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FieldML Example: Wheel With Varying Element Shapes And Functions

This FieldML 0.5 example defines a 12 element model of a wheel. The first 6 elements near the axis are wedge shaped (triangle extruded along a line) and their coordinates are interpolated using a triquadratic wedge basis with 18 nodes. An outer row of 6 cube shaped elements use 27-node triquadratic Lagrange interpolation for the coordinates field. Note that these shape names describe the bounds of the element chart coordinates, however their spatial coordinates (field) may be considerably distorted.

The wheel model is illustrated in the following figure:

../../_images/fieldml_wheel.png

FieldML wheel model with varying element shapes and functions. Nodes are drawn as spheres, and element numbers are shown at their centres.

The following sections discuss key parts of the FieldML document, and the full listing is given below.

Varying Element Shapes

Switching between wedge and cube shapes is achieved by lines 67 and 70-103 in the FieldML listing. In simpler examples all elements had the same shape, which was specified via a reference in the MeshType to a single shape bounds evaluator from the FieldML Standard Library. This example switches shape by element in a form that is efficient for large meshes, by:

  1. Defining an EnsembleType “mesh3d.shapeids” whose values will be mapped to the shapes in use;
  2. Defining a ParameterEvaluator “mesh3d.shapemap” and related DataSource mapping all elements to shapeids, which efficiently separates the bulk data;
  3. Marking in the MeshType that shapes are mapped by the PiecewiseEvaluator “mesh3d.shape” which switches shape evaluator by element indirectly via the element to shapeids map.

The definitions making up the varying shape map are extracted here:

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  <EnsembleType name="mesh3d.shapeids">
   <Members>
    <MemberRange min="1" max="2"/>
   </Members>
  </EnsembleType>
  <DataResource name="mesh3d.shapemap.data.resource">
   <DataResourceDescription>
    <DataResourceString>1 1 1 1 1 1 2 2 2 2 2 2 
    </DataResourceString>
   </DataResourceDescription>
   <ArrayDataSource name="mesh3d.shapemap.data.source" location="1" rank="1">
    <RawArraySize>12</RawArraySize>
    <ArrayDataSize>12</ArrayDataSize>
   </ArrayDataSource>
  </DataResource>
  <ParameterEvaluator name="mesh3d.shapemap" valueType="mesh3d.shapeids">
   <DenseArrayData data="mesh3d.shapemap.data.source">
    <DenseIndexes>
     <IndexEvaluator evaluator="mesh3d.argument.elements"/>
    </DenseIndexes>
   </DenseArrayData>
  </ParameterEvaluator>
  <PiecewiseEvaluator name="mesh3d.shape" valueType="boolean">
   <Bindings>
    <Bind argument="mesh3d.shapeids.argument" source="mesh3d.shapemap"/>
   </Bindings>
   <IndexEvaluators>
    <IndexEvaluator evaluator="mesh3d.shapeids.argument" indexNumber="1"/>
   </IndexEvaluators>
   <EvaluatorMap>
    <EvaluatorMapEntry value="1" evaluator="shape.unit.wedge12"/>
    <EvaluatorMapEntry value="2" evaluator="shape.unit.cube"/>
   </EvaluatorMap>
  </PiecewiseEvaluator>

Sparse Element Local-To-Global Node Maps

Because half the elements use 18-node triquadratic wedge interpolation and the other half use 27-node triquadratic Lagrange interpolation, there are 2 separate element field templates (“mesh3d.eft1” and “mesh3d.eft2”, respectively), and each requires a sparse element local-to-global node map defined over half of the elements.

Here is the local-to-global node map for 18-node triquadratic wedge interpolation:

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  <DataResource name="mesh3d.eft1.localtoglobalnodes.data.resource">
   <DataResourceDescription>
    <DataResourceString>
 1 8 9 10 2 3 1 51 52 53 45 46 44 94 95 96 88 89 87
 2 10 11 12 3 4 1 53 54 55 46 47 44 96 97 98 89 90 87
 3 12 13 14 4 5 1 55 56 57 47 48 44 98 99 100 90 91 87
 4 14 15 16 5 6 1 57 58 59 48 49 44 100 101 102 91 92 87
 5 16 17 18 6 7 1 59 60 61 49 50 44 102 103 104 92 93 87
 6 18 19 8 7 2 1 61 62 51 50 45 44 104 105 94 93 88 87
    </DataResourceString>
   </DataResourceDescription>
   <ArrayDataSource name="mesh3d.eft1.localtoglobalnodes.key.data.source" location="1" rank="2">
    <RawArraySize>6 19</RawArraySize>
    <ArrayDataSize>6 1</ArrayDataSize>
   </ArrayDataSource>
   <ArrayDataSource name="mesh3d.eft1.localtoglobalnodes.data.source" location="1" rank="2">
    <RawArraySize>6 19</RawArraySize>
    <ArrayDataOffset>0 1</ArrayDataOffset>
    <ArrayDataSize>6 18</ArrayDataSize>
   </ArrayDataSource>
  </DataResource>
  <ParameterEvaluator name="mesh3d.eft1.localtoglobalnodes" valueType="nodes">
   <DOKArrayData keyData="mesh3d.eft1.localtoglobalnodes.key.data.source" valueData="mesh3d.eft1.localtoglobalnodes.data.source">
    <DenseIndexes>
     <IndexEvaluator evaluator="mesh3d.eft1.nodes.argument"/>
    </DenseIndexes>
    <SparseIndexes>
     <IndexEvaluator evaluator="mesh3d.argument.elements"/>
    </SparseIndexes>
   </DOKArrayData>
  </ParameterEvaluator>

The first column of lines 301-306 is the element number, and the numbers beside them are the identifiers of the global nodes corresponding to the 18 local nodes for each element. The two ArrayDataSource definitions extract these into 6x1 ‘key’ and 6x18 ‘data’ arrays, used by the sparse Dictionary of Keys representation of the ParameterEvaluator “mesh3d.eft1.localtoglobalnodes”. The SparseIndexes for the ParameterEvaluator indicate which indexes are in each record of the key array, and in what order. The corresponding dense array record contains a complete set of values for all permutations of listed DenseIndexes. In this case the map is sparse over elements, but for those elements it has a global node for every local node.

Mesh Field Template with Varying Element Functions

Lines 105-109 define an index corresponding to each element field template, useful for mapping to them.

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  <EnsembleType name="mesh3d.eftIndexes">
   <Members>
    <MemberRange min="1" max="2"/>
   </Members>
  </EnsembleType>

Lines 642-670 define a map from elements to the above element field template indexes, and the final PiecewiseEvaluator “mesh3d.fieldtemplate1” defines the mesh field template via this indirect map, delegating to the appropriate element field template based on which index is returned.

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  <DataResource name="mesh3d.fieldtemplate1.eftmap.data.resource">
   <DataResourceDescription>
    <DataResourceString>1 1 1 1 1 1 2 2 2 2 2 2 
    </DataResourceString>
   </DataResourceDescription>
   <ArrayDataSource name="mesh3d.fieldtemplate1.eftmap.data.source" location="1" rank="1">
    <RawArraySize>12</RawArraySize>
    <ArrayDataSize>12</ArrayDataSize>
   </ArrayDataSource>
  </DataResource>
  <ParameterEvaluator name="mesh3d.fieldtemplate1.eftmap" valueType="mesh3d.eftIndexes">
   <DenseArrayData data="mesh3d.fieldtemplate1.eftmap.data.source">
    <DenseIndexes>
     <IndexEvaluator evaluator="mesh3d.argument.elements"/>
    </DenseIndexes>
   </DenseArrayData>
  </ParameterEvaluator>
  <PiecewiseEvaluator name="mesh3d.fieldtemplate1" valueType="real.1d">
   <Bindings>
    <Bind argument="mesh3d.eftIndexes.argument" source="mesh3d.fieldtemplate1.eftmap"/>
   </Bindings>
   <IndexEvaluators>
    <IndexEvaluator evaluator="mesh3d.eftIndexes.argument" indexNumber="1"/>
   </IndexEvaluators>
   <EvaluatorMap>
    <EvaluatorMapEntry value="1" evaluator="mesh3d.eft1.evaluator"/>
    <EvaluatorMapEntry value="2" evaluator="mesh3d.eft2.evaluator"/>
   </EvaluatorMap>
  </PiecewiseEvaluator>

The data ensures “eft1” is delegated to in elements 1-6, and “eft2” is delegated to in elements 7-12. This representation achieves the desired separation of bulk data. Note that direct delegation to element field templates from the element would make the XML serialisation impractically large for large models.

Complete FieldML Wheel Model

Following is the full FieldML 0.5 wheel model document:

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<?xml version="1.0" encoding="ISO-8859-1"?>
<Fieldml version="0.5.0" xsi:noNamespaceSchemaLocation="http://www.fieldml.org/resources/xml/0.5/FieldML_0.5.xsd" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xlink="http://www.w3.org/1999/xlink">
 <Region name="/">
  <Import xlink:href="http://www.fieldml.org/resources/xml/0.5/FieldML_Library_0.5.xml" region="library">
   <ImportType localName="real.1d" remoteName="real.1d"/>
   <ImportType localName="boolean" remoteName="boolean"/>
   <ImportEvaluator localName="shape.unit.wedge12" remoteName="shape.unit.wedge12"/>
   <ImportEvaluator localName="shape.unit.cube" remoteName="shape.unit.cube"/>
   <ImportEvaluator localName="interpolator.3d.unit.triquadraticWedge12" remoteName="interpolator.3d.unit.triquadraticWedge12"/>
   <ImportType localName="parameters.3d.unit.triquadraticWedge12" remoteName="parameters.3d.unit.triquadraticWedge12"/>
   <ImportEvaluator localName="parameters.3d.unit.triquadraticWedge12.argument" remoteName="parameters.3d.unit.triquadraticWedge12.argument"/>
   <ImportType localName="parameters.3d.unit.triquadraticWedge12.component" remoteName="parameters.3d.unit.triquadraticWedge12.component"/>
   <ImportEvaluator localName="parameters.3d.unit.triquadraticWedge12.component.argument" remoteName="parameters.3d.unit.triquadraticWedge12.component.argument"/>
   <ImportEvaluator localName="chart.3d.argument" remoteName="chart.3d.argument"/>
   <ImportEvaluator localName="interpolator.3d.unit.triquadraticLagrange" remoteName="interpolator.3d.unit.triquadraticLagrange"/>
   <ImportType localName="parameters.3d.unit.triquadraticLagrange" remoteName="parameters.3d.unit.triquadraticLagrange"/>
   <ImportEvaluator localName="parameters.3d.unit.triquadraticLagrange.argument" remoteName="parameters.3d.unit.triquadraticLagrange.argument"/>
   <ImportType localName="parameters.3d.unit.triquadraticLagrange.component" remoteName="parameters.3d.unit.triquadraticLagrange.component"/>
   <ImportEvaluator localName="parameters.3d.unit.triquadraticLagrange.component.argument" remoteName="parameters.3d.unit.triquadraticLagrange.component.argument"/>
   <ImportType localName="coordinates.rc.3d" remoteName="coordinates.rc.3d"/>
   <ImportType localName="coordinates.rc.3d.component" remoteName="coordinates.rc.3d.component"/>
   <ImportEvaluator localName="coordinates.rc.3d.component.argument" remoteName="coordinates.rc.3d.component.argument"/>
  </Import>
  <EnsembleType name="nodes">
   <Members>
    <MemberRange min="1" max="129"/>
   </Members>
  </EnsembleType>
  <EnsembleType name="node_derivatives">
   <Members>
    <MemberRange min="1" max="8"/>
   </Members>
  </EnsembleType>
  <ConstantEvaluator name="node_derivatives.value" value="1" valueType="node_derivatives"/>
  <ConstantEvaluator name="node_derivatives.d_ds1" value="2" valueType="node_derivatives"/>
  <ConstantEvaluator name="node_derivatives.d_ds2" value="3" valueType="node_derivatives"/>
  <ConstantEvaluator name="node_derivatives.d2_ds1ds2" value="4" valueType="node_derivatives"/>
  <ConstantEvaluator name="node_derivatives.d_ds3" value="5" valueType="node_derivatives"/>
  <ConstantEvaluator name="node_derivatives.d2_ds1ds3" value="6" valueType="node_derivatives"/>
  <ConstantEvaluator name="node_derivatives.d2_ds2ds3" value="7" valueType="node_derivatives"/>
  <ConstantEvaluator name="node_derivatives.d3_ds1ds2ds3" value="8" valueType="node_derivatives"/>
  <EnsembleType name="node_versions">
   <Members>
    <MemberRange min="1" max="1"/>
   </Members>
  </EnsembleType>
  <ConstantEvaluator name="node_versions.1" value="1" valueType="node_versions"/>
  <ArgumentEvaluator name="nodes.argument" valueType="nodes"/>
  <ArgumentEvaluator name="node_derivatives.argument" valueType="node_derivatives"/>
  <ArgumentEvaluator name="node_versions.argument" valueType="node_versions"/>
  <ArgumentEvaluator name="nodes.parameters" valueType="real.1d">
   <Arguments>
    <Argument name="nodes.argument"/>
    <Argument name="node_derivatives.argument"/>
    <Argument name="node_versions.argument"/>
   </Arguments>
  </ArgumentEvaluator>
  <MeshType name="mesh3d">
   <Elements name="elements">
    <Members>
     <MemberRange min="1" max="12"/>
    </Members>
   </Elements>
   <Chart name="xi">
    <Components name="mesh3d.xi.components" count="3"/>
   </Chart>
   <Shapes evaluator="mesh3d.shape"/>
  </MeshType>
  <ArgumentEvaluator name="mesh3d.argument" valueType="mesh3d"/>
  <EnsembleType name="mesh3d.shapeids">
   <Members>
    <MemberRange min="1" max="2"/>
   </Members>
  </EnsembleType>
  <DataResource name="mesh3d.shapemap.data.resource">
   <DataResourceDescription>
    <DataResourceString>1 1 1 1 1 1 2 2 2 2 2 2 
    </DataResourceString>
   </DataResourceDescription>
   <ArrayDataSource name="mesh3d.shapemap.data.source" location="1" rank="1">
    <RawArraySize>12</RawArraySize>
    <ArrayDataSize>12</ArrayDataSize>
   </ArrayDataSource>
  </DataResource>
  <ParameterEvaluator name="mesh3d.shapemap" valueType="mesh3d.shapeids">
   <DenseArrayData data="mesh3d.shapemap.data.source">
    <DenseIndexes>
     <IndexEvaluator evaluator="mesh3d.argument.elements"/>
    </DenseIndexes>
   </DenseArrayData>
  </ParameterEvaluator>
  <PiecewiseEvaluator name="mesh3d.shape" valueType="boolean">
   <Bindings>
    <Bind argument="mesh3d.shapeids.argument" source="mesh3d.shapemap"/>
   </Bindings>
   <IndexEvaluators>
    <IndexEvaluator evaluator="mesh3d.shapeids.argument" indexNumber="1"/>
   </IndexEvaluators>
   <EvaluatorMap>
    <EvaluatorMapEntry value="1" evaluator="shape.unit.wedge12"/>
    <EvaluatorMapEntry value="2" evaluator="shape.unit.cube"/>
   </EvaluatorMap>
  </PiecewiseEvaluator>
  <ArgumentEvaluator name="mesh3d.shapeids.argument" valueType="mesh3d.shapeids"/>
  <EnsembleType name="mesh3d.eftIndexes">
   <Members>
    <MemberRange min="1" max="2"/>
   </Members>
  </EnsembleType>
  <ArgumentEvaluator name="mesh3d.eftIndexes.argument" valueType="mesh3d.eftIndexes"/>
  <EnsembleType name="mesh3d.eft1.nodes">
   <Members>
    <MemberRange min="1" max="18"/>
   </Members>
  </EnsembleType>
  <ArgumentEvaluator name="mesh3d.eft1.nodes.argument" valueType="mesh3d.eft1.nodes"/>
  <ArgumentEvaluator name="mesh3d.eft1.nodeparameters.argument" valueType="real.1d">
   <Arguments>
    <Argument name="node_derivatives.argument"/>
    <Argument name="node_versions.argument"/>
    <Argument name="mesh3d.eft1.nodes.argument"/>
   </Arguments>
  </ArgumentEvaluator>
  <ConstantEvaluator name="mesh3d.eft1.nodes.1" value="1" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.2" value="2" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.3" value="3" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.4" value="4" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.5" value="5" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.6" value="6" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.7" value="7" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.8" value="8" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.9" value="9" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.10" value="10" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.11" value="11" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.12" value="12" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.13" value="13" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.14" value="14" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.15" value="15" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.16" value="16" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.17" value="17" valueType="mesh3d.eft1.nodes"/>
  <ConstantEvaluator name="mesh3d.eft1.nodes.18" value="18" valueType="mesh3d.eft1.nodes"/>
  <ReferenceEvaluator name="mesh3d.eft1.nodeparameters.node1.value.v1" evaluator="mesh3d.eft1.nodeparameters.argument" valueType="real.1d">
   <Bindings>
    <Bind argument="mesh3d.eft1.nodes.argument" source="mesh3d.eft1.nodes.1"/>
    <Bind argument="node_derivatives.argument" source="node_derivatives.value"/>
    <Bind argument="node_versions.argument" source="node_versions.1"/>
   </Bindings>
  </ReferenceEvaluator>
  <ReferenceEvaluator name="mesh3d.eft1.nodeparameters.node2.value.v1" evaluator="mesh3d.eft1.nodeparameters.argument" valueType="real.1d">
   <Bindings>
    <Bind argument="mesh3d.eft1.nodes.argument" source="mesh3d.eft1.nodes.2"/>
    <Bind argument="node_derivatives.argument" source="node_derivatives.value"/>
    <Bind argument="node_versions.argument" source="node_versions.1"/>
   </Bindings>
  </ReferenceEvaluator>
  <ReferenceEvaluator name="mesh3d.eft1.nodeparameters.node3.value.v1" evaluator="mesh3d.eft1.nodeparameters.argument" valueType="real.1d">
   <Bindings>
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