[n][-1], results_fbar_true[n][-1])
Reference
-
E.A. de Souza Neto, D. Perić, M. Dutko, D.R.J. Owen, Design of simple low order finite elements for large strain analysis of nearly incompressible solids, International Journal of Solids and Structures, Volume 33, Issues 20–22, 1996, Pages 3277-3296.
-
T. Elguedj, Y. Bazilevs, V.M. Calo, T.J.R. Hughes (2008), $\bar\B$ and $\bar{\text F}$ 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.
This article was written by Wenqing Wang & Thomas Nagel. If you are missing something or you find an error please let us know.
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Last revision: July 3, 2024