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# 11. `mesh` — Finite element meshes in pyFormex.

This module defines the Mesh class, which can be used to describe discrete geometrical models like those used in Finite Element models. It also contains some useful functions to create such models.

`Classes defined in module mesh`

class `mesh.``Mesh`(coords=None, elems=None, prop=None, eltype=None)

  A Mesh is a discrete geometrical model defined by nodes and elements.

  In the Mesh geometrical data model, the coordinates of all the points are gathered in a single twodimensional array with shape (ncoords,3). The individual geometrical elements are then described by indices into the coordinates array.

  This model has some advantages over the Formex data model (which stores all the points of all the elements by their coordinates):

  - a more compact storage, because coordinates of coinciding points are not repeated,
  - faster connectivity related algorithms.

  The downside is that geometry generating algorithms are far more complex and possibly slower.

  In pyFormex we therefore mostly use the Formex data model when creating geometry, but when we come to the point of exporting the geometry to file (and to other programs), a Mesh data model may be more adequate.

  The Mesh data model has at least the following attributes:

  - coords: (ncoords,3) shaped Coords object, holding the coordinates of all points in the Mesh;
  - elems: (nelems,nplex) shaped Connectivity object, defining the elements by indices into the Coords array. All values in elems should be in the range 0 <= value < ncoords.
  - prop: an array of element property numbers, default None.
  - eltype: an element type (a subclass of `Element`) or the name of an Element type, or None (default). If eltype is None, the eltype of the elems Connectivity table is used, and if that is missing, a default eltype is derived from the plexitude, by a call to `elements.elementType()`. In most cases the eltype can be set automatically. The user can override the default value, but an error will occur if the element type does not exist or does not match the plexitude.

  A Mesh can be initialized by its attributes (coords,elems,prop,eltype) or by a single geometric object that provides a toMesh() method.

  If only an element type is provided, a unit sized single element Mesh of that type is created. Without parameters, an empty Mesh is created.

  `setType`(eltype=None)

    Set the eltype from a character string.

    This function allows the user to change the element type of the Mesh. The input is a character string with the name of one of the element defined in elements.py. The function will only allow to set a type matching the plexitude of the Mesh.

    This method is seldom needed, because the applications should normally set the element type at creation time.

  `elType`()

    Return the element type of the Mesh.

  `elName`()

    Return the element name of the Mesh.

  `setNormals`(normals=None)

    Set/Remove the normals of the mesh.

  `getProp`()

    Return the properties as a numpy array (ndarray)

  `maxProp`()

    Return the highest property value used, or None

  `propSet`()

    Return a list with unique property values.

  `shallowCopy`(prop=None)

    Return a shallow copy.

    A shallow copy of a Mesh is a Mesh object using the same data arrays as the original Mesh. The only thing that can be changed is the property array. This is a convenient method to use the same Mesh with different property attributes.

  `toFormex`()

    Convert a Mesh to a Formex.

    The Formex inherits the element property numbers and eltype from the Mesh. Node property numbers however can not be translated to the Formex data model.

  `toMesh`()

    Convert to a Mesh.

    This just returns the Mesh object itself. It is provided as a convenience for use in functions that want work on different Geometry types.

  `toSurface`()

    Convert a Mesh to a TriSurface.

    Only Meshes of level 2 (surface) and 3 (volume) can be converted to a TriSurface. For a level 3 Mesh, the border Mesh is taken first. A level 2 Mesh is converted to element type ‘tri3’ and then to a TriSurface. The resulting TriSurface is only fully equivalent with the input Mesh if the latter has element type ‘tri3’.

    On success, returns a TriSurface corresponding with the input Mesh. If the Mesh can not be converted to a TriSurface, an error is raised.

  `toCurve`()

    Convert a Mesh to a Curve.

    If the element type is one of ‘line*’ types, the Mesh is converted to a Curve. The type of the returned Curve is dependent on the element type of the Mesh:

    - ‘line2’: PolyLine,
    - ‘line3’: BezierSpline (degree 2),
    - ‘line4’: BezierSpline (degree 3)

    This is equivalent with

        self.toFormex().toCurve()

    Any other type will raise an exception.

  `nedges`()

    Return the number of edges.

    This returns the number of rows that would be in getEdges(), without actually constructing the edges. The edges are not fused!

  `info`()

    Return short info about the Mesh.

    This includes only the shape of the coords and elems arrays.

  `report`(full=True)

    Create a report on the Mesh shape and size.

    The report always contains the number of nodes, number of elements, plexitude, dimensionality, element type, bbox and size. If full==True(default), it also contains the nodal coordinate list and element connectivity table. Because the latter can be rather bulky, they can be switched off. (Though numpy will limit the printed output).

    TODO: We should add an option here to let numpy print the full tables.

  `centroids`()

    Return the centroids of all elements of the Mesh.

    The centroid of an element is the point whose coordinates are the mean values of all points of the element. The return value is a Coords object with nelems points.

  `bboxes`()

    Returns the bboxes of all elements in the Mesh.

    Returns a coords with shape (nelems,2,3). Along the axis 1 are stored the minimal and maximal values of the Coords in each of the elements of the Mesh.

  `getCoords`()

    Get the coords data.

    Returns the full array of coordinates stored in the Mesh object. Note that this may contain points that are not used in the mesh. `compact()` will remove the unused points.

  `getElems`()

    Get the elems data.

    Returns the element connectivity data as stored in the object.

  `getLowerEntities`(level=-1, unique=False)

    Get the entities of a lower dimensionality.

    If the element type is defined in the `elements` module, this returns a Connectivity table with the entities of a lower dimensionality. The full list of entities with increasing dimensionality 0,1,2,3 is:

        ['points', 'edges', 'faces', 'cells' ]

    If level is negative, the dimensionality returned is relative to that of the caller. If it is positive, it is taken absolute. Thus, for a Mesh with a 3D element type, getLowerEntities(-1) returns the faces, while for a 2D element type, it returns the edges. For both meshes however, getLowerEntities(+1) returns the edges.

    By default, all entities for all elements are returned and common entities will appear multiple times. Specifying unique=True will return only the unique ones.

    The return value may be an empty table, if the element type does not have the requested entities (e.g. the ‘point’ type). If the eltype is not defined, or the requested entity level is outside the range 0..3, the return value is None.

  `getNodes`()

    Return the set of unique node numbers in the Mesh.

    This returns only the node numbers that are effectively used in the connectivity table. For a compacted Mesh, it is equivalent to `arange(self.nelems)`. This function also stores the result internally so that future requests can return it without the need for computing it again.

  `getPoints`()

    Return the nodal coordinates of the Mesh.

    This returns only those points that are effectively used in the connectivity table. For a compacted Mesh, it is equal to the coords attribute.

  `getEdges`()

    Return the unique edges of all the elements in the Mesh.

    This is a convenient function to create a table with the element edges. It is equivalent to `self.getLowerEntities(1,unique=True)`, but this also stores the result internally so that future requests can return it without the need for computing it again.

  `getFaces`()

    Return the unique faces of all the elements in the Mesh.

    This is a convenient function to create a table with the element faces. It is equivalent to `self.getLowerEntities(2,unique=True)`, but this also stores the result internally so that future requests can return it without the need for computing it again.

  `getCells`()

    Return the cells of the elements.

    This is a convenient function to create a table with the element cells. It is equivalent to `self.getLowerEntities(3,unique=True)`, but this also stores the result internally so that future requests can return it without the need for computing it again.

  `getElemEdges`()

    Defines the elements in function of its edges.

    This returns a Connectivity table with the elements defined in function of the edges. It returns the equivalent of `self.elems.insertLevel(self.elType().getEntities(1))` but as a side effect it also stores the definition of the edges and the returned element to edge connectivity in the attributes edges, resp. elem_edges.

  `getFreeEntities`(level=-1, return_indices=False)

    Return the border of the Mesh.

    Returns a Connectivity table with the free entities of the specified level of the Mesh. Free entities are entities that are only connected with a single element.

    If return_indices==True, also returns an (nentities,2) index for inverse lookup of the higher entity (column 0) and its local lower entity number (column 1).

  `getFreeEntitiesMesh`(level=-1, compact=True)

    Return a Mesh with lower entities.

    Returns a Mesh representing the lower entities of the specified level. If the Mesh has property numbers, the lower entities inherit the property of the element to which they belong.

    By default, the resulting Mesh is compacted. Compaction can be switched off by setting compact=False.

  `getBorder`(return_indices=False)

    Return the border of the Mesh.

    This returns a Connectivity table with the border of the Mesh. The border entities are of a lower hierarchical level than the mesh itself. These entities become part of the border if they are connected to only one element.

    If return_indices==True, it returns also an (nborder,2) index for inverse lookup of the higher entity (column 0) and its local border part number (column 1).

    This is a convenient shorthand for

        self.getFreeEntities(level=-1,return_indices=return_indices)

  `getBorderMesh`(compact=True)

    Return a Mesh with the border elements.

    The returned Mesh is of the next lower hierarchical level and contains all the free entitites of that level. If the Mesh has property numbers, the border elements inherit the property of the element to which they belong.

    By default, the resulting Mesh is compacted. Compaction can be switched off by setting compact=False.

    This is a convenient shorthand for

        self.getFreeEntitiesMesh(level=-1,compact=compact)

  `getBorderElems`()

    Return the elements that are on the border of the Mesh.

    This returns a list with the numbers of the elements that are on the border of the Mesh. Elements are considered to be at the border if they contain at least one complete element of the border Mesh (i.e. an element of the first lower hierarchical level). Thus, in a volume Mesh, elements only touching the border by a vertex or an edge are not considered border elements.

  `getBorderNodes`()

    Return the nodes that are on the border of the Mesh.

    This returns a list with the numbers of the nodes that are on the border of the Mesh.

  `peel`(nodal=False)

    Return a Mesh with the border elements removed.

    If nodal is True all elements connected to a border node are removed. If nodal is False, it is a convenient shorthand for

        self.cselect(self.getBorderElems())

  `getFreeEdgesMesh`(compact=True)

    Return a Mesh with the free edge elements.

    The returned Mesh is of the hierarchical level 1 (no mather what the level of the parent Mesh is) and contains all the free entitites of that level. If the Mesh has property numbers, the border elements inherit the property of the element to which they belong.

    By default, the resulting Mesh is compacted. Compaction can be switched off by setting compact=False.

    This is a convenient shorthand for

        self.getFreeEntitiesMesh(level=1,compact=compact)

  `adjacency`(level=0, diflevel=-1)

    Create an element adjacency table.

    Two elements are said to be adjacent if they share a lower entity of the specified level. The level is one of the lower entities of the mesh.

    Parameters:

    - level: hierarchy of the geometric items connecting two elements: 0 = node, 1 = edge, 2 = face. Only values of a lower hierarchy than the elements of the Mesh itself make sense.
    - diflevel: if >= level, and smaller than the hierarchy of self.elems, elements that have a connection of this level are removed. Thus, in a Mesh with volume elements, self.adjacency(0,1) gives the adjacency of elements by a node but not by an edge.

    Returns an Adjacency with integers specifying for each element its neighbours connected by the specified geometrical subitems.

  `frontWalk`(level=0, startat=0, frontinc=1, partinc=1, maxval=-1)

    Visit all elements using a frontal walk.

    In a frontal walk a forward step is executed simultanuously from all the elements in the current front. The elements thus reached become the new front. An element can be reached from the current element if both are connected by a lower entity of the specified level. Default level is ‘point’.

    Parameters:

    - level: hierarchy of the geometric items connecting two elements: 0 = node, 1 = edge, 2 = face. Only values of a lower hierarchy than the elements of the Mesh itself make sense. There are no connections on the upper level.

    The remainder of the parameters are like in `Adjacency.frontWalk()`.

    Returns an array of integers specifying for each element in which step the element was reached by the walker.

  `maskedEdgeFrontWalk`(mask=None, startat=0, frontinc=1, partinc=1, maxval=-1)

    Perform a front walk over masked edge connections.

    This is like frontWalk(level=1), but allows to specify a mask to select the edges that are used as connectors between elements.

    Parameters:

    - mask: Either None or a boolean array or index flagging the nodes which are to be considered connectors between elements. If None, all nodes are considered connections.

    The remainder of the parameters are like in `Adjacency.frontWalk()`.

  `partitionByConnection`(level=0, startat=0, sort='number', nparts=-1)

    Detect the connected parts of a Mesh.

    The Mesh is partitioned in parts in which all elements are connected. Two elements are connected if it is possible to draw a continuous (poly)line from a point in one element to a point in the other element without leaving the Mesh. The partitioning is returned as a integer array having a value for ech element corresponding to the part number it belongs to.

    By default the parts are sorted in decreasing order of the number of elements. If you specify nparts, you may wish to switch off the sorting by specifying sort=’‘.

  `splitByConnection`(level=0, startat=0, sort='number')

    Split the Mesh into connected parts.

    Returns a list of Meshes that each form a connected part. By default the parts are sorted in decreasing order of the number of elements.

  `largestByConnection`(level=0)

    Return the largest connected part of the Mesh.

    This is equivalent with, but more efficient than

        self.splitByConnection(level)[0]

  `growSelection`(sel, mode='node', nsteps=1)

    Grow a selection of a surface.

    p is a single element number or a list of numbers. The return value is a list of element numbers obtained by growing the front nsteps times. The mode argument specifies how a single frontal step is done:

    - ‘node’ : include all elements that have a node in common,
    - ‘edge’ : include all elements that have an edge in common.
  `partitionByAngle`(**arg)

    Partition a surface Mesh by the angle between adjacent elements.

    The Mesh is partitioned in parts bounded by the sharp edges in the surface. The arguments and return value are the same as in `TriSurface.partitionByAngle()`.

    Currently this only works for ‘tri3’ and ‘quad4’ type Meshes. Also, the ‘quad4’ partitioning method currently only works correctly if the quads are nearly planar.

  `nodeConnections`()

    Find and store the elems connected to nodes.

  `nNodeConnected`()

    Find the number of elems connected to nodes.

  `edgeConnections`()

    Find and store the elems connected to edges.

  `nEdgeConnected`()

    Find the number of elems connected to edges.

  `nodeAdjacency`()

    Find the elems adjacent to each elem via one or more nodes.

  `nNodeAdjacent`()

    Find the number of elems which are adjacent by node to each elem.

  `edgeAdjacency`()

    Find the elems adjacent to elems via an edge.

  `nEdgeAdjacent`()

    Find the number of adjacent elems.

  `nonManifoldNodes`()

    Return the non-manifold nodes of a Mesh.

    Non-manifold nodes are nodes where subparts of a mesh of level >= 2 are connected by a node but not by an edge.

    Returns an integer array with a sorted list of non-manifold node numbers. Possibly empty (always if the dimensionality of the Mesh is lower than 2).

  `nonManifoldEdges`()

    Return the non-manifold edges of a Mesh.

    Non-manifold edges are edges where subparts of a mesh of level 3 are connected by an edge but not by an face.

    Returns an integer array with a sorted list of non-manifold edge numbers. Possibly empty (always if the dimensionality of the Mesh is lower than 3).

    As a side effect, this constructs the list of edges in the object. The definition of the nonManifold edges in tgerms of the nodes can thus be got from

        self.edges[self.nonManifoldEdges()]

  `nonManifoldEdgeNodes`()

    Return the non-manifold edge nodes of a Mesh.

    Non-manifold edges are edges where subparts of a mesh of level 3 are connected by an edge but not by an face.

    Returns an integer array with a sorted list of numbers of nodes on the non-manifold edges. Possibly empty (always if the dimensionality of the Mesh is lower than 3).

  `fuse`(**kargs)

    Fuse the nodes of a Meshes.

    All nodes that are within the tolerance limits of each other are merged into a single node.

    The merging operation can be tuned by specifying extra arguments that will be passed to `Coords:fuse()`.

  `matchCoords`(mesh, **kargs)

    Match nodes of Mesh with nodes of self.

    This is a convenience function equivalent to:

        self.coords.match(mesh.coords,**kargs)

    See also `Coords.match()`

  `matchCentroids`(mesh, **kargs)

    Match elems of Mesh with elems of self.

    self and Mesh are same eltype meshes and are both without duplicates.

    Elems are matched by their centroids.

  `compact`()

    Remove unconnected nodes and renumber the mesh.

    Returns a mesh where all nodes that are not used in any element have been removed, and the nodes are renumbered to a compacter scheme.

    Example:

        >>> x = Coords([[i] for i in arange(5)])
        >>> M = Mesh(x,[[0,2],[1,4],[4,2]])
        >>> M = M.compact()
        >>> print(M.coords)
        [[ 0.  0.  0.]
         [ 1.  0.  0.]
         [ 2.  0.  0.]
         [ 4.  0.  0.]]
        >>> print(M.elems)
        [[0 2]
         [1 3]
         [3 2]]
        >>> M = Mesh(x,[[0,2],[1,3],[3,2]])
        >>> M = M.compact()
        >>> print(M.coords)
        [[ 0.  0.  0.]
         [ 1.  0.  0.]
         [ 2.  0.  0.]
         [ 3.  0.  0.]]
        >>> print(M.elems)
        [[0 2]
         [1 3]
         [3 2]]

  `select`(selected, compact=True)

    Return a Mesh only holding the selected elements.

    Parameters:

    - selected: an object that can be used as an index in the elems array, such as
      - a single element number
      - a list, or array, of element numbers
      - a bool array of length self.nelems(), where True values flag the elements to be selected
    - compact: boolean. If True (default), the returned Mesh will be compacted, i.e. the unused nodes are removed and the nodes are renumbered from zero. If False, returns the node set and numbers unchanged.

    Returns a Mesh (or subclass) with only the selected elements.

    See `cselect()` for the complementary operation.

  `cselect`(selected, compact=True)

    Return a mesh without the selected elements.

    Parameters:

    - selected: an object that can be used as an index in the elems array, such as
      - a single element number
      - a list, or array, of element numbers
      - a bool array of length self.nelems(), where True values flag the elements to be selected
    - compact: boolean. If True (default), the returned Mesh will be compacted, i.e. the unused nodes are removed and the nodes are renumbered from zero. If False, returns the node set and numbers unchanged.

    Returns a Mesh with all but the selected elements.

    This is the complimentary operation of `select()`.

  `avgNodes`(nodsel, wts=None)

    Create average nodes from the existing nodes of a mesh.

    nodsel is a local node selector as in `selectNodes()` Returns the (weighted) average coordinates of the points in the selector as (nelems*nnod,3) array of coordinates, where nnod is the length of the node selector. wts is a 1-D array of weights to be attributed to the points. Its length should be equal to that of nodsel.

  `meanNodes`(nodsel)

    Create nodes from the existing nodes of a mesh.

    nodsel is a local node selector as in `selectNodes()` Returns the mean coordinates of the points in the selector as (nelems*nnod,3) array of coordinates, where nnod is the length of the node selector.

  `addNodes`(newcoords, eltype=None)

    Add new nodes to elements.

    newcoords is an (nelems,nnod,3) or`(nelems*nnod,3)` array of coordinates. Each element gets exactly nnod extra nodes from this array. The result is a Mesh with plexitude self.nplex() + nnod.

  `addMeanNodes`(nodsel, eltype=None)

    Add new nodes to elements by averaging existing ones.

    nodsel is a local node selector as in `selectNodes()` Returns a Mesh where the mean coordinates of the points in the selector are added to each element, thus increasing the plexitude by the length of the items in the selector. The new element type should be set to correct value.

  `selectNodes`(nodsel, eltype=None)

    Return a mesh with subsets of the original nodes.

    nodsel is an object that can be converted to a 1-dim or 2-dim array. Examples are a tuple of local node numbers, or a list of such tuples all having the same length. Each row of nodsel holds a list of local node numbers that should be retained in the new connectivity table.

  `withProp`(val)

    Return a Mesh which holds only the elements with property val.

    val is either a single integer, or a list/array of integers. The return value is a Mesh holding all the elements that have the property val, resp. one of the values in val. The returned Mesh inherits the matching properties.

    If the Mesh has no properties, a copy with all elements is returned.

  `withoutProp`(val)

    Return a Mesh without the elements with property val.

    This is the complementary method of Mesh.withProp(). val is either a single integer, or a list/array of integers. The return value is a Mesh holding all the elements that do not have the property val, resp. one of the values in val. The returned Mesh inherits the matching properties.

    If the Mesh has no properties, a copy with all elements is returned.

  `connectedTo`(nodes)

    Return a Mesh with the elements connected to the specified node(s).

    nodes: int or array_like, int.

    Return a Mesh with all the elements from the original that contain at least one of the specified nodes.

  `notConnectedTo`(nodes)

    Return a Mesh with the elements not connected to the given node(s).

    nodes: int or array_like, int.

    Returns a Mesh with all the elements from the original that do not contain any of the specified nodes.

  `hits`(entities, level)

    Count the lower entities from a list connected to the elements.

    entities: a single number or a list/array of entities level: 0 or 1 or 2 if entities are nodes or edges or faces, respectively.

    The numbering of the entities corresponds to self.insertLevel(level). Returns an (nelems,) shaped int array with the number of the entities from the list that are contained in each of the elements. This method can be used in selector expressions like:

        self.select(self.hits(entities,level) > 0)

  `splitRandom`(n, compact=True)

    Split a Mesh in n parts, distributing the elements randomly.

    Returns a list of n Mesh objects, constituting together the same Mesh as the original. The elements are randomly distributed over the subMeshes.

    By default, the Meshes are compacted. Compaction may be switched off for efficiency reasons.

  `reverse`(sel=None)

    Return a Mesh where the elements have been reversed.

    Reversing an element has the following meaning:

    - for 1D elements: reverse the traversal direction,
    - for 2D elements: reverse the direction of the positive normal,
    - for 3D elements: reverse inside and outside directions of the element’s border surface. This also changes the sign of the elementt’s volume.

    The `reflect()` method by default calls this method to undo the element reversal caused by the reflection operation.

    Parameters:

    -sel: a selector (index or True/False array)

  `reflect`(dir=0, pos=0.0, reverse=True, **kargs)

    Reflect the coordinates in one of the coordinate directions.

    Parameters:

    - dir: int: direction of the reflection (default 0)
    - pos: float: offset of the mirror plane from origin (default 0.0)
    - reverse: boolean: if True, the `Mesh.reverse()` method is called after the reflection to undo the element reversal caused by the reflection of its coordinates. This will in most cases have the desired effect. If not however, the user can set this to False to skip the element reversal.
  `convert`(totype, fuse=False)

    Convert a Mesh to another element type.

    Converting a Mesh from one element type to another can only be done if both element types are of the same dimensionality. Thus, 3D elements can only be converted to 3D elements.

    The conversion is done by splitting the elements in smaller parts and/or by adding new nodes to the elements.

    Not all conversions between elements of the same dimensionality are possible. The possible conversion strategies are implemented in a table. New strategies may be added however.

    The return value is a Mesh of the requested element type, representing the same geometry (possibly approximatively) as the original mesh.

    If the requested conversion is not implemented, an error is raised.

    Warning

    Conversion strategies that add new nodes may produce double nodes at the common border of elements. The `fuse()` method can be used to merge such coincident nodes. Specifying fuse=True will also enforce the fusing. This option become the default in future.

  `convertRandom`(choices)

    Convert choosing randomly between choices

    Returns a Mesh obtained by converting the current Mesh by a randomly selected method from the available conversion type for the current element type.

  `subdivide`(*ndiv, **kargs)

    Subdivide the elements of a Mesh.

    Parameters:

    - ndiv: specifies the number (and place) of divisions (seeds) along the edges of the elements. Accepted type and value depend on the element type of the Mesh. Currently implemented:
      - ‘tri3’: ndiv is a single int value specifying the number of divisions (of equal size) for each edge.
      - ‘quad4’: ndiv is a sequence of two int values nx,ny, specifying the number of divisions along the first, resp. second parametric direction of the element
      - ‘hex8’: ndiv is a sequence of three int values nx,ny,nz specifying the number of divisions along the first, resp. second and the third parametric direction of the element
    - fuse: bool, if True (default), the resulting Mesh is completely fused. If False, the Mesh is only fused over each individual element of the original Mesh.

    Returns a Mesh where each element is replaced by a number of smaller elements of the same type.

    Note

    This is currently only implemented for Meshes of type ‘tri3’ and ‘quad4’ and ‘hex8’ and for the derived class ‘TriSurface’.

  `reduceDegenerate`(eltype=None)

    Reduce degenerate elements to lower plexitude elements.

    This will try to reduce the degenerate elements of the mesh to elements of a lower plexitude. If a target element type is given, only the matching reduce scheme is tried. Else, all the target element types for which a reduce scheme from the Mesh eltype is available, will be tried.

    The result is a list of Meshes of which the last one contains the elements that could not be reduced and may be empty. Property numbers propagate to the children.

  `splitDegenerate`(autofix=True)

    Split a Mesh in degenerate and non-degenerate elements.

    If autofix is True, the degenerate elements will be tested against known degeneration patterns, and the matching elements will be transformed to non-degenerate elements of a lower plexitude.

    The return value is a list of Meshes. The first holds the non-degenerate elements of the original Mesh. The last holds the remaining degenerate elements. The intermediate Meshes, if any, hold elements of a lower plexitude than the original. These may still contain degenerate elements.

  `removeDegenerate`(eltype=None)

    Remove the degenerate elements from a Mesh.

    Returns a Mesh with all degenerate elements removed.

  `removeDuplicate`(permutations=True)

    Remove the duplicate elements from a Mesh.

    Duplicate elements are elements that consist of the same nodes, by default in no particular order. Setting permutations=False will only consider elements with the same nodes in the same order as duplicates.

    Returns a Mesh with all duplicate elements removed.

  `renumber`(order='elems')

    Renumber the nodes of a Mesh in the specified order.

    order is an index with length equal to the number of nodes. The index specifies the node number that should come at this position. Thus, the order values are the old node numbers on the new node number positions.

    order can also be a predefined value that will generate the node index automatically:

    - ‘elems’: the nodes are number in order of their appearance in the Mesh connectivity.
    - ‘random’: the nodes are numbered randomly.
    - ‘front’: the nodes are numbered in order of their frontwalk.
  `reorder`(order='nodes')

    Reorder the elements of a Mesh.

    Parameters:

    - order: either a 1-D integer array with a permutation of `arange(self.nelems())`, specifying the requested order, or one of the following predefined strings:
      - ‘nodes’: order the elements in increasing node number order.
      - ‘random’: number the elements in a random order.
      - ‘reverse’: number the elements in reverse order.

    Returns a Mesh equivalent with self but with the elements ordered as specified.

    See also: `Connectivity.reorder()`

  `renumberElems`(order='nodes')

    Reorder the elements of a Mesh.

    Parameters:

    - order: either a 1-D integer array with a permutation of `arange(self.nelems())`, specifying the requested order, or one of the following predefined strings:
      - ‘nodes’: order the elements in increasing node number order.
      - ‘random’: number the elements in a random order.
      - ‘reverse’: number the elements in reverse order.

    Returns a Mesh equivalent with self but with the elements ordered as specified.

    See also: `Connectivity.reorder()`

  `connect`(coordslist, div=1, degree=1, loop=False, eltype=None)

    Connect a sequence of toplogically congruent Meshes into a hypermesh.

    Parameters:

    - coordslist: either a list of Coords objects, or a list of Mesh objects or a single Mesh object.

      If Mesh objects are given, they should (all) have the same element type as self. Their connectivity tables will not be used though. They will only serve to construct a list of Coords objects by taking the coords attribute of each of the Meshes. If only a single Mesh was specified, self.coords will be added as the first Coords object in the list.

      All Coords objects in the coordslist (either specified or constructed from the Mesh objects), should have the exact same shape as self.coords. The number of Coords items in the list should be a multiple of degree, plus 1.

      Each of the Coords in the final coordslist is combined with the connectivity table, element type and property numbers of self to produce a list of toplogically congruent meshes. The return value is the hypermesh obtained by connecting each consecutive slice of (degree+1) of these meshes. The hypermesh has a dimensionality that is one higher than the original Mesh (i.e. points become lines, lines become surfaces, surfaces become volumes). The resulting elements will be of the given degree in the direction of the connection.

      Notice that unless a single Mesh was specified as coordslist, the coords of self are not used. In many cases however self or self.coords will be one of the items in the specified coordslist.

    - degree: degree of the connection. Currently only degree 1 and 2 are supported.

      - If degree is 1, every Coords from the coordslist is connected with hyperelements of a linear degree in the connection direction.

      - If degree is 2, quadratic hyperelements are created from one Coords item and the next two in the list. Note that all Coords items should contain the same number of nodes, even for higher order elements where the intermediate planes contain less nodes.

        Currently, degree=2 is not allowed when coordslist is specified as a single Mesh.

    - loop: if True, the connections with loop around the list and connect back to the first. This is accomplished by adding the first Coords item back at the end of the list.

    - div: Either an integer, or a sequence of float numbers (usually in the range ]0.0..1.0]) or a list of sequences of the same length of the connecting list of coordinates. In the latter case every sequence inside the list can either be a float sequence (usually in the range ]0.0..1.0]) or it contains one integer (e.g [[4],[0.3,1]]). This should only be used for degree==1.

      With this parameter the generated elements can be further subdivided along the connection direction. If an int is given, the connected elements will be divided into this number of elements along the connection direction. If a sequence of float numbers is given, the numbers specify the relative distance along the connection direction where the elements should end. If the last value in the sequence is not equal to 1.0, there will be a gap between the consecutive connections. If a list of sequences is given, every consecutive element of the coordinate list is connected using the corresponding sequence in the list(1-length integer of float sequence specified as before).

    - eltype: the element type of the constructed hypermesh. Normally, this is set automatically from the base element type and the connection degree. If a different element type is specified, a final conversion to the requested element type is attempted.

  `extrude`(n, step=1.0, dir=0, degree=1, eltype=None)

    Extrude a Mesh in one of the axes directions.

    Returns a new Mesh obtained by extruding the given Mesh over n steps of length step in direction of axis dir.

  `revolve`(n, axis=0, angle=360.0, around=None, loop=False, eltype=None)

    Revolve a Mesh around an axis.

    Returns a new Mesh obtained by revolving the given Mesh over an angle around an axis in n steps, while extruding the mesh from one step to the next. This extrudes points into lines, lines into surfaces and surfaces into volumes.

  `sweep`(path, eltype=None, **kargs)

    Sweep a mesh along a path, creating an extrusion

    Returns a new Mesh obtained by sweeping the given Mesh over a path. The returned Mesh has double plexitude of the original.

    This method accepts all the parameters of `coords.sweepCoords()`, with the same meaning. Usually, you will need to at least set the normal parameter. The eltype parameter can be used to set the element type on the returned Meshes.

    This operation is similar to the extrude() method, but the path can be any 3D curve.

  `smooth`(iterations=1, lamb=0.5, k=0.1, edg=True, exclnod=[], exclelem=[], weight=None)

    Return a smoothed mesh.

    Smoothing algorithm based on lowpass filters.

    If edg is True, the algorithm tries to smooth the outer border of the mesh seperately to reduce mesh shrinkage.

    Higher values of k can reduce shrinkage even more (up to a point where the mesh expands), but will result in less smoothing per iteration.

    - exclnod: It contains a list of node indices to exclude from the smoothing. If exclnod is ‘border’, all nodes on the border of the mesh will be unchanged, and the smoothing will only act inside. If exclnod is ‘inner’, only the nodes on the border of the mesh will take part to the smoothing.
    - exclelem: It contains a list of elements to exclude from the smoothing. The nodes of these elements will not take part to the smoothing. If exclnod and exclelem are used at the same time the union of them will be exluded from smoothing.
    -weight : it is a string that can assume 2 values inversedistance and
      distance. It allows to specify the weight of the adjancent points according to their distance to the point
  classmethod `concatenate`(clas, meshes, **kargs)

    Concatenate a list of meshes of the same plexitude and eltype

    All Meshes in the list should have the same plexitude. Meshes with plexitude are ignored though, to allow empty Meshes to be added in.

    Merging of the nodes can be tuned by specifying extra arguments that will be passed to `Coords:fuse()`.

    If any of the meshes has property numbers, the resulting mesh will inherit the properties. In that case, any meshes without properties will be assigned property 0. If all meshes are without properties, so will be the result.

    This is a class method, and should be invoked as follows:

        Mesh.concatenate([mesh0,mesh1,mesh2])

  `test`(nodes='all', dir=0, min=None, max=None, atol=0.0)

    Flag elements having nodal coordinates between min and max.

    This function is very convenient in clipping a Mesh in a specified direction. It returns a 1D integer array flagging (with a value 1 or True) the elements having nodal coordinates in the required range. Use where(result) to get a list of element numbers passing the test. Or directly use clip() or cclip() to create the clipped Mesh

    The test plane can be defined in two ways, depending on the value of dir. If dir == 0, 1 or 2, it specifies a global axis and min and max are the minimum and maximum values for the coordinates along that axis. Default is the 0 (or x) direction.

    Else, dir should be compaitble with a (3,) shaped array and specifies the direction of the normal on the planes. In this case, min and max are points and should also evaluate to (3,) shaped arrays.

    nodes specifies which nodes are taken into account in the comparisons. It should be one of the following:

    - a single (integer) point number (< the number of points in the Formex)
    - a list of point numbers
    - one of the special strings: ‘all’, ‘any’, ‘none’

    The default (‘all’) will flag all the elements that have all their nodes between the planes x=min and x=max, i.e. the elements that fall completely between these planes. One of the two clipping planes may be left unspecified.

  `clip`(t, compact=True)

    Return a Mesh with all the elements where t>0.

    t should be a 1-D integer array with length equal to the number of elements of the Mesh. The resulting Mesh will contain all elements where t > 0.

  `cclip`(t, compact=True)

    This is the complement of clip, returning a Mesh where t<=0.

  `clipAtPlane`(p, n, nodes='any', side='+')

    Return the Mesh clipped at plane (p,n).

    This is a convenience function returning the part of the Mesh at one side of the plane (p,n)

  `levelVolumes`()

    Return the level volumes of all elements in a Mesh.

    The level volume of an element is defined as:

    - the length of the element if the Mesh is of level 1,
    - the area of the element if the Mesh is of level 2,
    - the (signed) volume of the element if the Mesh is of level 3.

    The level volumes can be computed directly for Meshes of eltypes ‘line2’, ‘tri3’ and ‘tet4’ and will produce accurate results. All other Mesh types are converted to one of these before computing the level volumes. Conversion may result in approximation of the results. If conversion can not be performed, None is returned.

    If succesful, returns an (nelems,) float array with the level volumes of the elements. Returns None if the Mesh level is 0, or the conversion to the level’s base element was unsuccesful.

    Note that for level-3 Meshes, negative volumes will be returned for elements having a reversed node ordering.

  `lengths`()

    Return the length of all elements in a level-1 Mesh.

    For a Mesh with eltype ‘line2’, the lengths are exact. For other eltypes, a conversion to ‘line2’ is done before computing the lengths. This may produce an exact result, an approximated result or no result (if the conversion fails).

    If succesful, returns an (nelems,) float array with the lengths. Returns None if the Mesh level is not 1, or the conversion to ‘line2’ does not succeed.

  `areas`()

    Return the area of all elements in a level-2 Mesh.

    For a Mesh with eltype ‘tri3’, the areas are exact. For other eltypes, a conversion to ‘tri3’ is done before computing the areas. This may produce an exact result, an approximate result or no result (if the conversion fails).

    If succesful, returns an (nelems,) float array with the areas. Returns None if the Mesh level is not 2, or the conversion to ‘tri3’ does not succeed.

  `volumes`()

    Return the signed volume of all the mesh elements

    For a ‘tet4’ tetraeder Mesh, the volume of the elements is calculated as 1/3 * surface of base * height.

    For other Mesh types the volumes are calculated by first splitting the elements into tetraeder elements.

    The return value is an array of float values with length equal to the number of elements. If the Mesh conversion to tetraeder does not succeed, the return value is None.

  `length`()

    Return the total length of a Mesh.

    Returns the sum of self.lengths(), or 0.0 if the self.lengths() returned None.

  `area`()

    Return the total area of a Mesh.

    Returns the sum of self.areas(), or 0.0 if the self.areas() returned None.

  `volume`()

    Return the total volume of a Mesh.

    For a Mesh of level < 3, a value 0.0 is returned. For a Mesh of level 3, the volume is computed by converting its border to a surface and taking the volume inside that surface. It is equivalent with

        self.toSurface().volume()

    This is far more efficient than self.volumes().sum().

  `fixVolumes`()

    Reverse the elements with negative volume.

    Elements with negative volume may result from incorrect local node numbering. This method will reverse all elements in a Mesh of dimensionality 3, provide the volumes of these elements can be computed.

  `addNoise`(*args, **kargs)

    Apply ‘addNoise’ transformation to the Geometry object.

    See `coords.Coords.addNoise()` for details.

  `affine`(*args, **kargs)

    Apply ‘affine’ transformation to the Geometry object.

    See `coords.Coords.affine()` for details.

  `align`(*args, **kargs)

    Apply ‘align’ transformation to the Geometry object.

    See `coords.Coords.align()` for details.

  `bump`(*args, **kargs)

    Apply ‘bump’ transformation to the Geometry object.

    See `coords.Coords.bump()` for details.

  `bump1`(*args, **kargs)

    Apply ‘bump1’ transformation to the Geometry object.

    See `coords.Coords.bump1()` for details.

  `bump2`(*args, **kargs)

    Apply ‘bump2’ transformation to the Geometry object.

    See `coords.Coords.bump2()` for details.

  `centered`(*args, **kargs)

    Apply ‘centered’ transformation to the Geometry object.

    See `coords.Coords.centered()` for details.

  `cylindrical`(*args, **kargs)

    Apply ‘cylindrical’ transformation to the Geometry object.

    See `coords.Coords.cylindrical()` for details.

  `egg`(*args, **kargs)

    Apply ‘egg’ transformation to the Geometry object.

    See `coords.Coords.egg()` for details.

  `flare`(*args, **kargs)

    Apply ‘flare’ transformation to the Geometry object.

    See `coords.Coords.flare()` for details.

  `hyperCylindrical`(*args, **kargs)

    Apply ‘hyperCylindrical’ transformation to the Geometry object.

    See `coords.Coords.hyperCylindrical()` for details.

  `isopar`(*args, **kargs)

    Apply ‘isopar’ transformation to the Geometry object.

    See `coords.Coords.isopar()` for details.

  `map`(*args, **kargs)

    Apply ‘map’ transformation to the Geometry object.

    See `coords.Coords.map()` for details.

  `map1`(*args, **kargs)

    Apply ‘map1’ transformation to the Geometry object.

    See `coords.Coords.map1()` for details.

  `mapd`(*args, **kargs)

    Apply ‘mapd’ transformation to the Geometry object.

    See `coords.Coords.mapd()` for details.

  `position`(*args, **kargs)

    Apply ‘position’ transformation to the Geometry object.

    See `coords.Coords.position()` for details.

  `projectOnCylinder`(*args, **kargs)

    Apply ‘projectOnCylinder’ transformation to the Geometry object.

    See `coords.Coords.projectOnCylinder()` for details.

  `projectOnPlane`(*args, **kargs)

    Apply ‘projectOnPlane’ transformation to the Geometry object.

    See `coords.Coords.projectOnPlane()` for details.

  `projectOnSphere`(*args, **kargs)

    Apply ‘projectOnSphere’ transformation to the Geometry object.

    See `coords.Coords.projectOnSphere()` for details.

  `replace`(*args, **kargs)

    Apply ‘replace’ transformation to the Geometry object.

    See `coords.Coords.replace()` for details.

  `rollAxes`(*args, **kargs)

    Apply ‘rollAxes’ transformation to the Geometry object.

    See `coords.Coords.rollAxes()` for details.

  `rot`(*args, **kargs)

    Apply ‘rotate’ transformation to the Geometry object.

    See `coords.Coords.rotate()` for details.

  `rotate`(*args, **kargs)

    Apply ‘rotate’ transformation to the Geometry object.

    See `coords.Coords.rotate()` for details.

  `scale`(*args, **kargs)

    Apply ‘scale’ transformation to the Geometry object.

    See `coords.Coords.scale()` for details.

  `shear`(*args, **kargs)

    Apply ‘shear’ transformation to the Geometry object.

    See `coords.Coords.shear()` for details.

  `spherical`(*args, **kargs)

    Apply ‘spherical’ transformation to the Geometry object.

    See `coords.Coords.spherical()` for details.

  `superSpherical`(*args, **kargs)

    Apply ‘superSpherical’ transformation to the Geometry object.

    See `coords.Coords.superSpherical()` for details.

  `swapAxes`(*args, **kargs)

    Apply ‘swapAxes’ transformation to the Geometry object.

    See `coords.Coords.swapAxes()` for details.

  `toCylindrical`(*args, **kargs)

    Apply ‘toCylindrical’ transformation to the Geometry object.

    See `coords.Coords.toCylindrical()` for details.

  `toSpherical`(*args, **kargs)

    Apply ‘toSpherical’ transformation to the Geometry object.

    See `coords.Coords.toSpherical()` for details.

  `transformCS`(*args, **kargs)

    Apply ‘transformCS’ transformation to the Geometry object.

    See `coords.Coords.transformCS()` for details.

  `translate`(*args, **kargs)

    Apply ‘translate’ transformation to the Geometry object.

    See `coords.Coords.translate()` for details.

  `trl`(*args, **kargs)

    Apply ‘translate’ transformation to the Geometry object.

    See `coords.Coords.translate()` for details.

  `setProp`(prop=None, blocks=None)

    Create or destroy the property array for the Geometry.

    A property array is a rank-1 integer array with dimension equal to the number of elements in the Geometry. Each element thus has its own property number. These numbers can be used for any purpose. They play an import role when creating new geometry: new elements inherit the property number of their parent element. Properties are also preserved on most geometrical transformations.

    Because elements with different property numbers can be drawn in different colors, the property numbers are also often used to impose color.

    Parameters:

    - prop: a single integer value or a list/array of integer values. If the number of passed values is less than the number of elements, they wil be repeated. If you give more, they will be ignored.

      The special value ‘range’ will set the property numbers equal to the element number.

      A value None (default) removes the properties from the Geometry.

    - blocks: a single integer value or a list/array of integer values. If the number of passed values is less than the length of prop, they wil be repeated. If you give more, they will be ignored. Every prop will be repeated the corresponding number of times specified in blocks.

  `toProp`(prop)

    Converts the argument into a legal set of properties for the object.

    The conversion involves resizing the argument to a 1D array of length self.nelems(), and converting the data type to integer.

  `copy`()

    Return a deep copy of the Geometry object.

    The returned object is an exact copy of the input, but has all of its data independent of the former.

  `splitProp`(prop=None)

    Partition a Geometry (Formex/Mesh) according to the values in prop.

    Parameters:

    - prop: an int array with length self.nelems(), or None. If None, the prop attribute of the Geometry is used.

    Returns a list of Geometry objects of the same type as the input. Each object contains all the elements having the same value of prop. The number of objects in the list is equal to the number of unique values in prop. The list is sorted in ascending order of their prop value.

    It prop is None and the the object has no prop attribute, an empty list is returned.

  `resized`(size=1.0, tol=1e-05)

    Return a copy of the Geometry scaled to the given size.

    size can be a single value or a list of three values for the three coordinate directions. If it is a single value, all directions are scaled to the same size. Directions for which the geometry has a size smaller than tol times the maximum size are not rescaled.

  `write`(fil, sep=' ', mode='w')

    Write a Geometry to a .pgf file.

    If fil is a string, a file with that name is opened. Else fil should be an open file. The Geometry is then written to that file in a native format, using sep as separator between the coordinates. If fil is a string, the file is closed prior to returning.

`Functions defined in module mesh`

`mesh.``mergeNodes`(nodes, fuse=True, **kargs)

  Merge all the nodes of a list of node sets.

  Merging the nodes creates a single Coords object containing all nodes, and the indices to find the points of the original node sets in the merged set.

  Parameters:

  - nodes: a list of Coords objects, all having the same shape, except possibly for their first dimension
  - fuse: if True (default), coincident (or very close) points will be fused to a single point
  - **kargs: keyword arguments that are passed to the fuse operation

  Returns:

  - a Coords with the coordinates of all (unique) nodes,
  - a list of indices translating the old node numbers to the new. These numbers refer to the serialized Coords.

  The merging operation can be tuned by specifying extra arguments that will be passed to `Coords.fuse()`.

`mesh.``mergeMeshes`(meshes, fuse=True, **kargs)

  Merge all the nodes of a list of Meshes.

  Each item in meshes is a Mesh instance. The return value is a tuple with:

  - the coordinates of all unique nodes,
  - a list of elems corresponding to the input list, but with numbers referring to the new coordinates.

  The merging operation can be tuned by specifying extra arguments that will be passed to `Coords:fuse()`. Setting fuse=False will merely concatenate all the mesh.coords, but not fuse them.

`mesh.``unitAttractor`(x, e0=0.0, e1=0.0)

  Moves values in the range 0..1 closer to or away from the limits.

  - x: a list or array with values in the range 0.0 to 1.0.
  - e0, e1: attractor parameters for the start, resp. the end of the range. A value larger than zero will attract the points closer to the corresponding endpoint, while a negative value will repulse them. If two positive values are given, the middle part of the interval will become sparsely populated.

  Example:

      >>> set_printoptions(precision=4)
      >>> print(unitAttractor([0.,0.25,0.5,0.75,1.0],2.))
      [ 0.      0.0039  0.0625  0.3164  1.    ]

`mesh.``seed`(n, e0=0.0, e1=0.0)

  Create one-dimensional element seeds in a unit interval

  Returns parametric values along the unit interval in order to divide it in n elements, possibly of unequal length.

  Parameters:

  - n: number of elements (yielding n+1 parameter values).
  - e0, e1: attractor parameters for the start, resp. the end of the range. A value larger than zero will attract the points closer to the corresponding endpoint, while a negative value will repulse them. If two positive values are given, the middle part of the interval will become sparsely populated.

  Example:

      >>> set_printoptions(precision=4)
      >>> print(seed(5,2.,2.))
      [ 0.      0.0639  0.3362  0.6638  0.9361  1.    ]

`mesh.``gridpoints`(seed0, seed1=None, seed2=None)

  Create weigths for linear lines, quadrilateral and hexahedral elements coordinates

  Parameters:

  - ‘seed0’ : int or list of floats . It specifies divisions along the first parametric direction of the element
  - ‘seed1’ : int or list of floats . It specifies divisions along the second parametric direction of the element
  - ‘seed2’ : int or list of floats . It specifies divisions along the t parametric direction of the element

  If these parametes are integer values the divisions will be equally spaced between 0 and 1

`mesh.``quad4_wts`(seed0, seed1)

  Create weights for quad4 subdivision.

  Parameters:

  - ‘seed0’ : int or list of floats . It specifies divisions along the first parametric direction of the element
  - ‘seed1’ : int or list of floats . It specifies divisions along the second parametric direction of the element

  If these parametes are integer values the divisions will be equally spaced between 0 and 1

`mesh.``quadgrid`(seed0, seed1)

  Create a quadrilateral mesh of unit size with the specified seeds.

  The seeds are a monotonously increasing series of parametric values in the range 0..1. They define the positions of the nodes in the parametric directions 0, resp. 1. Normally, the first and last values of the seeds are 0., resp. 1., leading to a unit square grid.

  The seeds are usually generated with the seed() function.

`mesh.``hex8_wts`(seed0, seed1, seed2)

  Create weights for hex8 subdivision.

  Parameters:

  - ‘seed0’ : int or list of floats . It specifies divisions along the first parametric direction of the element
  - ‘seed1’ : int or list of floats . It specifies divisions along the second parametric direction of the element
  - ‘seed2’ : int or list of floats . It specifies divisions along the t parametric direction of the element

  If these parametes are integer values the divisions will be equally spaced between 0 and 1

`mesh.``hex8_els`(nx, ny, nz)

  Create connectivity table for hex8 subdivision.

`mesh.``rectangle`(L, W, nl, nw)

  Create a plane rectangular mesh of quad4 elements

  Parameters:

  - L,W: length,width of the rectangle
  - nl,nw: seeds for the elements along the length, width of the rectangle. They should one of the following:
    - an integer number, specifying the number of equally sized elements along that direction,
    - a tuple (n,) or (n,e0) or (n,e0,e1), to be used as parameters in the `mesh.seed()` function,
    - a list of float values in the range 0.0 to 1.0, specifying the relative position of the seeds. The values should be ordered and the first and last values should be 0.0 and 1.0.

`mesh.``rectangle_with_hole`(L, W, r, nr, nt, e0=0.0, eltype='quad4')

  Create a quarter of rectangle with a central circular hole.

  Parameters:

  - L,W: length,width of the (quarter) rectangle
  - r: radius of the hole
  - nr,nt: number of elements over radial,tangential direction
  - e0: concentration factor for elements in the radial direction

  Returns a Mesh

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