Mesh¶

class Mesh(*args)

Mesh.

Available constructors:

Mesh(dim=1)

Mesh(vertices)

Mesh(vertices, simplices)

Parameters: dim : int, The dimension of the vertices. By default, it creates only one vertex of dimension with components equal to 0. vertices : 2-d sequence of float Vertices’ coordinates in . simplices : 2-d sequence of int List of simplices defining the topology of the mesh. The simplex connects the vertices of indices in . In dimension 1, a simplex is an interval ; in dimension 2, it is a triangle .

Examples

>>> import openturns as ot
>>> # Define the vertices of the mesh
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [1.5, 1.0]]
>>> # Define the simplices of the mesh
>>> simplices = [[0, 1, 2], [1, 2, 3]]
>>> # Create the mesh of dimension 2
>>> mesh2d = ot.Mesh(vertices, simplices)


Methods

 ImportFromMSHFile() Import mesh from FreeFem 2-d mesh files. checkPointInSimplex(point, index) Check if a point is inside a simplex. checkPointInSimplexWithCoordinates(point, index) Check if a point is inside a simplex and returns its barycentric coordinates. computeP1Gram() Compute the P1 Lagrange finite element gram matrix of the mesh. computeSimplexVolume(index) Compute the volume of a given simplex. computeWeights() Compute an approximation of an integral defined over the mesh. contains(point) Check if the given point is inside of the domain. draw() Draw the mesh. draw1D() Draw the mesh of dimension 1. draw2D() Draw the mesh of dimension 2. draw3D(*args) Draw the bidimensional projection of the mesh. exportToVTKFile(fileName) Export the mesh to a VTK file. getClassName() Accessor to the object’s name. getDescription() Get the description of the vertices. getDimension() Get the dimension of the domain. getId() Accessor to the object’s id. getLowerBound() Get the lower bound of the domain. getName() Accessor to the object’s name. getNearestVertex(*args) Get the nearest vertex of a given point. getNearestVertexAndSimplexIndicesWithCoordinates(point) Get the index of the nearest vertex of a given point and the containing simplex if any, and returns its barycentric coordinates. getNearestVertexIndex(*args) Get the index of the nearest vertex of a given point. getNumericalVolume() Get the volume of the domain. getShadowedId() Accessor to the object’s shadowed id. getSimplex(index) Get the simplex of a given index. getSimplices() Get the simplices of the mesh. getSimplicesNumber() Get the number of simplices of the mesh. getUpperBound() Get the upper bound of the domain. getVertex(index) Get the vertex of a given index. getVertices() Get the vertices of the mesh. getVerticesNumber() Get the number of vertices of the mesh. getVerticesToSimplicesMap() Accessor to the map between vertices and simplices. getVisibility() Accessor to the object’s visibility state. getVolume() Get the geometric volume of the domain. hasName() Test if the object is named. hasVisibleName() Test if the object has a distinguishable name. isEmpty() Test whether the domain is empty or not. isNumericallyEmpty() Check if the domain is numerically empty. isRegular() Check if the mesh is regular (only for 1-d meshes). isValid() Check the mesh validity. numericallyContains(point) Check if the given point is inside of the discretization of the domain. setName(name) Accessor to the object’s name. setShadowedId(id) Accessor to the object’s shadowed id. setSimplices(simplices) Set the simplices of the mesh. setVertex(index, vertex) Set a vertex of a given index. setVertices(vertices) Set the vertices of the mesh. setVisibility(visible) Accessor to the object’s visibility state. streamToVTKFormat() Give a VTK representation of the mesh.
__init__(*args)

x.__init__(…) initializes x; see help(type(x)) for signature

static ImportFromMSHFile()

Import mesh from FreeFem 2-d mesh files.

Parameters: MSHFile : str A MSH ASCII file. mesh : Mesh Mesh defined in the file MSHFile.
checkPointInSimplex(point, index)

Check if a point is inside a simplex.

Parameters: point : sequence of float Point of dimension , the dimension of the vertices of the mesh. index : int Integer characterizes one simplex of the mesh. isInside : bool Flag telling whether point is inside the simplex of index index.

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> simplex = [[0, 1, 2]]
>>> mesh2d = ot.Mesh(vertices, simplex)
>>> # Create a point A inside the simplex
>>> pointA = [0.6, 0.3]
>>> print(mesh2d.checkPointInSimplex(pointA, 0))
True
>>> # Create a point B outside the simplex
>>> pointB = [1.1, 0.6]
>>> print(mesh2d.checkPointInSimplex(pointB, 0))
False

checkPointInSimplexWithCoordinates(point, index)

Check if a point is inside a simplex and returns its barycentric coordinates.

Parameters: point : sequence of float Point of dimension , the dimension of the vertices of the mesh. index : int Integer characterizes one simplex of the mesh. isInside : bool Flag telling whether point is inside the simplex of index index. coordinates : Point The barycentric coordinates of the given point wrt the vertices of the simplex .

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> simplex = [[0, 1, 2]]
>>> mesh2d = ot.Mesh(vertices, simplex)
>>> # Create a point A inside the simplex
>>> pointA = [0.6, 0.3]
>>> print(mesh2d.checkPointInSimplexWithCoordinates(pointA, 0))
[True, class=Point name=Unnamed dimension=3 values=[0.4,0.3,0.3]]
>>> # Create a point B outside the simplex
>>> pointB = [1.1, 0.6]
>>> print(mesh2d.checkPointInSimplexWithCoordinates(pointB, 0))
[False, class=Point name=Unnamed dimension=3 values=[-0.1,0.5,0.6]]

computeP1Gram()

Compute the P1 Lagrange finite element gram matrix of the mesh.

Returns: P1 Lagrange finite element gram matrix of the mesh.

Notes

The P1 Lagrange finite element space associated to a mesh with vertices is the space of piecewise-linear functions generated by the functions , where , for and the restriction of to any simplex is an affine function. The vertices that are not included into at least one simplex are not taken into account.

The gram matrix of the mesh is defined as the symmetric positive definite matrix whose generic element is given by:

This method is used in several algorithms related to stochastic process representation such as the Karhunen-Loeve decomposition.

Examples

>>> import openturns as ot
>>> # Define the vertices of the mesh
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [1.5, 1.0]]
>>> # Define the simplices of the mesh
>>> simplices = [[0, 1, 2], [1, 2, 3]]
>>> # Create the mesh of dimension 2
>>> mesh2d = ot.Mesh(vertices, simplices)
>>> print(mesh2d.computeP1Gram())
[[ 0.0416667 0.0208333 0.0208333 0         ]
[ 0.0208333 0.0625    0.03125   0.0104167 ]
[ 0.0208333 0.03125   0.0625    0.0104167 ]
[ 0         0.0104167 0.0104167 0.0208333 ]]

computeSimplexVolume(index)

Compute the volume of a given simplex.

Parameters: index : int Integer characterizes one simplex of the mesh. volume : float Volume of the simplex of index index.

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> simplex = [[0, 1, 2]]
>>> mesh2d = ot.Mesh(vertices, simplex)
>>> print(mesh2d.computeSimplexVolume(0))
0.5

computeWeights()

Compute an approximation of an integral defined over the mesh.

Returns: weights : Point Weights such that an integral of a function over the mesh is a weighted sum of its values at the vertices.
contains(point)

Check if the given point is inside of the domain.

Parameters: point : sequence of float Point with the same dimension as the current domain’s dimension. isInside : bool Flag telling whether the given point is inside of the domain.
draw()

Draw the mesh.

Returns: graph : Graph If the dimension of the mesh is 1, it draws the corresponding interval, using the draw1D() method; if the dimension is 2, it draws the triangular simplices, using the draw2D() method; if the dimension is 3, it projects the simplices on the plane of the two first components, using the draw3D() method with its default parameters, superposing the simplices.
draw1D()

Draw the mesh of dimension 1.

Returns: graph : Graph Draws the line linking the vertices of the mesh when the mesh is of dimension 1.

Examples

>>> import openturns as ot
>>> from openturns.viewer import View
>>> vertices = [[0.5], [1.5], [2.1], [2.7]]
>>> simplices = [[0, 1], [1, 2], [2, 3]]
>>> mesh1d = ot.Mesh(vertices, simplices)
>>> # Create a graph
>>> aGraph = mesh1d.draw1D()
>>> # Draw the mesh
>>> View(aGraph).show()

draw2D()

Draw the mesh of dimension 2.

Returns: graph : Graph Draws the edges of each simplex, when the mesh is of dimension 2.

Examples

>>> import openturns as ot
>>> from openturns.viewer import View
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [1.5, 1.0]]
>>> simplices = [[0, 1, 2], [1, 2, 3]]
>>> mesh2d = ot.Mesh(vertices, simplices)
>>> # Create a graph
>>> aGraph = mesh2d.draw2D()
>>> # Draw the mesh
>>> View(aGraph).show()

draw3D(*args)

Draw the bidimensional projection of the mesh.

Available usages:

draw3D(drawEdge=True, thetaX=0.0, thetaY=0.0, thetaZ=0.0, shading=False, rho=1.0)

Parameters: drawEdge : bool Tells if the edge of each simplex has to be drawn. thetaX : float Gives the value of the rotation along the X axis in radian. thetaY : float Gives the value of the rotation along the Y axis in radian. thetaZ : float Gives the value of the rotation along the Z axis in radian. rotation : SquareMatrix Operates a rotation on the mesh before its projection of the plane of the two first components. shading : bool Enables to give a visual perception of depth and orientation. rho : float, Contraction factor of the simplices. If , all the simplices are contracted and appear deconnected: some holes are created, which enables to see inside the mesh. If , the simplices keep their initial size and appear connected. If , each simplex is reduced to its gravity center. graph : Graph Draws the bidimensional projection of the mesh on the plane.

Examples

>>> import openturns as ot
>>> from openturns.viewer import View
>>> from math import cos, sin, pi
>>> vertices = [[0.0, 0.0, 0.0], [0.0, 0.0, 1.0], [0.0, 1.0, 0.0],
...             [0.0, 1.0, 1.0], [1.0, 0.0, 0.0], [1.0, 0.0, 1.0],
...             [1.0, 1.0, 0.0], [1.0, 1.0, 1.0]]
>>> simplices = [[0, 1, 2, 4], [3, 5, 6, 7],[1, 2, 3, 6],
...              [1, 2, 4, 6], [1, 3, 5, 6], [1, 4, 5, 6]]
>>> mesh3d = ot.Mesh(vertices, simplices)
>>> # Create a graph
>>> aGraph = mesh3d.draw3D()
>>> # Draw the mesh
>>> View(aGraph).show()
>>> rotation = ot.SquareMatrix(3)
>>> rotation[0, 0] = cos(pi / 3.0)
>>> rotation[0, 1] = sin(pi / 3.0)
>>> rotation[1, 0] = -sin(pi / 3.0)
>>> rotation[1, 1] = cos(pi / 3.0)
>>> rotation[2, 2] = 1.0
>>> # Create a graph
>>> aGraph = mesh3d.draw3D(True, rotation, True, 1.0)
>>> # Draw the mesh
>>> View(aGraph).show()

exportToVTKFile(fileName)

Export the mesh to a VTK file.

Parameters: myVTKFile.vtk : str Name of the created file which contains the mesh and the associated random values that can be visualized with the open source software Paraview.
getClassName()

Accessor to the object’s name.

Returns: class_name : str The object class name (object.__class__.__name__).
getDescription()

Get the description of the vertices.

Returns: description : str Description of the vertices.

Examples

>>> import openturns as ot
>>> mesh = ot.Mesh()
>>> vertices = ot.Sample([[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]])
>>> vertices.setDescription(['X', 'Y'])
>>> mesh.setVertices(vertices)
>>> print(mesh.getDescription())
[X,Y]

getDimension()

Get the dimension of the domain.

Returns: dim : int Dimension of the domain.
getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getLowerBound()

Get the lower bound of the domain.

Returns: lower : Point The lower bound of an axes-aligned bounding box of the domain.
getName()

Accessor to the object’s name.

Returns: name : str The name of the object.
getNearestVertex(*args)

Get the nearest vertex of a given point.

Parameters: point : sequence of float Point of dimension , the dimension of the vertices of the mesh. vertex : Point Coordinates of the nearest vertex of point.

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> simplices = [[0, 1, 2]]
>>> mesh2d = ot.Mesh(vertices, simplices)
>>> point = [0.9, 0.4]
>>> print(mesh2d.getNearestVertex(point))
[1,0]

getNearestVertexAndSimplexIndicesWithCoordinates(point)

Get the index of the nearest vertex of a given point and the containing simplex if any, and returns its barycentric coordinates.

Parameters: point : sequence of float Point of dimension , the dimension of the vertices of the mesh. indices : Indices Collecton of 1 or 2 integers, the first one being the index of the vertex the closest to the given point and the second one the index of the containing simplex if the given point is inside of the mesh. coordinates : Point The barycentric coordinates of the given point wrt the vertices of the containing simplex. It is of dimension 0 if the point is not contained into the mesh.

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> simplex = [[0, 1, 2]]
>>> mesh2d = ot.Mesh(vertices, simplex)
>>> # Create a point A inside the simplex
>>> pointA = [0.6, 0.3]
>>> print(mesh2d.getNearestVertexAndSimplexIndicesWithCoordinates(pointA))
[[1,0], class=Point name=Unnamed dimension=3 values=[0.4,0.3,0.3]]
>>> # Create a point B outside the simplex
>>> pointB = [1.1, 0.6]
>>> print(mesh2d.getNearestVertexAndSimplexIndicesWithCoordinates(pointB))
[[2], class=Point name=Unnamed dimension=0 values=[]]

getNearestVertexIndex(*args)

Get the index of the nearest vertex of a given point.

Parameters: point : sequence of float Point of dimension , the dimension of the vertices of the mesh. index : int Index of the simplex the nearest of point according to the Euclidean norm.

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> simplices = [[0, 1, 2]]
>>> mesh2d = ot.Mesh(vertices, simplices)
>>> point = [0.9, 0.4]
>>> print(mesh2d.getNearestVertexIndex(point))
1

getNumericalVolume()

Get the volume of the domain.

Returns: volume : float Volume of the underlying mesh which is the discretization of the domain. For now, by default, it is equal to the geometrical volume.
getShadowedId()

Accessor to the object’s shadowed id.

Returns: id : int Internal unique identifier.
getSimplex(index)

Get the simplex of a given index.

Parameters: index : int Index characterizing one simplex of the mesh. indices : Indices Indices defining the simplex of index index. The simplex relies the vertices of index in . In dimension 1, a simplex is an interval ; in dimension 2, it is a triangle .

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [1.5, 1.0]]
>>> simplices = [[0, 1, 2], [1, 2, 3]]
>>> mesh2d = ot.Mesh(vertices, simplices)
>>> print(mesh2d.getSimplex(0))
[0,1,2]
>>> print(mesh2d.getSimplex(1))
[1,2,3]

getSimplices()

Get the simplices of the mesh.

Returns: indicesCollection : collection of Indices List of indices defining all the simplices. The simplex relies the vertices of index in . In dimension 1, a simplex is an interval ; in dimension 2, it is a triangle .

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0], [1.5, 1.0]]
>>> simplices = [[0, 1, 2], [1, 2, 3]]
>>> mesh2d = ot.Mesh(vertices, simplices)
>>> print(mesh2d.getSimplices())
[[0,1,2],[1,2,3]]

getSimplicesNumber()

Get the number of simplices of the mesh.

Returns: number : int Number of simplices of the mesh.
getUpperBound()

Get the upper bound of the domain.

Returns: upper : Point The upper bound of an axes-aligned bounding box of the domain.
getVertex(index)

Get the vertex of a given index.

Parameters: index : int Index characterizing one vertex of the mesh. vertex : Point Coordinates in of the vertex of index index, where is the dimension of the vertices of the mesh.

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> simplices = [[0, 1, 2]]
>>> mesh2d = ot.Mesh(vertices, simplices)
>>> print(mesh2d.getVertex(1))
[1,0]
>>> print(mesh2d.getVertex(0))
[0,0]

getVertices()

Get the vertices of the mesh.

Returns: vertices : Sample Coordinates in of the vertices, where is the dimension of the vertices of the mesh.

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> simplices = [[0, 1, 2]]
>>> mesh2d = ot.Mesh(vertices, simplices)
>>> print(mesh2d.getVertices())
0 : [ 0 0 ]
1 : [ 1 0 ]
2 : [ 1 1 ]

getVerticesNumber()

Get the number of vertices of the mesh.

Returns: number : int Number of vertices of the mesh.
getVerticesToSimplicesMap()

Accessor to the map between vertices and simplices.

Returns: verticesSimplicesMap : IndicesCollection For each vertex, list the vertices indices it belongs to.
getVisibility()

Accessor to the object’s visibility state.

Returns: visible : bool Visibility flag.
getVolume()

Get the geometric volume of the domain.

Returns: volume : float Geometrical volume of the domain.
hasName()

Test if the object is named.

Returns: hasName : bool True if the name is not empty.
hasVisibleName()

Test if the object has a distinguishable name.

Returns: hasVisibleName : bool True if the name is not empty and not the default one.
isEmpty()

Test whether the domain is empty or not.

Returns: isInside : bool True if the interior of the geometric domain is empty.
isNumericallyEmpty()

Check if the domain is numerically empty.

Returns: isInside : bool Flag telling whether the domain is numerically empty, i.e. if its numerical volume is inferior or equal to (defined in the ResourceMap: = DomainImplementation-SmallVolume).

Examples

>>> import openturns as ot
>>> domain = ot.Domain([1.0, 2.0], [1.0, 2.0])
>>> print(domain.isNumericallyEmpty())
True

isRegular()

Check if the mesh is regular (only for 1-d meshes).

Returns: isRegular : bool Tells if the mesh is regular or not.

Examples

>>> import openturns as ot
>>> vertices = [[0.5], [1.5], [2.4], [3.5]]
>>> simplices = [[0, 1], [1, 2], [2, 3]]
>>> mesh1d = ot.Mesh(vertices, simplices)
>>> print(mesh1d.isRegular())
False
>>> vertices = [[0.5], [1.5], [2.5], [3.5]]
>>> mesh1d = ot.Mesh(vertices, simplices)
>>> print(mesh1d.isRegular())
True

isValid()

Check the mesh validity.

Returns: validity : bool Tells if the mesh is valid i.e. if there is non-overlaping simplices, no unused vertex, no simplices with duplicate vertices and no coincident vertices.
numericallyContains(point)

Check if the given point is inside of the discretization of the domain.

Parameters: point : sequence of float Point with the same dimension as the current domain’s dimension. isInside : bool Flag telling whether the point is inside the discretized domain associated to the domain. For now, by default, the discretized domain is equal to the geometrical domain.
setName(name)

Accessor to the object’s name.

Parameters: name : str The name of the object.
setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters: id : int Internal unique identifier.
setSimplices(simplices)

Set the simplices of the mesh.

Parameters: indices : 2-d sequence of int List of indices defining all the simplices. The simplex relies the vertices of index in . In dimension 1, a simplex is an interval ; in dimension 2, it is a triangle .

Examples

>>> import openturns as ot
>>> mesh = ot.Mesh()
>>> simplices = [[0, 1, 2], [1, 2, 3]]
>>> mesh.setSimplices(simplices)

setVertex(index, vertex)

Set a vertex of a given index.

Parameters: index : int Index of the vertex to set. vertex : sequence of float Cordinates in of the vertex of index index, where is the dimension of the vertices of the mesh.

Examples

>>> import openturns as ot
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> simplices = [[0, 1, 2]]
>>> mesh = ot.Mesh(vertices, simplices)
>>> vertex = [0.0, 0.5]
>>> mesh.setVertex(0, vertex)
>>> print(mesh.getVertices())
0 : [ 0   0.5 ]
1 : [ 1   0   ]
2 : [ 1   1   ]

setVertices(vertices)

Set the vertices of the mesh.

Parameters: vertices : 2-d sequence of float Cordinates in of the vertices, where is the dimension of the vertices of the mesh.

Examples

>>> import openturns as ot
>>> mesh = ot.Mesh()
>>> vertices = [[0.0, 0.0], [1.0, 0.0], [1.0, 1.0]]
>>> mesh.setVertices(vertices)

setVisibility(visible)

Accessor to the object’s visibility state.

Parameters: visible : bool Visibility flag.
streamToVTKFormat()

Give a VTK representation of the mesh.

Returns: stream : str VTK representation of the mesh.