.. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:here  to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_probabilistic_modeling_stochastic_processes_plot_mesh_creation.py: Creation of a mesh ================== In this example we will demonstrate how to create a mesh :math:\mathcal{M} associated to a domain :math:\mathcal{D} \in \mathbb{R}^n. A mesh is defined from vertices in :math:\mathbb{R}^n and a topology that connects the vertices: the simplices. The simplex :math:Indices([i_1,\dots, i_{n+1}]) relies the vertices of index :math:(i_1,\dots, i_{n+1}) in :math:\mathbb{N}^n. In dimension 1, a simplex is an interval :math:Indices([i_1,i_2]); in dimension 2, it is a triangle :math:Indices([i_1,i_2, i_3]). The library enables to easily create a mesh which is a box of dimension :math:d=1 or :math:d=2 regularly meshed in all its directions, thanks to the object IntervalMesher. Consider :math:X: \Omega \times \mathcal{D} \rightarrow \mathbb{R}^d a multivariate stochastic process of dimension :math:d, where :math:\mathcal{D} \in \mathbb{R}^n. The mesh :math:\mathcal{M} is a discretization of the domain :math:\mathcal{D}. .. 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) Define a one dimensional mesh .. code-block:: default vertices = [[0.5], [1.5], [2.1], [2.7]] simplicies = [[0, 1], [1, 2], [2, 3]] mesh1D = ot.Mesh(vertices, simplicies) graph1 = mesh1D.draw() graph1.setTitle('One dimensional mesh') view = viewer.View(graph1) .. image:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_mesh_creation_001.png :alt: One dimensional mesh :class: sphx-glr-single-img Define a bi dimensional mesh .. code-block:: default vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [1.5, 1.0], [2.0, 1.5], [0.5, 1.5]] simplicies = [[0, 1, 2], [1, 2, 3], [2, 3, 4], [2, 4, 5], [0, 2, 5]] mesh2D = ot.Mesh(vertices, simplicies) graph2 = mesh2D.draw() graph2.setTitle('Bidimensional mesh') graph2.setLegendPosition('bottomright') view = viewer.View(graph2) .. image:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_mesh_creation_002.png :alt: Bidimensional mesh :class: sphx-glr-single-img Case 2: Define a mesh wich is regularly meshed box in dimension 1 or 2 .. code-block:: default # Define the number of interval in each direction of the box myIndices = [5, 10] myMesher = ot.IntervalMesher(myIndices) # Create the mesh of the box [0., 2.] * [0., 4.] lowerBound=[0., 0.] upperBound=[2., 4.] myInterval = ot.Interval(lowerBound, upperBound) myMeshBox = myMesher.build(myInterval) mygraph3 = myMeshBox.draw() mygraph3.setTitle('Bidimensional mesh on a box') view = viewer.View(mygraph3) .. image:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_mesh_creation_003.png :alt: Bidimensional mesh on a box :class: sphx-glr-single-img Case 3: Define a mesh wich is regularly meshed box and Transform it through a fuunction .. code-block:: default myIndices = [20, 20] mesher = ot.IntervalMesher(myIndices) # r in [1., 2.] and theta in (0., pi] lowerBound2=[1.0, 0.0] upperBound2=[2.0, m.pi] myInterval = ot.Interval(lowerBound2, upperBound2) meshBox2 = mesher.build(myInterval) # define the mapping function f = ot.SymbolicFunction(['r', 'theta'], ['r*cos(theta)', 'r*sin(theta)']) oldVertices = meshBox2.getVertices() newVertices = f(oldVertices) meshBox2.setVertices(newVertices) graphMappedBox = meshBox2.draw() graphMappedBox.setTitle('Mapped box mesh') view = viewer.View(graphMappedBox) .. image:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_mesh_creation_004.png :alt: Mapped box mesh :class: sphx-glr-single-img Create a mesh of dimension 2: the heart .. code-block:: default def meshHeart(ntheta, nr): # First, build the nodes nodes = ot.Sample(0, 2) nodes.add([0.0, 0.0]) for j in range(ntheta): theta = (m.pi * j) / ntheta if (abs(theta - 0.5 * m.pi) < 1e-10): rho = 2.0 elif (abs(theta) < 1e-10) or (abs(theta-m.pi) < 1e-10): rho = 0.0 else: absTanTheta = abs(m.tan(theta)) rho = absTanTheta**(1.0 / absTanTheta) + m.sin(theta) cosTheta = m.cos(theta) sinTheta = m.sin(theta) for k in range(nr): tau = (k + 1.0) / nr r = rho * tau nodes.add([r * cosTheta, r * sinTheta - tau]) # Second, build the triangles triangles = [] ## First heart for j in range(ntheta): triangles.append([0, 1 + j * nr, 1 + ((j + 1) % ntheta)* nr]) # Other hearts for j in range(ntheta): for k in range(nr-1): i0 = k + 1 + j * nr i1 = k + 2 + j * nr i2 = k + 2 + ((j + 1) % ntheta) * nr i3 = k + 1 + ((j + 1) % ntheta) * nr triangles.append([i0, i1, i2%(nr*ntheta)]) triangles.append([i0, i2, i3%(nr*ntheta)]) return ot.Mesh(nodes, triangles) mesh4 = meshHeart(48, 16) graphMesh = mesh4.draw() graphMesh.setTitle('Bidimensional mesh') graphMesh.setLegendPosition('') view = viewer.View(graphMesh) plt.show() .. image:: /auto_probabilistic_modeling/stochastic_processes/images/sphx_glr_plot_mesh_creation_005.png :alt: Bidimensional mesh :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 2.259 seconds) .. _sphx_glr_download_auto_probabilistic_modeling_stochastic_processes_plot_mesh_creation.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_mesh_creation.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: plot_mesh_creation.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _