.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_reliability/reliability_analysis/plot_estimate_probability_form.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_reliability_reliability_analysis_plot_estimate_probability_form.py: Use the FORM - SORM algorithms ============================== .. GENERATED FROM PYTHON SOURCE LINES 7-18 In this example we estimate a failure probability with the `FORM` algorithm on the :ref:`cantilever beam ` example. More precisely, we show how to use the associated results: - the design point in both physical and standard space, - the probability estimation according to the FORM approximation, and the following SORM ones: Tvedt, Hohenbichler and Breitung, - the Hasofer reliability index and the generalized ones evaluated from the Breitung, Tvedt and Hohenbichler approximations, - the importance factors defined as the normalized director factors of the design point in the :math:`U`-space - the sensitivity factors of the Hasofer reliability index and the FORM probability, - the coordinates of the mean point in the standard event space. See :ref:`FORM ` and :ref:`SORM ` for theoretical details. .. GENERATED FROM PYTHON SOURCE LINES 20-22 Model definition ---------------- .. GENERATED FROM PYTHON SOURCE LINES 24-28 .. code-block:: Python from openturns.usecases import cantilever_beam import openturns as ot import openturns.viewer as otv .. GENERATED FROM PYTHON SOURCE LINES 29-30 We load the model from the usecases module : .. GENERATED FROM PYTHON SOURCE LINES 30-32 .. code-block:: Python cb = cantilever_beam.CantileverBeam() .. GENERATED FROM PYTHON SOURCE LINES 33-34 We use the input parameters distribution from the data class : .. GENERATED FROM PYTHON SOURCE LINES 34-37 .. code-block:: Python distribution = cb.distribution distribution.setDescription(["E", "F", "L", "I"]) .. GENERATED FROM PYTHON SOURCE LINES 38-39 We define the model .. GENERATED FROM PYTHON SOURCE LINES 39-41 .. code-block:: Python model = cb.model .. GENERATED FROM PYTHON SOURCE LINES 42-43 Create the event whose probability we want to estimate. .. GENERATED FROM PYTHON SOURCE LINES 45-50 .. code-block:: Python vect = ot.RandomVector(distribution) G = ot.CompositeRandomVector(model, vect) event = ot.ThresholdEvent(G, ot.Greater(), 0.3) event.setName("deviation") .. GENERATED FROM PYTHON SOURCE LINES 51-53 FORM Analysis ------------- .. GENERATED FROM PYTHON SOURCE LINES 55-56 Define a solver, here we use a :class:`~openturns.MultiStart` optimization based on :class:`~openturns.Cobyla` .. GENERATED FROM PYTHON SOURCE LINES 56-64 .. code-block:: Python startingSample = distribution.getSample(10) optimAlgo = ot.MultiStart(ot.Cobyla(), startingSample) optimAlgo.setMaximumCallsNumber(1000) optimAlgo.setMaximumAbsoluteError(1.0e-4) optimAlgo.setMaximumRelativeError(1.0e-4) optimAlgo.setMaximumResidualError(1.0e-4) optimAlgo.setMaximumConstraintError(1.0e-4) .. GENERATED FROM PYTHON SOURCE LINES 65-66 Run FORM .. GENERATED FROM PYTHON SOURCE LINES 66-70 .. code-block:: Python algo = ot.FORM(optimAlgo, event) algo.run() result = algo.getResult() .. GENERATED FROM PYTHON SOURCE LINES 71-73 Analysis of the results ----------------------- .. GENERATED FROM PYTHON SOURCE LINES 75-76 Probability .. GENERATED FROM PYTHON SOURCE LINES 76-78 .. code-block:: Python result.getEventProbability() .. rst-class:: sphx-glr-script-out .. code-block:: none 1.0916739674554962e-06 .. GENERATED FROM PYTHON SOURCE LINES 79-80 Hasofer reliability index .. GENERATED FROM PYTHON SOURCE LINES 80-82 .. code-block:: Python result.getHasoferReliabilityIndex() .. rst-class:: sphx-glr-script-out .. code-block:: none 4.735667964445085 .. GENERATED FROM PYTHON SOURCE LINES 83-84 Design point in the standard U* space. .. GENERATED FROM PYTHON SOURCE LINES 86-88 .. code-block:: Python print(result.getStandardSpaceDesignPoint()) .. rst-class:: sphx-glr-script-out .. code-block:: none [-0.665624,4.31233,1.23025,-1.36885] .. GENERATED FROM PYTHON SOURCE LINES 89-90 Design point in the physical X space. .. GENERATED FROM PYTHON SOURCE LINES 92-94 .. code-block:: Python print(result.getPhysicalSpaceDesignPoint()) .. rst-class:: sphx-glr-script-out .. code-block:: none [6.56566e+10,458.962,2.58907,1.34804e-07] .. GENERATED FROM PYTHON SOURCE LINES 95-96 Importance factors .. GENERATED FROM PYTHON SOURCE LINES 96-99 .. code-block:: Python graph = result.drawImportanceFactors() view = otv.View(graph) .. image-sg:: /auto_reliability/reliability_analysis/images/sphx_glr_plot_estimate_probability_form_001.svg :alt: Importance Factors from Design Point - deviation :srcset: /auto_reliability/reliability_analysis/images/sphx_glr_plot_estimate_probability_form_001.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 100-105 .. code-block:: Python marginalSensitivity, otherSensitivity = result.drawHasoferReliabilityIndexSensitivity() marginalSensitivity.setLegends(["E", "F", "L", "I"]) marginalSensitivity.setLegendPosition("bottom") view = otv.View(marginalSensitivity) .. image-sg:: /auto_reliability/reliability_analysis/images/sphx_glr_plot_estimate_probability_form_002.svg :alt: Hasofer Reliability Index Sensitivities - Marginal parameters - deviation :srcset: /auto_reliability/reliability_analysis/images/sphx_glr_plot_estimate_probability_form_002.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 106-111 .. code-block:: Python marginalSensitivity, otherSensitivity = result.drawEventProbabilitySensitivity() marginalSensitivity.setLegends(["E", "F", "L", "I"]) marginalSensitivity.setLegendPosition("bottom") view = otv.View(marginalSensitivity) .. image-sg:: /auto_reliability/reliability_analysis/images/sphx_glr_plot_estimate_probability_form_003.svg :alt: FORM - Event Probability Sensitivities - Marginal parameters - deviation :srcset: /auto_reliability/reliability_analysis/images/sphx_glr_plot_estimate_probability_form_003.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 112-113 Error history .. GENERATED FROM PYTHON SOURCE LINES 113-119 .. code-block:: Python optimResult = result.getOptimizationResult() graphErrors = optimResult.drawErrorHistory() graphErrors.setLegendPosition("bottom") graphErrors.setYMargin(0.0) view = otv.View(graphErrors) .. image-sg:: /auto_reliability/reliability_analysis/images/sphx_glr_plot_estimate_probability_form_004.svg :alt: Error history :srcset: /auto_reliability/reliability_analysis/images/sphx_glr_plot_estimate_probability_form_004.svg :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 120-121 Get additional results with SORM .. GENERATED FROM PYTHON SOURCE LINES 121-125 .. code-block:: Python algo = ot.SORM(optimAlgo, event) algo.run() sorm_result = algo.getResult() .. GENERATED FROM PYTHON SOURCE LINES 126-127 Reliability index with Breitung approximation .. GENERATED FROM PYTHON SOURCE LINES 127-129 .. code-block:: Python sorm_result.getGeneralisedReliabilityIndexBreitung() .. rst-class:: sphx-glr-script-out .. code-block:: none 4.914716102889629 .. GENERATED FROM PYTHON SOURCE LINES 130-131 ... with Hohenbichler approximation .. GENERATED FROM PYTHON SOURCE LINES 131-133 .. code-block:: Python sorm_result.getGeneralisedReliabilityIndexHohenbichler() .. rst-class:: sphx-glr-script-out .. code-block:: none 4.920092507179259 .. GENERATED FROM PYTHON SOURCE LINES 134-135 .. with Tvedt approximation .. GENERATED FROM PYTHON SOURCE LINES 135-137 .. code-block:: Python sorm_result.getGeneralisedReliabilityIndexTvedt() .. rst-class:: sphx-glr-script-out .. code-block:: none 4.923406200459035 .. GENERATED FROM PYTHON SOURCE LINES 138-139 SORM probability of the event with Breitung approximation .. GENERATED FROM PYTHON SOURCE LINES 139-141 .. code-block:: Python sorm_result.getEventProbabilityBreitung() .. rst-class:: sphx-glr-script-out .. code-block:: none 4.4455602114028477e-07 .. GENERATED FROM PYTHON SOURCE LINES 142-143 ... with Hohenbichler approximation .. GENERATED FROM PYTHON SOURCE LINES 143-145 .. code-block:: Python sorm_result.getEventProbabilityHohenbichler() .. rst-class:: sphx-glr-script-out .. code-block:: none 4.325165899286397e-07 .. GENERATED FROM PYTHON SOURCE LINES 146-147 ... with Tvedt approximation .. GENERATED FROM PYTHON SOURCE LINES 147-149 .. code-block:: Python sorm_result.getEventProbabilityTvedt() .. rst-class:: sphx-glr-script-out .. code-block:: none 4.252532653059983e-07 .. GENERATED FROM PYTHON SOURCE LINES 150-152 FORM analysis with finite difference gradient --------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 154-156 When the considered function has no analytical expression, the gradient may not be known. In this case, a constant step finite difference gradient definition may be used. .. GENERATED FROM PYTHON SOURCE LINES 158-175 .. code-block:: Python def cantilever_beam_python(X): E, F, L, II = X return [F * L**3 / (3 * E * II)] cbPythonFunction = ot.PythonFunction(4, 1, func=cantilever_beam_python) epsilon = [1e-8] * 4 # Here, a constant step of 1e-8 is used for every dimension gradStep = ot.ConstantStep(epsilon) cbPythonFunction.setGradient( ot.CenteredFiniteDifferenceGradient(gradStep, cbPythonFunction.getEvaluation()) ) G = ot.CompositeRandomVector(cbPythonFunction, vect) event = ot.ThresholdEvent(G, ot.Greater(), 0.3) event.setName("deviation") .. GENERATED FROM PYTHON SOURCE LINES 176-179 However, given the different nature of the model variables, a blended (variable) finite difference step may be preferable: The step depends on the location in the input space .. GENERATED FROM PYTHON SOURCE LINES 179-187 .. code-block:: Python gradStep = ot.BlendedStep(epsilon) cbPythonFunction.setGradient( ot.CenteredFiniteDifferenceGradient(gradStep, cbPythonFunction.getEvaluation()) ) G = ot.CompositeRandomVector(cbPythonFunction, vect) event = ot.ThresholdEvent(G, ot.Greater(), 0.3) event.setName("deviation") .. GENERATED FROM PYTHON SOURCE LINES 188-189 We can then run the FORM analysis in the same way as before: .. GENERATED FROM PYTHON SOURCE LINES 189-193 .. code-block:: Python algo = ot.FORM(optimAlgo, event) algo.run() result = algo.getResult() .. GENERATED FROM PYTHON SOURCE LINES 194-195 .. code-block:: Python otv.View.ShowAll() .. _sphx_glr_download_auto_reliability_reliability_analysis_plot_estimate_probability_form.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_estimate_probability_form.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_estimate_probability_form.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_estimate_probability_form.zip `