.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_functional_modeling/field_functions/plot_vertexvalue_function.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_functional_modeling_field_functions_plot_vertexvalue_function.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_functional_modeling_field_functions_plot_vertexvalue_function.py:


Vertex value function
=====================

.. GENERATED FROM PYTHON SOURCE LINES 6-29

A vertex value function
:math:`f_{vertexvalue}: \mathcal{D} \times \mathbb{R}^d \rightarrow \mathcal{D} \times \mathbb{R}^q` is a
particular field function that lets invariant the mesh of a field
and defined by a function :math:`h : \mathbb{R}^n \times \mathbb{R}^d  \rightarrow \mathbb{R}^q` such that:

.. math::
   \begin{aligned} f_{vertexvalue}(\underline{t}, \underline{x})=(\underline{t}, h(\underline{t},\underline{x}))\end{aligned}

Let's note that the input dimension of :math:`f_{vertexvalue}` still design the
dimension of :math:`\underline{x}` : :math:`d`. Its output dimension is equal to :math:`q`.

The creation of the *VertexValueFunction* object requires the
function :math:`h` and the integer :math:`n`: the dimension of the
vertices of the mesh :math:`\mathcal{M}`.

This example illustrates the creation of a field from the function
:math:`h:\mathbb{R}\times\mathbb{R}^2` such as:

.. math::
   \begin{aligned}
      h(\underline{t}, \underline{x})=(t+x_1^2+x_2^2)
   \end{aligned}


.. GENERATED FROM PYTHON SOURCE LINES 32-36

.. code-block:: Python

    import openturns as ot

    ot.Log.Show(ot.Log.NONE)








.. GENERATED FROM PYTHON SOURCE LINES 37-38

Create a mesh

.. GENERATED FROM PYTHON SOURCE LINES 38-41

.. code-block:: Python

    N = 100
    mesh = ot.RegularGrid(0.0, 1.0, N)








.. GENERATED FROM PYTHON SOURCE LINES 42-43

Create the function that acts the values of the mesh

.. GENERATED FROM PYTHON SOURCE LINES 43-45

.. code-block:: Python

    h = ot.SymbolicFunction(["t", "x1", "x2"], ["t+x1^2+x2^2"])








.. GENERATED FROM PYTHON SOURCE LINES 46-47

Create the field function

.. GENERATED FROM PYTHON SOURCE LINES 47-49

.. code-block:: Python

    f = ot.VertexValueFunction(h, mesh)








.. GENERATED FROM PYTHON SOURCE LINES 50-51

Evaluate f

.. GENERATED FROM PYTHON SOURCE LINES 51-59

.. code-block:: Python

    inF = ot.Normal(2).getSample(N)
    outF = f(inF)

    # print input/output at first 10 mesh nodes
    txy = mesh.getVertices()
    txy.stack(inF)
    txy.stack(outF)
    txy[:10]





.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <table>
      <tr><td></td><th>t</th><th>X0</th><th>X1</th><th>y0</th></tr>
      <tr><th>0</th><td>0</td><td>0.02772026</td><td>0.1697782</td><td>0.02959305</td></tr>
      <tr><th>1</th><td>1</td><td>-0.730191</td><td>-0.05513265</td><td>1.536219</td></tr>
      <tr><th>2</th><td>2</td><td>-0.4083213</td><td>0.6251446</td><td>2.557532</td></tr>
      <tr><th>3</th><td>3</td><td>-0.03512366</td><td>-1.551634</td><td>5.408803</td></tr>
      <tr><th>4</th><td>4</td><td>0.06106403</td><td>0.5456437</td><td>4.301456</td></tr>
      <tr><th>5</th><td>5</td><td>1.012743</td><td>0.3416257</td><td>6.142357</td></tr>
      <tr><th>6</th><td>6</td><td>1.0148</td><td>1.861681</td><td>10.49568</td></tr>
      <tr><th>7</th><td>7</td><td>-1.89551</td><td>0.68089</td><td>11.05657</td></tr>
      <tr><th>8</th><td>8</td><td>0.5093272</td><td>-0.3883912</td><td>8.410262</td></tr>
      <tr><th>9</th><td>9</td><td>-0.878559</td><td>0.1473901</td><td>9.79359</td></tr>
    </table>
    </div>
    <br />
    <br />


.. _sphx_glr_download_auto_functional_modeling_field_functions_plot_vertexvalue_function.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_vertexvalue_function.ipynb <plot_vertexvalue_function.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_vertexvalue_function.py <plot_vertexvalue_function.py>`