XDMF Model and Format

Here the XDMF3 Model and Format is described. See Xdmf2 Model and Format Archive for the previous version.

The need for a standardized method to exchange scientific data between High Performance Computing codes and tools lead to the development of the eXtensible Data Model and Format (XDMF). Uses for XDMF range from a standard format used by HPC codes to take advantage of widely used visualization programs like ParaView[1] and EnSight[2], to a mechanism for performing coupled calculations using multiple, previously stand alone codes.

XDMF categorizes data by two main attributes; size and function. Data can be Light (typically less than about a thousand values) or Heavy (megabytes, terabytes, etc.). In addition to raw values, data can refer to Format (rank and dimensions of an array) or Model (how that data is to be used. i.e. XYZ coordinates vs. Vector components).

XDMF uses XML to store Light data and to describe the data Model. Either HDF5[3] or binary files can be used to store Heavy data. The data Format is stored redundantly in both XML and HDF5. This allows tools to parse XML to determine the resources that will be required to access the Heavy data. For the binary Heavy data option, the xml must list a filename where the binary data is stored.

While not required, a C++ API is provided to read and write XDMF data. This API has also been wrapped so it is available from popular languages like Python, Tcl, and Java. The API is not necessary in order to produce or consume XDMF data. Currently several HPC codes that already produced HDF5 data, use native text output to produce the XML necessary for valid XDMF.

XML

The eXtensible Markup Language (XML) format is widely used for many purposes and is well documented at many sites. There are numerous open source parsers available for XML. The XDMF API takes advantage of the libxml2 parser to provide the necessary functionality. Without going into too much detail, XDMF views XML as a "personalized HTML" with some special rules. It it case sensitive and is made of three major components: elements, entities, and processing information. In XDMF the element is the most important component. Additionally XDMF takes advantage of two major extensions to XML: XInclude and XPath.

Elements

Elements follow the basic form:

<ElementTag 
    AttributeName="AttributeValue"
    AttributeName="AttributeValue"
    ... >
   CData
</ElementTag>

Each element begins with a <tag> and ends with a </tag>. Optionally there can be several "Name=Value" pairs which convey additional information. Between the <tag> and the </tag> there can be other <tag></tag> pairs and/or character data (CData). CData is typically where the values are stored; like the actual text in an HTML document. The XML parser in the XDMF API parses the XML file and builds a tree structure in memory to describe its contents. This tree can be queried, modified, and then "serialized" back into XML.

Comments in XML start with a "<!--" and end with a "-->". So <!--This is a Comment -->.

XML is said to be "well formed" if it is syntactically correct. That means all of the quotes match, all elements have end elements, etc. XML is said to be "valid" if it conforms to the Schema or DTD defined at the head of the document. For example, the schema might specify that element type A can contain element B but not element C. Verifying that the provided XML is well formed and/or valid are functions typically performed by the XML parser.

XInclude

As opposed to entity references in XML (see below), XInclude allows for the inclusion of files that are not well formed XML. This means that with XInclude the included file could be well formed XML or perhaps a flat text file of values. The syntax looks like this :

<Xdmf Version="2.0" xmlns:xi="[http://www.w3.org/2001/XInclude]">
  <xi:include href="Example3.xmf"/>
</Xdmf>

the xmlns:xi establishes a namespace xi. Then anywhere within the Xdmf element, xi:include will pull in the URL.

XPath

This allows for elements in the XML document and the API to reference specific elements in a document. For example:

The first Grid in the first Domain

/Xdmf/Domain/Grid

The tenth Grid .... XPath is one based.

/Xdmf/Domain/Grid[10]

The first grid with an attribute Name which has a value of "Copper Plate"

/Xdmf/Domain/Grid[@Name="Copper Plate"]

All valid XDMF must appear between the <Xdmf> and the </Xdmf> tags. So a minimal (empty) XDMF XML file would be:

<?xml version="1.0" ?>
<!DOCTYPE Xdmf SYSTEM "Xdmf.dtd" []>
<Xdmf Version="2.0">
</Xdmf>

While there exists an Xdmf DTD and a Schema they are only necessary for validating parsers. For performance reasons, validation is typically disabled.

Entities

In addition to Xinclude and XPath, which allow for references to data outside the actual XMDF, XML's basic substitution mechanism of entities can be used to render the XDMF document more readable. For instance, once an entity alias has been defined in the header via

<?xml version="1.0" ?>
<!DOCTYPE Xdmf SYSTEM "Xdmf.dtd" [
  <!ENTITY cellDimsZYX "45 30 120">
]>

the text in double quotes is substituted for the entity reference &cellDimsZXY; (note the trailing semicolon) whenever the parser encounters the latter in the remaining part of the document.

XDMF Elements

The organization of XDMF begins with the Xdmf element. So that parsers can distinguish from previous versions of XDMF, there exists a Version attribute (currently at 3.0). Any element in XDMF can have a Name attribute or have a Reference attribute. The Name attribute becomes important for grids while the Reference attribute is used to take advantage of the XPath facility (more detail on this later). Xdmf elements contain one or more Domain elements (computational domain). There is seldom motivation to have more than one Domain.

A Domain can have one or more Grid elements. Each Grid contains a Topology, Geometry, and zero or more Attribute elements. Topology specifies the connectivity of the grid while Geometry specifies the location of the grid nodes. Attribute elements are used to specify values such as scalars and vectors that are located at the node, edge, face, cell center, or grid center.

To specify actual values for connectivity, geometry, or attributes, XDMF defines a DataItem element. A DataItem can provide the actual values or provide the physical storage (which is typically an HDF5 file).

XdmfItem

There are six different types of DataItems :

1. Uniform - this is the default. A single array of values.
2. Collection - a one dimension array of DataItems
3. Tree - a hierarchical structure of DataItems
4. HyperSlab - contains two data items. The first selects the start, stride and count indexes of the second DataItem.
5. Coordinates - contains two DataItems. The first selects the parametric coordinates of the second DataItem.
6. Function - calculates an expression.

Uniform

The simplest type is Uniform that specifies a single array. As with all XDMF elements, there are reasonable defaults wherever possible. So the simplest DataItem would be :

<DataItem Dimensions="3">
    1.0 2.0 3.0
</DataItem>

Since no ItemType has been specified, Uniform has been assumed. The default Format is XML and the default NumberType is a 32 bit floating point value. So the fully qualified DataItem for the same data would be:

<DataItem ItemType="Uniform"
          Format="XML"
          NumberType="Float" 
          Precision="4"
          Rank="1" 
          Dimensions="3">
  1.0 2.0 3.0
</DataItem>

Since it is only practical to store a small amount of data values in the XML, production codes typically write their data to HDF5 and specify the location in XML. HDF5 is a hierarchical, self describing data format. So an application can open an HDF5 file without any prior knowledge of the data and determine the dimensions and number type of all the arrays stored in the file. XDMF requires that this information also be stored redundantly in the XML so that applications need not have access to the actual heavy data in order to determine storage requirements.

For example, suppose an application stored a three dimensional array of pressure values at each iteration into an HDF5 file. The XML might be :

<DataItem ItemType="Uniform"
          Format="HDF"
          NumberType="Float" 
          Precision="8"
          Dimensions="64 128 256">
  OutputData.h5:/Results/Iteration 100/Part 2/Pressure
</DataItem>

Alternatively, an application may store its data in binary files. In this case the XML might be.

<DataItem ItemType="Uniform"
          Format="Binary"
          Dimensions="64 128 256">
  PressureFile.bin
</DataItem>

Dimensions are specified with the slowest varying dimension first (i.e. KJI order). The HDF filename can be fully qualified, if it is not it is assumed to be located in the current directory or the same directory as the XML file.

Collection and Tree

Collections are Trees with only a single level. This is such a frequent occurrence that it was decided to make a Collection a separate type in case the application can optimize access. Collections and Trees have DataItem elements as children. The leaf nodes are Uniform DataItem elements:

<DataItem Name="Tree Example" ItemType="Tree">
    <DataItem ItemType="Tree">
        <DataItem Name="Collection 1" ItemType="Collection">
            <DataItem Dimensions="3">
                1.0 2.0 3.0
            </DataItem>
            <DataItem Dimensions="4">
                4 5 6 7
            </DataItem>
        </DataItem>
    </DataItem>
    <DataItem Name="Collection 2" ItemType="Collection">
        <DataItem Dimensions="3">
            7 8  9
        </DataItem>
        <DataItem Dimensions="4">
            10 11 12 13
        </DataItem>
    </DataItem>
    <DataItem ItemType="Uniform"
              Format="HDF"
              NumberType="Float" 
              Precision="8"
              Dimensions="64 128 256">
        OutputData.h5:/Results/Iteration 100/Part 2/Pressure
    </DataItem>
</DataItem>

This DataItem is a tree with three children. The first child is another tree that contains a collection of two uniform DataItem elements. The second child is a collection with two uniform DataItem elements. The third child is a uniform DataItem.

HyperSlab

A HyperSlab specifies a subset of some other DataItem. The slab is specified by giving the start, stride, and count of the values in each of the target DataItem dimensions. For example, given a dataset MyData.h5:/XYZ that is 100x200x300x3, we could describe a region starting at [0,0,0,0], ending at [50, 100, 150, 2] that includes every other plane of data with the HyperSlab DataItem

<DataItem ItemType="HyperSlab"
          Dimensions="25 50 75 3"
          Type="HyperSlab">
    <DataItem Dimensions="3 4" 
              Format="XML">
        0 0 0 0 
        2 2 2 1 
        25 50 75 3
    </DataItem>
    <DataItem
        Name="Points"
        Dimensions="100 200 300 3"
        Format="HDF">
        MyData.h5:/XYZ
    </DataItem>
</DataItem>

Notice that the first DataItem specified Start, Stride and Count for each dimension of the second DataItem.

Coordinate

Suppose, instead that we only wish to specify the first Y data value from the DataItem and the last X value. This can be accomplished by providing the parametric coordinates of the desired values and using the Coordinates ItemType.

<DataItem ItemType="Coordinate"
          Dimensions="2"
          Type="Coordinate">
    <DataItem Dimensions="2 4"
              Format="XML">
        0 0 0 1
        99 199 299 0
    </DataItem>
    <DataItem Name="Points"
              Dimensions="100 200 300 3"
              Format="HDF">
        MyData.h5:/XYZ
    </DataItem>
</DataItem>

The first Y value is index 1 of item 0,0,0 while the last X value is index 0 of item 99, 199, 299. The dimensionality of the specified coordinates must match that of the target DataItem.

Function

Function ItemType specifies some operation on the children DataItem elements. The elements are referenced by $X where X is the zero based index of the child. For example, the following DataItem would add the two children DataItem elements together in a value by value operation resulting in the values 5.1, 7.2 and 9.3 :

<DataItem ItemType="Function" 
          Function="$0 + $1"
          Dimensions="3">
    <DataItem Dimensions="3">
        1.0 2.0 3.0
    </DataItem>
    <DataItem Dimensions="3">
        4.1 5.2 6.3
    </DataItem>
</DataItem>

The function description can be arbitrarily complex and contain SIN, COS, TAN, ACOS, ASIN, ATAN, LOG, EXP, ABS, and SQRT. In addition, there are the JOIN() and WHERE() expressions. JOIN can concatenate or interlace arrays while WHERE() can extract values where some condition is true. In the following examples we take advantage of the XPath facility to reference DataItem elements that have been previously specified :

Add the value 10 to every element

<DataItem Name="MyFunction" 
          ItemType="Function"
          Function="10 + $0">
    <DataItem Reference="/Xdmf/DataItem[1]" />
</DataItem> 

Multiply two arrays (element by element) and take the absolute value

<DataItem ItemType="Function"
          Function="ABS($0 * $1)">
    <DataItem Reference="/Xdmf/DataItem[1]" />
    <DataItem Reference="/Xdmf/DataItem[2]" />
</DataItem>

Select element 5 thru 15 from the first DataItem

<DataItem ItemType="Function"
          Function="$0[5:15]">
    <DataItem Reference="/Xdmf/DataItem[1]" />
</DataItem>

Concatenate two arrays

<DataItem ItemType="Function"
          Function="JOIN($0 ; $1)">
    <DataItem Reference="/Xdmf/DataItem[1]" />
    <DataItem Reference="/Xdmf/DataItem[2]" />
</DataItem>

Interlace 3 arrays (Useful for describing vectors from scalar data)

<DataItem ItemType="Function"
          Function="JOIN($0 , $1, $2)">
    <DataItem Reference="/Xdmf/DataItem[1]" />
    <DataItem Reference="/Xdmf/DataItem[2]" /> 
    <DataItem Reference="/Xdmf/DataItem[3]" />
</DataItem>

Grid

The DataItem element is used to define the data format portion of XDMF. It is sufficient to specify fairly complex data structures in a portable manner. The data model portion of XDMF begins with the Grid element. A Grid is a container for information related to 2D and 3D points, structured or unstructured connectivity, and assigned values.

The Grid element now has a GridType attribute. Valid GridTypes are :

1. Uniform - a homogeneous single grid (i.e. a pile of triangles)
2. Collection - an array of Uniform grids all with the same Attributes
3. Tree - a hierarchical group
4. SubSet - a portion of another Grid

Uniform, Collection, and Tree

Uniform Grid elements are the simplest type and must contain a Topology and Geometry element. If GridType is Collection, a CollectionType can be specified as either "Spatial" or "Temporal". Just like the DataItem element, Tree and Collection Grid elements contain other Grid elements as children:

<Grid Name="Car Wheel" GridType="Tree">
    <Grid Name="Tire" GridType="Uniform">
        <Topology> ... </Topology>
        <Geometry> ... </Geometry>
    </Grid>
    <Grid Name="Lug Nuts" GridType="Collection">
        <Grid Name="Lug Nut 0" GridType="Uniform">
            <Topology> ... </Topology>
            <Geometry> ... </Geometry>
        </Grid>
        <Grid Name="Lug Nut 1" GridType="Uniform">
            <Topology> ... </Topology>
            <Geometry> ... </Geometry>
        </Grid>
        <Grid Name="Lug Nut 2" GridType="Uniform">
            <Topology> ... </Topology>
            <Geometry> ... </Geometry>
        </Grid>
    </Grid>
    .
    .
    . 

SubSet

A SubSet GridType is used to define a portion of another grid or define new attributes on the grid. This allows users to share the geometry and topology of another grid, the attributes from the original grid are not assigned. The Section attribute of a SubSet can be DataItem or All :

<Grid Name="Portion" GridType="Subset" Section="DataItem">
    <!-- Select 2 cells from another grid. 
         Which 2 are defined by the DataItem -->
    <DataItem 
        DataType="Int"
        Dimensions="2"
        Format="XML">
        0 2
    </DataItem>
    <Grid Name="Target" Reference="XML">
        /Xdmf/Domain/Grid[@Name="Main Grid"]
    </Grid>
    <Attribute Name="New Values" Center="Cell">
        <DataItem Format="XML" Dimensions="2">
            100 150
        </DataItem>
    </Attribute>
</Grid>

Or

<Grid Name="Portion" GridType="Subset" Section="All">
    <!-- Select the entire grid and add an  attribute -->
    <Grid Name="Target" Reference="XML">
        /Xdmf/Domain/Grid[@Name="Main Grid"]
    </Grid>
    <Attribute Name="New Values" Center="Cell">
        <DataItem Format="XML" Dimensions="3">
            100 150 200
        </DataItem>
    </Attribute>
</Grid>

Topology

The Topology element describes the general organization of the data. This is the part of the computational grid that is invariant with rotation, translation, and scale. For structured grids, the connectivity is implicit. For unstructured grids, if the connectivity differs from the standard, an Order may be specified. Currently, the following Topology cell types are defined :

Mixed Topology Cell Number *Needs number of cells.

Linear

• Polyvertex - a group of unconnected points
• Polyline - a group of line segments
• Polygon
• Triangle
• Quadrilateral
• Tetrahedron
• Pyramid
• Wedge
• Hexahedron

Quadratic

• Edge_3 - Quadratic line with 3 nodes
• Tri_6
• Quad_8
• Tet_10
• Pyramid_13
• Wedge_15
• Hex_20

Arbitrary

Mixed topology enables to define a mixture of unstructured cells (see picture to right).

A cell defined in a mixed topology is made of a type (integer) and a list of associated data (integer) describing the cell.

Cell types can be:

• 1 - POLYVERTEX
• 2 - POLYLINE
• 3 - POLYGON
• 4 - TRIANGLE
• 5 - QUADRILATERAL
• 6 - TETRAHEDRON
• 7 - PYRAMID
• 8 - WEDGE
• 9 - HEXAHEDRON
• 16 - POLYHEDRON
• 34 - EDGE_3
• 35 - QUADRILATERAL_9
• 36 - TRIANGLE_6
• 37 - QUADRILATERAL_8
• 38 - TETRAHEDRON_10
• 39 - PYRAMID_13
• 40 - WEDGE_15
• 41 - WEDGE_18
• 48 - HEXAHEDRON_20
• 49 - HEXAHEDRON_24
• 50 - HEXAHEDRON_27

The list of data describing each cell depends on the cell type.

If the cell is neither a polygon nor a polyhedron, the list of data describing each cell is the list of indices of each node of the cell.

If the cell is a polygon, the list of data describing each polygon is:

• number of nodes the polygon is made of
• the (ordered) list of each index of each node

If the cell is a polyhedron, the list of data describing each polyhedron is:

• number of faces the polyhedron is made of
• for each face: the number of nodes of the face, and, the (ordered) list of each index of each node

Here is an example of an octahedron (polyhedron):

<Domain>
  <Grid Name="Octahedron" GridType="Uniform">
    <Topology TopologyType="Mixed">
      <DataItem Dimensions="26" NumberType="Int" Precision="4" Format="XML">
        <!-- polyhedron cell type = 16, 8 faces, each face = 3 points, 3 point indices -->
        16
         8
         3 0 1 4
         3 0 1 5
         3 1 2 4
         3 1 2 5
         3 2 3 4
         3 2 3 5
         3 3 0 4
         3 3 0 5
      </DataItem>
    </Topology>
    <Geometry GeometryType="XYZ">
      <DataItem Rank="2" Dimensions="6 3" NumberType="Float" Precision="8" Format="XML">
        -1. -1.  0.
        -1.  1.  0.
         1.  1.  0.
         1. -1.  0.
         0.  0.  1.
         0.  0. -1.
      </DataItem>
    </Geometry>
  </Grid>
</Domain>

Structured

• 2DSMesh - Curvilinear
• 2DRectMesh - Axis are perpendicular
• 2DCoRectMesh - Axis are perpendicular and spacing is constant
• 3DSMesh
• 3DRectMesh
• 3DCoRectMesh

There is a NodesPerElement attribute for the cell types where it is not implicit. For example, to define a group of Octagons, set TopologyType="Polygon" and NodesPerElement="8". For structured grid topologies, the connectivity is implicit. For unstructured topologies the Topology element must contain a DataItem that defines the connectivity:

<Topology TopologyType="Quadrilateral" NumberOfElements="2">
    <DataItem Format="XML" DataType="Int" Dimensions="2 4">
        0 1 2 3
        1 6 7 2
    </DataItem>
</Topology>

The connectivity defines the indexes into the XYZ geometry that define the cell. In this example, the two quads share an edge defined by the line from node 1 to node 2. A Topology element can define Dimensions or NumberOfElements; this is just added for clarity. Here NumberOfElements defines the number of cells we have (3: 1 Tet, 1 Poly, and 1 Hex) and the dimensions denotes the total number of items used to describe the topology (total of 20).

Mixed topologies must define the cell type of every element. If that cell type does not have an implicit number of nodes, that must also be specified. Some cell types will also need to denote how many node points are needed to define the shape. In this example, we define a topology of three cells consisting of a Tet (cell type 6) a Polygon (cell type 3) and a Hex (cell type 9):

<Topology TopologyType="Mixed" NumberOfElements="3" >
    <DataItem Format="XML" DataType="Int" Dimensions="20">
        6 0 1 2 7
        3 4 4 5 6 7
        9 8 9 10 11 12 13 14 15
    </DataItem>
</Topology> 

Notice that the Polygon must define the number of nodes (4) before its connectivity. The cell type numbers are defined in the API documentation.

The BaseOffset attribute for specifying connectivity indices starting at a value other than 0 (e.g., the default starting array index in Fortran is 1) was available in the original XDMF implementation but is not available in XDMF3.

Geometry

The Geometry element describes the XYZ values of the mesh. The important attribute here is the organization of the points. The default is XYZ; an X,Y, and Z for each point starting at parametric index 0. Possible organizations are :

• XYZ - Interlaced locations
• XY - Z is set to 0.0
• X_Y_Z - X,Y, and Z are separate arrays
• VXVYVZ - Three arrays, one for each axis
• ORIGIN_DXDYDZ - Six Values : Ox,Oy,Oz + Dx,Dy,Dz
• ORIGIN_DXDY - Four Values : Ox,Oy + Dx,Dy

The following Geometry element defines 8 points :

<Geometry GeometryType="XYZ">
    <DataItem Format="XML" Dimensions="2 4 3">
        0.0    0.0    0.0
        1.0    0.0    0.0
        1.0    1.0    0.0
        0.0    1.0    0.0 
        0.0    0.0    2.0
        1.0    0.0    2.0
        1.0    1.0    2.0
        0.0    1.0    2.0
    </DataItem>
</Geometry>

Together with the Grid and Topology element we now have enough to define a full XDMF XML file (that can be visualized in VisIt or ParaView) that defines two quadrilaterals that share an edge (notice not all points are used):

An unstructured mesh consisting of two quadrilaterals that share an edge (visualized using VisIt)

<?xml version="1.0" ?>
<!DOCTYPE Xdmf SYSTEM "Xdmf.dtd" []>
<Xdmf Version="2.0" xmlns:xi="[http://www.w3.org/2001/XInclude]">
    <Domain>
        <Grid Name="Two Quads">
            <Topology TopologyType="Quadrilateral" NumberOfElements="2">
                <DataItem Format="XML" DataType="Int" Dimensions="2 4">
                    0 1 2 3
                    1 6 7 2
                </DataItem>
            </Topology>
            <Geometry GeometryType="XYZ">
                <DataItem Format="XML" Dimensions="2 4 3">
                    0.0    0.0    0.0
                    1.0    0.0    0.0
                    1.0    1.0    0.0
                    0.0    1.0    0.0
                    0.0    0.0    2.0
                    1.0    0.0    2.0
                    1.0    1.0    2.0
                    0.0    1.0    2.0
                </DataItem>
            </Geometry>
        </Grid>
    </Domain>
</Xdmf>

It is valid to have DataItem elements to be direct children of the Xdmf or Domain elements. This could be useful if several Grids share the same Geometry but have separate Topology:

An unstructured mesh consisting of two quadrilaterals that share an edge (visualized using ParaView)

<?xml version="1.0" ?>
<!DOCTYPE Xdmf SYSTEM "Xdmf.dtd" []>
<Xdmf Version="2.0" xmlns:xi="[http://www.w3.org/2001/XInclude]">
    <Domain>
        <DataItem Name="Point Data" Format="XML" Dimensions="2 4 3">
            0.0    0.0    0.0
            1.0    0.0    0.0
            1.0    1.0    0.0
            0.0    1.0    0.0
            0.0    0.0    2.0
            1.0    0.0    2.0
            1.0    1.0    2.0
            0.0    1.0    2.0
        </DataItem>
        <Grid Name="Two Quads">
            <Topology Type="Quadrilateral" NumberOfElements="2" >
                <DataItem Format="XML" 
                          DataType="Int"
                          Dimensions="2 4">
                    0 1 2 3
                    1 6 7 2
                </DataItem>
            </Topology>
            <Geometry GeometryType="XYZ">
                <DataItem Reference="XML">
                    /Xdmf/Domain/DataItem[@Name="Point Data"]
                </DataItem>
            </Geometry>
        </Grid>
    </Domain>
</Xdmf>

Attribute

The Attribute element defines values associated with the mesh. There are three attributes which defines what type of data is associated with the mesh.

AttributeType specifies a rank of data stored on mesh, it could be one of:

• Scalar
• Vector
• Tensor - 9 values expected
• Tensor6 - a symmetrical tensor
• Matrix - an arbitrary NxM matrix

Center defines where data are centered and it could take values from:

• Node
• Edge
• Face
• Cell
• Grid
• Other

A Grid centered Attribute might be something like "Material Type" where the value is constant everywhere in the grid. Edge and Face centered values are defined, but do not map well to many visualization systems (ParaView, for example, ignores them at present). Edge and Face values are defined for each Cell in turn, using VTK ordering. The same Face or Edge may be visited multiple times, with duplicate values. Other centered attributed is currently used only with FiniteElementFunction ItemType. FiniteElementFunction has data centered on nodes, midpoints, centroids, etc. and combination of them. For this reason the centering is called Other.

ItemType determines if and how data should be treated in a special way. There is only one currently supported ItemType

• FiniteElementFunction

and only in the case of FiniteElementFunction there are additional attributes ElementFamily with options:

• CG
• DG
• Q
• DQ
• RT

and ElementDegree taking arbitrary integer value and ElementCell with options:

• interval
• triangle
• tetrahedron
• quadrilateral
• hexahedron

Attribute with ItemType="FiniteElementFunction" must contain at least 2 children DataItem elements. They define in the following order(!):

• indices to degrees of freedom values (in finite element codes also called dofmap)
• degrees of freedom values

First DataItem defines indices to degrees of freedom values. These are cell-wise indices to the second DataItem array.

Second DataItem defines values of degrees of freedom. These are actual values that define finite element function. In the language of finite element theory, degrees of freedom are dual mappings which define the finite element. Their values are evaluations of these duals on the finite element function. For ElementFamily="CG/DG/Q/DQ" these values are actual values of the function at nodes. Nodes coincide with nodes of VTK cell types, the only thing different is their ordering. (Non-trivial) Permutations from XDMF's FiniteElementFunction order to VTK order are:

• ElementFamily="CG/DG", ElementDegree="2" and ElementCell="triangle" maps to VTK_QUADRATIC_TRIANGLE nodes as {0, 1, 2, 5, 3, 4}
• ElementFamily="CG/DG", ElementDegree="2" and ElementCell="tetrahedron" maps to VTK_QUADRATIC_TETRA nodes as {0, 1, 2, 3, 9, 6, 8, 7, 5, 4}
• ElementFamily="Q/DQ", ElementDegree="1" and ElementCell="quadrilateral" maps to VTK_QUAD nodes as (lexicographic) {0, 1, 3, 2}
• ElementFamily="Q/DQ", ElementDegree="2" and ElementCell="quadrilateral" maps to VTK_BIQUADRATIC_QUAD nodes as (lexicographic) {0, 1, 4, 3, 2, 7, 5, 6, 8}

VTK ordering of nodes is depicted in VTK "File Formats" manual. XDMF ordering is inspired with finite element code FEniCS (www.fenicsproject.org and femtable.org).

Examples of usage of Attribute:

Typically simple Attributes are assigned on the Node :

<Attribute Name="Node Values" Center="Node">
    <DataItem Format="XML" Dimensions="6 4">
        100 200 300 400
        500 600 600 700
        800 900 1000 1100
        1200 1300 1400 1500
        1600 1700 1800 1900
        2000 2100 2200 2300
    </DataItem>
</Attribute>

Or assigned to the cell centers :

<Attribute Name="Cell Values" Center="Cell">
    <DataItem Format="XML" Dimensions="3">
        3000 2000 1000
    </DataItem>
</Attribute>

An example of Scalar FiniteElementFunction of family "CG" and degree "1" living on 2D simplicial mesh is (it is function x*y on [1, 1] square mesh):

<Attribute ItemType="FiniteElementFunction" ElementFamily="CG" ElementDegree="1" ElementCell="triangle" Name="u" Center="Other" AttributeType="Scalar">
  <DataItem Dimensions="8 3" NumberType="UInt" Format="XML">
    3 6 4
    3 1 4
    6 8 7
    6 4 7
    1 4 2
    1 0 2
    4 7 5
    4 2 5
  </DataItem>
  <DataItem Dimensions="9 1" NumberType="Float" Format="XML">
    0
    0
    0.5
    0
    0.25
    1
    0
    0.5
    0
  </DataItem>
</Attribute>

There are 8 cells on which the function is stored. From the first DataItem we can see, that for the first cell first three values for degrees of freedom are on indices 3 6 4 of the second array. They take values 0 0 0.25. Because the function family is CG and degree is 1 we know, that these values are values of piecewise linear function on triangle's vertices. The ordering of these values is the same as VTK ordering, if it wasn't we would need to apply permutations defined above.

An example of Vector FiniteElementFunction of family "CG" and degree "1" living on 2D simplicial mesh is (it is function (x, y) on [1, 1] square mesh):

<Attribute ItemType="FiniteElementFunction" ElementFamily="CG" ElementDegree="1" ElementCell="triangle" Name="u" Center="Other" AttributeType="Vector">
  <DataItem Dimensions="8 6" NumberType="UInt" Format="XML">
    6 12 8 7 13 9
    6 2 8 7 3 9
    12 16 14 13 17 15
    12 8 14 13 9 15
    2 8 4 3 9 5 
    2 0 4 3 1 5
    8 14 10 9 15 11
    8 4 10 9 5 11
  </DataItem>
  <DataItem Dimensions="18 1" NumberType="Float" Format="XML">
    0
    1
    0
    0.5
    0.5
    1
    0
    0
    0.5
    0.5
    1
    1
    0.5
    0
    1
    0.5
    1
    0
  </DataItem>
</Attribute>

Unlike in the AttributeType="Scalar" case the function has now 6 values of degrees of freedom per each cell. It corresponds to two components of each of three nodal values. There are 6 degrees of freedom indexed 6 12 8 7 13 9 with values 0 0.5 0.5 0 0 0.5 for the first cell. Because AttributeType="Vector" and mesh has topological dimension 2 we expect vector to have 2 components. These 6 values iterates first through nodes, then through vector components. It means that that function has values (0, 0) (0.5, 0) (0.5, 0.5).

Set

The Set element may be used to define Attributes on a subset of entities of a Grid.

Valid SetTypes are:

• Node
• Edge
• Face
• Cell

Set may have multiple DataItems. The last DataItem always provides the indices of the entity. If SetType is Face or Edge, the First DataItem defines the Cell indices, and subsequent Dataitems define the Cell-local Face or Edge indices. It is therefore possible to define multiple values on Edges or Faces. The following example defines values on Face 0 of Cells 1,2,3 and 4.

<Set Name="Ids" SetType="Face">
    <DataItem Format="XML" Dimensions="4" >
        1 2 3 4
    </DataItem>
    <DataItem Format="XML" Dimensions="4" >
        0 0 0 0
    </DataItem> 
    <Attribute Name="Example" Center="Face">
        <DataItem Format="XML" Dimensions="4" >
            100.0 110.0 100.0 200.0
        </DataItem>
    </Attribute>
</Set>

Time

The Time element is a child of the Grid element and specifies the temporal information for the grid.

Xdmf3 only supports a single time value for each grid. Older versions allow use of different TimeTypes.

The type of time element is defined by the TimeType attribute of the element. Valid TimeTypes are :

• Single - A single time value for the entire grid
• HyperSlab - Start, Stride, Count
• List - A list of discrete times
• Range - Min, Max

So in the simplest form, specifying a single time:

<Time Value="0.1" />

For a more complex situation, consider a grid with GridType="Collection" which contains a set 100 children grids written every 1usec of simulation time:

<Time TimeType="HyperSlab">
    <DataItem Format="XML" NumberType="Float" Dimensions="3">
        0.0 0.000001 100
    </DataItem>
</Time>

For a collection of grids not written at regular intervals:

<Time TimeType="List">
    <DataItem Format="XML" NumberType="Float" Dimensions="7">
        0.0 0.1 0.5 1.0 1.1 10.0 100.5
    </DataItem>
</Time>

For data which is valid not at a discrete time but within a range:

<Time TimeType="Range">
    <DataItem Format="XML" NumberType="Float" Dimensions="2">
        0.0 0.5
    </DataItem>
</Time>

Information

There is regularly code or system specific information that needs to be stored with the data that does not map to the current data model. There is an Information element. This is intended for application specific information that can be ignored. A good example might be the bounds of a grid for use in visualization. Information elements have a Name and Value attribute. If Value is nonexistent the value is in the CDATA of the element:

<Information Name="XBounds" Value="0.0 10.0"/>
<Information Name="Bounds"> 0.0 10.0 100.0 110.0 200.0 210.0 </Information>

Several items can be addressed using the Information element like time, units, descriptions, etc. without polluting the XDMF schema. If some of these get used extensively they may be promoted to XDMF elements in the future.

XML Element (Xdmf ClassName) and Default XML Attributes

Default values are shown in red.

• Attribute (XdmfAttribute)

 Name            (no default)
 AttributeType   Scalar | Vector | Tensor | Tensor6 | Matrix | GlobalID
 Center          Node | Cell | Grid | Face | Edge | Other
 ItemType        (no default) | FiniteElementFunction
 ElementFamily   (no default) | CG | DG | Q | DQ | RT (Only Meaningful if ItemType="FiniteElementFunction")
 ElementDegree   (no default) (Only Meaningful if ItemType="FiniteElementFunction")
 ElementCell     (no default) | interval | triangle | tetrahedron | quadrilateral | hexahedron (Only Meaningful if ItemType="FiniteElementFunction")
 

• DataItem (XdmfDataItem)

 Name            (no default)
 ItemType        Uniform | Collection | tree | HyperSlab | coordinates | Function
 Dimensions      (no default) in KJI Order
 NumberType      Float | Int | UInt | Char | UChar
 Precision       1 | 2 (Int or UInt only) |4 | 8
 Format          XML | HDF | Binary
 Endian          Native | Big | Little (applicable only to Binary format)
 Compression     Raw|Zlib|BZip2 (applicable only to Binary format and depend on xdmf configuration)
 Seek            0 (number of bytes to skip, applicable only to Binary format with Raw compression)

• Domain (XdmfDomain)

 Name            (no default)

• Geometry (XdmfGeometry)

 GeometryType     XYZ | XY | X_Y_Z | VxVyVz | Origin_DxDyDz | Origin_DxDy

• Grid (XdmfGrid)

 Name             (no default)
 GridType         Uniform | Collection | Tree | Subset
 CollectionType   Spatial | Temporal (Only Meaningful if GridType="Collection")
 Section          DataItem | All  (Only Meaningful if GridType="Subset")

• Information (XdmfInformation)

 Name              (no default)
 Value             (no default)

• Xdmf (XdmfRoot)

 Version            Current Version | *

• Topology (XdmfTopology)

 Name               (no default)
 TopologyType       Polyvertex | Polyline | Polygon |
                    Triangle | Quadrilateral | Tetrahedron | Pyramid| Wedge | Hexahedron |
                    Edge_3 | Triangle_6 | Quadrilateral_8 | Tetrahedron_10 | Pyramid_13 |
                    Wedge_15 | Hexahedron_20 |
                    Mixed |
                    2DSMesh | 2DRectMesh | 2DCoRectMesh |
                    3DSMesh | 3DRectMesh | 3DCoRectMesh
 NodesPerElement    (no default) Only Important for Polyvertex, Polygon and Polyline
 NumberOfElements   (no default)
     OR
 Dimensions         (no default)
 Order              each cell type has its own default

• Time

TimeType            Single | HyperSlab | List | Range
Value               (no default - Only valid for TimeType="Single")