.. only:: html

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

        Click :ref:`here <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
=====================

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}



.. code-block:: default

    from __future__ import print_function
    import openturns as ot
    import openturns.viewer as viewer
    from matplotlib import pylab as plt
    import math as m
    ot.Log.Show(ot.Log.NONE)








Create a mesh


.. code-block:: default

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








Create the function that acts the values of the mesh


.. code-block:: default

    h = ot.SymbolicFunction(['t', 'x1', 'x2'],  ['t+x1^2+x2^2'])








Create the field function


.. code-block:: default

    f = ot.VertexValueFunction(h, mesh)








Evaluate f


.. code-block:: default

    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

    <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>0.9387123</TD><TD>0.3737019</TD><TD>1.020834</TD></TR>
    <TR><TH>1</TH><TD>1</TD><TD>-0.450642</TD><TD>-0.02880025</TD><TD>1.203908</TD></TR>
    <TR><TH>2</TH><TD>2</TD><TD>-0.5263226</TD><TD>0.5003629</TD><TD>2.527378</TD></TR>
    <TR><TH>3</TH><TD>3</TD><TD>-1.794296</TD><TD>0.6534551</TD><TD>6.646503</TD></TR>
    <TR><TH>4</TH><TD>4</TD><TD>0.257082</TD><TD>-0.5985328</TD><TD>4.424333</TD></TR>
    <TR><TH>5</TH><TD>5</TD><TD>1.538375</TD><TD>-2.040349</TD><TD>11.52962</TD></TR>
    <TR><TH>6</TH><TD>6</TD><TD>-0.3003127</TD><TD>-0.3696585</TD><TD>6.226835</TD></TR>
    <TR><TH>7</TH><TD>7</TD><TD>0.4155438</TD><TD>-0.06083329</TD><TD>7.176377</TD></TR>
    <TR><TH>8</TH><TD>8</TD><TD>1.205391</TD><TD>-1.452755</TD><TD>11.56346</TD></TR>
    <TR><TH>9</TH><TD>9</TD><TD>-1.157649</TD><TD>-0.9877864</TD><TD>11.31587</TD></TR>
    </TABLE>
    <br />
    <br />


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  0.002 seconds)


.. _sphx_glr_download_auto_functional_modeling_field_functions_plot_vertexvalue_function.py:


.. only :: html

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



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

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



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

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


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_