Note

Click here to download the full example code

Solution verification of the analytic Tan open-hole model

Assess the discretization error of the :class:TanOpenHole model with the DiscretizationSolutionVerification tool, which is based on the Richardson extrapolation ^{1 2}, and compare two field-derived quantities:

• sigma_xx_probe: sigma_xx bilinearly interpolated at a fixed point just outside the hole. It converges smoothly as the grid is refined.
• sigma_xx_peak: the maximum of sigma_xx over the grid nodes. Because the node that happens to fall closest to the stress concentration jumps around as the grid changes, this quantity is non-monotone -- a sawtooth. Richardson extrapolation cannot be computed for a sawtooth, so the tool will fall back to a palliative estimator.

From the analytic solution, the model computes grid nodes evaluations and interpolates a field. Any value at a grid node is therefore exact, but coarse meshing degrades any other value querried from the discrete field. We look at two of them, plus the model's own native scalar outputs as a baseline.

... note::

This model is here used an illustrative use-case case for solution verification,
despite having a working principle different from
what is usually prone to solution verification.

from __future__ import annotations

import logging

import numpy as np
import plotly.graph_objects as go
from gemseo.utils.directory_creator import DirectoryNamingMethod
from numpy import atleast_1d
from pandas import DataFrame
from plotly.subplots import make_subplots

from vimseo import EXAMPLE_RUNS_DIR
from vimseo.api import activate_logger
from vimseo.api import create_model
from vimseo.core.model_result import ModelResult
from vimseo.core.model_settings import IntegratedModelSettings
from vimseo.tools.verification.solution_verification import (
    DiscretizationSolutionVerification,
)
from vimseo.utilities.datasets import dataframe_to_dataset

First we set the logger level:

activate_logger(level=logging.INFO)

The model to verify is the analytic Tan open-hole model, loaded in tension:

model_name = "TanOpenHole"
load_case = "Tension"
model = create_model(
    model_name,
    load_case,
    model_options=IntegratedModelSettings(
        directory_archive_root=EXAMPLE_RUNS_DIR / "archive/solution_verification",
        directory_scratch_root=EXAMPLE_RUNS_DIR / "scratch/solution_verification",
        cache_file_path=EXAMPLE_RUNS_DIR
        / f"caches/solution_verification/{model_name}_{load_case}_cache.hdf",
    ),
)

Out:

    INFO - 09:18:30: Found 6 entries in the cache file : C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\model_runs\caches\solution_verification\TanOpenHole_Tension_cache.hdf node : node

We define a set of grid resolutions (points per direction), from coarse to fine:

grid_resolutions = [25, 33, 50, 66, 100, 200]

Helper function to place two plotly figures side by side in a single figure.

def side_by_side(fig_left, fig_right, left_title, right_title, y_title):
    """Place the traces of two plotly figures in a single 1x2 subplot figure."""
    combined = make_subplots(rows=1, cols=2, subplot_titles=(left_title, right_title))
    for trace in fig_left.data:
        combined.add_trace(trace, row=1, col=1)
    for trace in fig_right.data:
        # Avoid duplicating the shared legend entries in the right-hand panel.
        trace.showlegend = False
        combined.add_trace(trace, row=1, col=2)
    combined.update_xaxes(title_text="dx")
    combined.update_yaxes(title_text=y_title, row=1, col=1)
    # ``make_subplots`` centres each panel title over its half-width column, so
    # keep them short (see call sites) and a touch smaller than the default so
    # they do not run into each other in the middle gutter.
    combined.update_annotations(font_size=13)
    return combined

Generate the convergence data for the four quantities of interest. The model is executed once for each grid resolution, and the four quantities (the two field-derived ones and the two native hole-edge stresses) are collected in a single table.

model.execute({"grid_resolution": atleast_1d(grid_resolutions[0])})
input_data = model.get_input_data()
length = float(input_data["length"][0])
width = float(input_data["width"][0])
radius = float(input_data["radius"][0])
probe_x = 0.5 * length
probe_y = 0.5 * width + radius + 3.0

dx_values = []
probe_stresses = []
peak_stresses = []
sigma_xx_r_values = []
sigma_xx_d0_values = []
model_results = {}
for grid_resolution in grid_resolutions:
    output_data = model.execute({"grid_resolution": atleast_1d(grid_resolution)})
    result = ModelResult.from_data(
        {"outputs": output_data, "inputs": model.get_input_data()},
        model=model,
        load_fields=True,
    )
    field = result.fields["flux"][0]
    dx_values.append(float(output_data["dx"][0]))
    probe_stresses.append(field.probe("sigma_xx", probe_x, probe_y))
    peak_stresses.append(float(np.nanmax(field.point_data["sigma_xx"])))
    sigma_xx_r_values.append(float(output_data["sigma_xx_r"][0]))
    sigma_xx_d0_values.append(float(output_data["sigma_xx_d0"][0]))
    model_results[grid_resolution] = result

convergence_table = DataFrame({
    "dx": dx_values,
    "sigma_xx_probe": probe_stresses,
    "sigma_xx_peak": peak_stresses,
    "sigma_xx_r": sigma_xx_r_values,
    "sigma_xx_d0": sigma_xx_d0_values,
})
print(convergence_table)

Out:

         dx  sigma_xx_probe  sigma_xx_peak   sigma_xx_r  sigma_xx_d0
0  3.333333     1312.692749    1998.087574  3137.629147   2094.95564
1  2.500000     1298.612676    1998.087574  3137.629147   2094.95564
2  1.632653     1296.809439    1753.841182  3137.629147   2094.95564
3  1.230769     1295.471224    1816.365910  3137.629147   2094.95564
4  0.808081     1295.369808    1937.173531  3137.629147   2094.95564
5  0.402010     1294.023377    2038.068734  3137.629147   2094.95564

The tool consumes an IODataset. We assemble it from the convergence table with the dataframe_to_dataset helper, using the name{group} naming convention to place dx in the input group and the four stresses in the output group:

dataset = dataframe_to_dataset(
    convergence_table.rename(
        columns={
            "dx": "dx{inputs}",
            "sigma_xx_probe": "sigma_xx_probe{outputs}",
            "sigma_xx_peak": "sigma_xx_peak{outputs}",
            "sigma_xx_r": "sigma_xx_r{outputs}",
            "sigma_xx_d0": "sigma_xx_d0{outputs}",
        }
    )
)

Out:

C:\Users\sebastien.bocquet\Dev\vimseo\src\vimseo\utilities\datasets.py:379: FutureWarning:

Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]`

C:\Users\sebastien.bocquet\Dev\vimseo\src\vimseo\utilities\datasets.py:379: FutureWarning:

Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]`

C:\Users\sebastien.bocquet\Dev\vimseo\src\vimseo\utilities\datasets.py:379: FutureWarning:

Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]`

C:\Users\sebastien.bocquet\Dev\vimseo\src\vimseo\utilities\datasets.py:379: FutureWarning:

Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]`

C:\Users\sebastien.bocquet\Dev\vimseo\src\vimseo\utilities\datasets.py:379: FutureWarning:

Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]`

A smoothly converging quantity

We first verify the smooth probe stress. It converges monotonically, so the power-law fit recovers the converged stress with a small residual. The three-point Richardson cross-validation may still return nan here: it is fragile as soon as one grid triplet is ill-conditioned, which is one of the motivations for the robust estimators.

verificator = DiscretizationSolutionVerification(
    directory_naming_method=DirectoryNamingMethod.NUMBERED,
    working_directory="DiscretizationSolutionVerification_probe",
)
verificator.execute(
    simulated_data=dataset,
    element_size_variable_name="dx",
    abscissa_name="dx",
    output_name="sigma_xx_probe",
)
extrapolation = verificator.result.extrapolation
print("Richardson q_extrap:", extrapolation["q_extrap"])
print(
    f"Power-law fit: q_converged={extrapolation['q_converged_fit']:.2f}, "
    f"order={extrapolation['order_fit']:.2f}, rmse={extrapolation['fit_rmse']:.2g}"
)
print(
    f"Robust median: q_converged={extrapolation['q_converged_robust']:.2f} "
    f"+/- {extrapolation['q_converged_robust_band']:.2g}"
)
# The tool selects Richardson when available and falls back to a palliative
# otherwise; ``q_converged_method`` says which one was used.
print(
    f"Selected converged value: {extrapolation['q_converged']:.2f} "
    f"(method: {extrapolation['q_converged_method']})"
)

Out:

C:\Users\sebastien.bocquet\Dev\vimseo\.tox\doc\Lib\site-packages\pydantic\main.py:387: UserWarning:

Pydantic serializer warnings:
  Expected `str` but got `tuple` with value `(FieldInfo(annotation=Non... assess convergence.'),)` - serialized value may not be as expected

    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_probe
    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_probe\CustomDOETool
 WARNING - 09:18:31: Richardson extrapolation could not be computed (returned nan); falling back to the palliative 'power_law_fit (palliative)' for output 'sigma_xx_probe'. This is not the reference Richardson method.
Richardson q_extrap: nan
Power-law fit: q_converged=1294.77, order=4.00, rmse=0.92
Robust median: q_converged=1295.37 +/- 0.15
Selected converged value: 1294.77 (method: power_law_fit (palliative))

The convergence-fit plot shows the sampled stress, the fitted power law and the converged-value estimates at a null element size:

figures = verificator.plot_results(
    verificator.result,
    save=False,
    show=True,
    directory_path=verificator.working_directory,
)
figures["convergence_fit"]

Out:

    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_probe
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_probe
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_probe
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_probe
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A sawtooth quantity

We now verify the peak stress. Its non-monotone (sawtooth) behaviour makes the three-point Richardson extrapolation fail (nan), so the Richardson-based indicators are unavailable:

verificator_peak = DiscretizationSolutionVerification(
    directory_naming_method=DirectoryNamingMethod.NUMBERED,
    working_directory="DiscretizationSolutionVerification_peak",
)
verificator_peak.execute(
    simulated_data=dataset,
    element_size_variable_name="dx",
    abscissa_name="dx",
    output_name="sigma_xx_peak",
)
extrapolation_peak = verificator_peak.result.extrapolation
print(
    "Richardson q_extrap (nan expected for sawtooth):", extrapolation_peak["q_extrap"]
)
# The residual and band are now much larger than for the smooth probe, correctly
# flagging that this converged value is far less trustworthy.
print(
    f"Power-law fit: q_converged={extrapolation_peak['q_converged_fit']:.2f}, "
    f"order={extrapolation_peak['order_fit']:.2f}, rmse={extrapolation_peak['fit_rmse']:.2g}"
)
print(
    f"Robust median: q_converged={extrapolation_peak['q_converged_robust']:.2f} "
    f"+/- {extrapolation_peak['q_converged_robust_band']:.2g}"
)
# Richardson failed here, so the tool falls back to a palliative and says so.
print(
    f"Selected converged value: {extrapolation_peak['q_converged']:.2f} "
    f"(method: {extrapolation_peak['q_converged_method']})"
)

Out:

C:\Users\sebastien.bocquet\Dev\vimseo\.tox\doc\Lib\site-packages\pydantic\main.py:387: UserWarning:

Pydantic serializer warnings:
  Expected `str` but got `tuple` with value `(FieldInfo(annotation=Non... assess convergence.'),)` - serialized value may not be as expected

    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_peak
    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_peak\CustomDOETool
 WARNING - 09:18:31: Richardson extrapolation could not be computed (returned nan); falling back to the palliative 'power_law_fit (palliative)' for output 'sigma_xx_peak'. This is not the reference Richardson method.
Richardson q_extrap (nan expected for sawtooth): nan
Power-law fit: q_converged=1897.80, order=4.00, rmse=96
Robust median: q_converged=1937.17 +/- 1.5e+02
Selected converged value: 1897.80 (method: power_law_fit (palliative))

Two plots are produced. The convergence-fit plot (on all grid points) stays informative: it overlays the power-law fit (with its fitted order and residual) and the model-free median of the finest grids (with its uncertainty band) on the sawtooth data:

figures_peak = verificator_peak.plot_results(
    verificator_peak.result,
    save=False,
    show=True,
    directory_path=verificator_peak.working_directory,
)
figures_peak["convergence_fit"]

Out:

    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_peak
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_peak
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_peak
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_peak
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Discussion: smooth versus sawtooth

Putting the two diagnostics side by side for the two outputs tells a consistent story about whether the mesh is in the asymptotic (converged) regime.

Cross-validation plot (smooth probe on the left, sawtooth peak on the right). For the smooth probe stress the leave-one-grid-out folds almost overlap and all extrapolate to nearly the same value, so the cross-validation band is tight: dropping any grid barely moves the estimate, a sign that the converged value is trustworthy. For the sawtooth peak stress the folds scatter and extrapolate to markedly different values, giving a wide band: the estimate depends heavily on which grids are used.

side_by_side(
    figures["convergence_cross_validation"],
    figures_peak["convergence_cross_validation"],
    "smooth probe",
    "sawtooth peak",
    "sigma_xx",
)

Relative-error plot (smooth probe on the left, sawtooth peak on the right). For the probe the relative error with respect to the converged value decreases monotonically as the element size shrinks -- the expected asymptotic behaviour. For the peak it oscillates and does not settle, showing that the quantity never enters the asymptotic regime, so no meaningful discretization order (and hence no reliable Richardson extrapolation) exists.

side_by_side(
    figures["relative_error_versus_element_size"],
    figures_peak["relative_error_versus_element_size"],
    "smooth probe",
    "sawtooth peak",
    "relative error",
)

The two diagnostics agree, and they agree with the fit residual and the robust band reported earlier: the probe stress is converged and its palliative value is reliable, whereas the peak stress is dominated by mesh sampling noise and its "converged value" should be treated with caution. In practice, a wide cross-validation band or a non-monotone relative error is the signal to refine the mesh further (or to pick a smoother quantity of interest) before trusting the extrapolation.

Native analytic outputs: a mesh-independent baseline

The model's native scalar outputs sigma_xx_r and sigma_xx_d0 are the hole-edge stresses evaluated directly on the analytic Tan solution (at the hole radius, and one stress-point distance d0 beyond it). They do not read the discretised field at all, so they are exactly mesh-independent. They were collected in the same loop as the field-derived quantities above, so no extra model runs are needed here -- we just verify them from the same dataset. Verifying them is a useful sanity check and a third reference behaviour, next to the smooth and sawtooth field quantities.

native_verificators = {}
for native_output in ("sigma_xx_r", "sigma_xx_d0"):
    verificator_native = DiscretizationSolutionVerification(
        directory_naming_method=DirectoryNamingMethod.NUMBERED,
        working_directory=f"DiscretizationSolutionVerification_{native_output}",
    )
    verificator_native.execute(
        simulated_data=dataset,
        element_size_variable_name="dx",
        abscissa_name="dx",
        output_name=native_output,
    )
    native_verificators[native_output] = verificator_native
    native_extrapolation = verificator_native.result.extrapolation
    print(
        f"{native_output}: converged={native_extrapolation['q_converged']:.4f} "
        f"(method: {native_extrapolation['q_converged_method']})"
    )

Out:

C:\Users\sebastien.bocquet\Dev\vimseo\.tox\doc\Lib\site-packages\pydantic\main.py:387: UserWarning:

Pydantic serializer warnings:
  Expected `str` but got `tuple` with value `(FieldInfo(annotation=Non... assess convergence.'),)` - serialized value may not be as expected

    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_r
    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_r\CustomDOETool
sigma_xx_r: converged=3137.6291 (method: richardson)
C:\Users\sebastien.bocquet\Dev\vimseo\.tox\doc\Lib\site-packages\pydantic\main.py:387: UserWarning:

Pydantic serializer warnings:
  Expected `str` but got `tuple` with value `(FieldInfo(annotation=Non... assess convergence.'),)` - serialized value may not be as expected

    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_d0
    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_d0\CustomDOETool
sigma_xx_d0: converged=2094.9556 (method: richardson)

Because the values are constant, Richardson succeeds trivially and returns the constant as the converged value (no palliative needed). The convergence-fit plots are flat, side by side (sigma_xx_r on the left, sigma_xx_d0 on the right):

native_figures = {
    name: verificator.plot_results(
        verificator.result,
        save=False,
        show=True,
        directory_path=verificator.working_directory,
    )
    for name, verificator in native_verificators.items()
}
side_by_side(
    native_figures["sigma_xx_r"]["convergence_fit"],
    native_figures["sigma_xx_d0"]["convergence_fit"],
    "sigma_xx_r",
    "sigma_xx_d0",
    "sigma_xx",
)

Out:

    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_r
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_r
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_r
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_d0
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_d0
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    INFO - 09:18:31: Working directory is C:\Users\sebastien.bocquet\Dev\vimseo\docs\runnable_examples\05_solution_verification\DiscretizationSolutionVerification_sigma_xx_d0
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The relative-error plots sit at zero: these outputs carry no discretization error, in contrast with the field-sampled quantities above. This is exactly the behaviour expected of an analytical model output, and a reassuring baseline for the solution-verification workflow.

side_by_side(
    native_figures["sigma_xx_r"]["relative_error_versus_element_size"],
    native_figures["sigma_xx_d0"]["relative_error_versus_element_size"],
    "sigma_xx_r",
    "sigma_xx_d0",
    "relative error",
)

Comparison of the fields

Beyond the scalar convergence, the whole sigma_xx field can be compared between the coarsest and the finest grid. Both are exact at their nodes, but the finer grid resolves the stress concentration around the hole far better:

coarse = model_results[grid_resolutions[0]].fields["flux"][0]
fine = model_results[grid_resolutions[-1]].fields["flux"][0]
x_coarse, y_coarse, z_coarse = coarse.to_structured_grid("sigma_xx")
x_fine, y_fine, z_fine = fine.to_structured_grid("sigma_xx")

color_min = float(np.nanmin([np.nanmin(z_coarse), np.nanmin(z_fine)]))
color_max = float(np.nanmax([np.nanmax(z_coarse), np.nanmax(z_fine)]))

fig = make_subplots(
    rows=1,
    cols=2,
    shared_yaxes=True,
    subplot_titles=(
        f"coarse grid ({grid_resolutions[0]})",
        f"fine grid ({grid_resolutions[-1]})",
    ),
)
fig.add_trace(
    go.Heatmap(
        x=x_coarse, y=y_coarse, z=z_coarse.T, coloraxis="coloraxis", name="coarse"
    ),
    row=1,
    col=1,
)
fig.add_trace(
    go.Heatmap(x=x_fine, y=y_fine, z=z_fine.T, coloraxis="coloraxis", name="fine"),
    row=1,
    col=2,
)
fig.update_layout(
    title={
        "text": "sigma_xx field around the hole",
        "x": 0.5,
        "xanchor": "center",
        "y": 0.98,
        "yanchor": "top",
    },
    # Extra top margin so the centred figure title clears the two panel titles
    # that ``make_subplots`` pins just above the plot area.
    margin={"t": 110},
    coloraxis={
        "colorscale": "Viridis",
        "cmin": color_min,
        "cmax": color_max,
        "colorbar": {"title": "sigma_xx"},
    },
)
# Equal aspect ratio so the hole stays circular in both panels:
fig.update_yaxes(scaleanchor="x", row=1, col=1)
fig.update_yaxes(scaleanchor="x2", row=1, col=2)
fig

Total running time of the script: ( 0 minutes 0.901 seconds)

Download Python source code: plot_tan_open_hole_convergence.py

Download Jupyter notebook: plot_tan_open_hole_convergence.ipynb

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1. Lewis Fry Richardson. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society A, 210:307–357, 1911. doi:https://doi.org/10.1098/rsta.1911.0009. ↩

2. Petr Krysl. Confidence intervals for richardson extrapolation in solid mechanics. J. Verif. Valid. Uncert., 7(3):031005, 2022. doi:https://doi.org/10.1115/1.4055728. ↩