examples

Author: jschueller

doc/_static/beam.svg

@@ -3,27 +3,14 @@
 =====================================
 """
 # %%
-# We are interested in the evolution of an epidemic through a
-# population (for instance, Covid!). We have an epidemiologic model describing
-# the epidemic dynamics. More precisely, we focus on the evolution of the number
-# of people infected by the disease.
-#
-# .. image:: /_static/epid.png
-#    :scale: 80 %
-#    :alt: alternate text
-#    :align: center
-#
-# --------
-#
-# See the epidemiological model :doc:`here<../../fmus/epid>`.
-#
-# --------
-#
+# This example shows how to create a metamodel of the
+# :doc:`epidemiological model<../../examples/model_description>`.
 #
 # **To decrease the model simulation costs, let's create a metamodel.**
 #
-# Metamodeling a model which output depends on time is a difficult problem. We
-# will combine two methods: Karhunen-Loeve dimension reduction should precede the Kriging metamodeling.
+# Metamodeling a model with time-dependent output is a difficult problem. We
+# will combine two methods: Karhunen-Loeve dimension reduction should precede
+# the Kriging metamodeling.
 #
 # We will proceed the following way:
 #
@@ -61,20 +48,21 @@
 )
 
 # %%
-# We create a Monte-Carlo design of experiment, on which we
-# simulate the FMU.
+# We create a Monte-Carlo design of experiment, on which we simulate the FMU.
 # The simulation inputs and outputs will be used to train the metamodel.
 
 inputLaw = ot.Uniform(1.5, 2.5)
-inputSample = inputLaw.getSample(30)
+inputSample = inputLaw.getSample(10)
 outputFMUSample = function(inputSample)
 
 graph = outputFMUSample.draw().getGraph(0, 0)
 graph.setTitle("FMU simulations")
 graph.setXTitle("Time")
 graph.setYTitle("Number of infected")
-graph.setLegends([f"{line[0]:.3f}" for line in inputSample[:15]] + ["_"] * 15)
-view = otv.View(graph, legend_kw={"title": "infection rate", "loc": "upper left"})
+graph.setLegendFontSize(6.)
+graph.setLegends([f"{line[0]:.2f}" for line in inputSample[:10]])
+view = otv.View(graph,
+                legend_kw={"title": "infection rate", "loc": "upper left"})
 
 # %%
 # We define a function to visualize the upcoming Karhunen-Loeve modes.
@@ -133,7 +121,9 @@ def drawKL(scaledKL, KLev, mesh, title="Scaled KL modes"):
 # We metamodel the Karhunen-Loeve coefficients using ordinary Kriging.
 dim = inputSample.getDimension()  # only 1 input dimension
 univb = ot.ConstantBasisFactory(dim).build()  # univariate basis
-coll = [ot.AggregatedFunction([univb.build(i)] * n_mode) for i in range(univb.getSize())]
+coll = [ot.AggregatedFunction(
+    [univb.build(i)] * n_mode) for i in range(univb.getSize())]
+
 basis = ot.Basis(coll)  # multivariate basis
 covarianceModel = ot.SquaredExponential(dim)
 covarianceModel = ot.TensorizedCovarianceModel([covarianceModel] * n_mode)
@@ -145,8 +135,8 @@ def drawKL(scaledKL, KLev, mesh, title="Scaled KL modes"):
 metamodel = result.getMetaModel()
 
 # %%
-# We have created all pieces for a "PointToField" metamodel. Let put these
-# pieces together:
+# We have created all pieces for a "PointToField" metamodel.
+# Let put these pieces together:
 
 
 def globalMetamodel(sample):
@@ -225,12 +215,11 @@ def globalMetamodel(sample):
 # ----------------------
 #
 # The ``globalMetamodel`` (computationnally faster than the FMU) created with
-# the above script can now be used as a computationnally much cheaper
-# substitute to the FMU for
+# the above script can now be used as a faster substitute to the FMU for
 #
-# - `sensitivity analysis <openturns.github.io/openturns/latest/auto_reliability_sensitivity/index.html#sensitivity-analysis>`_,
-# - `parameter inference <openturns.github.io/openturns/latest/auto_calibration/index.html#bayesian-calibration>`_,
-# - `estimate a failure probability <openturns.github.io/openturns/latest/auto_reliability_sensitivity/index.html#reliability>`_,
+# - `sensitivity analysis <https://openturns.github.io/openturns/latest/auto_sensitivity_analysis/index.html>`_,
+# - `parameter inference <https://openturns.github.io/openturns/latest/auto_calibration/index.html>`_,
+# - `estimate a failure probability <https://openturns.github.io/openturns/latest/auto_reliability/index.html>`_,
 #
 # etc.
 

doc/examples/application/plot_metamodel.py

@@ -3,7 +3,7 @@
 ==================================================
 """
 # %%
-# A load is applied to a :doc:`cantilever beam<../model_description>`.
+# A load is applied to a :doc:`cantilever beam<../../examples/model_description>`.
 # The beam must not exceed a bending of 30 cm. The load (F), beam
 # Young's modulus (E), length (L) and section modulus (I) are uncertain.
 #

doc/examples/application/plot_reliability.py

@@ -55,3 +55,11 @@ Heat exchanger example
    :maxdepth: 1
 
    ../auto_example/f2f/index.rst
+
+Model descriptions
+------------------
+
+.. toctree::
+   :maxdepth: 1
+   
+   model_description

doc/examples/index.rst

@@ -1,7 +1,57 @@
+Models description
+==================
+
+Examples used in this documentation rely on several models.
+
+Following sections show how they works.
+
+The cantilever beam model
+-------------------------
+
+.. image:: /_static/beam.svg
+  :scale: 100 %
+  :alt: alternate text
+  :align: center
+
+We consider a cantilever beam defined by its Young’s modulus ``E``, its length ``L`` and its section modulus ``I``.
+One end of the cantilever beam is fixed in a wall and we apply a concentrated
+bending load ``F`` at the other end of the beam, resulting in a deviation ``Y``.
+The mechanical equation ruling the deviation is
+
+
+.. raw:: html
+
+   <div style="margin-left:20px;width:300px;height:120px;">
+.. math::
+
+   Y = \frac{FL^3}{3EI}
+
+.. raw:: html
+
+   </div>
+
+
+This model is implemented in Modelica language:
+
+.. code::
+
+   model deviation
+     output Real y;
+     input Real E (start=3.0e7);
+     input Real F (start=3.0e4);
+     input Real L (start=250);
+     input Real I (start=400);
+   equation
+     y=(F*L^3)/(3*E*I);
+   end deviation;
+
+
+
+
 The epidemiological model
-=========================
+--------------------------
 
-The epidemiological model describes epidemics which propagate through human contact.
+This model describes epidemics which propagate through human contact.
 An isolated population is considered, whose total number is constant.
 The people are divided in three categories:
 
@@ -10,17 +60,20 @@ The people are divided in three categories:
 * the Removed (who are either dead or immune).
 
 The disease can only be propagated from Infected to Susceptibles.
-This happens at a rate called ``infection rate`` (:math:`\beta`).
-An Infected becomes Removed after an infection duration (:math:`\gamma`) corresponding to the inverse of the ``healing_rate``.
+This happens at a rate called ``infection rate`` :math:`\beta`.
+An Infected becomes Removed after an infection duration :math:`\gamma` corresponding to the inverse of the ``healing_rate``.
 
-.. image:: ../_static/epid.png
-   :scale: 80 %
+.. image:: /_static/epid.png
+   :scale: 50 %
    :alt: alternate text
    :align: center
 
 The evolution of the number of Susceptibles, Infected and Removed over
 time writes:
 
+
+
+
 .. math::
 
    \begin{aligned}
@@ -29,6 +82,9 @@ time writes:
    \frac{\partial R}{\partial t}(t) &= \gamma I(t)
    \end{aligned}
 
+
+
+
 This model is implemented in Modelica language. The default simulation time is 50 units of time (days for instance).
 
 .. code::

doc/fmus/epid.rst doc/examples/model_description.rstdoc/fmus/epid.rst renamed to doc/examples/model_description.rst

@@ -45,11 +45,6 @@ The core features of `otfmi <https://github.com/openturns/otfmi>`__ are:
 
    application/application
 
-.. toctree::
-   :maxdepth: 1
-
-   fmus/index
-
 .. toctree::
    :maxdepth: 1