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Meeting Minutes - 11 April 2000


WARNING : Work in Progress

Latest Version :
http://www.physiome.org.nz/fieldml/pages/minutes.html
This Version :
http://www.physiome.org.nz/fieldml/archive/2000/04/minutes_20000411.html
Authors :
Warren Hedley (Biomedical Engineering Research Group, University of Auckland)
Contributors :

Contents

Introduction

In this edition of the FieldML meeting minutes, we jump straight into a proposal for certain parts of a specification for FieldML version 1.0. In particular, we re-visit element bases and the traditional nodal method of field description, that was first proposed in the 12 December 1999 minutes. Although clearly inferior to and less flexible than the general linear map method, demand dictates that we discuss this method before heading on to the real finite element method.

Element Bases

Finally, the definitive treatise on defining element bases in FieldML. The <element> element (whoa, this is getting confusing) will be probably be the only (XML) element that appears in both the MeshML and FieldML parts of RegionML. When it appears in FieldML, it's most important property is the specification of the element basis functions that will be used for field interpolation, and this is used in both the traditional nodal and the general linear methods of field description - a variation is used in the natural field description.

The root of a basis function set definition is the <basis_function_set> element, which must contain a list of <basis_function> elements, as shown in Figure 1.


<basis_function_set>
  <basis_function>Lagrange_1</basis_function>
  <basis_function>cubic_Hermite</basis_function>
  <basis_function>monomial_2</basis_function>
</basis_function_set>

Figure 1 An example of a basis function set definition.

The basis function sets that a FieldML processor is required to recognise are :

  • "Lagrange_n" : the set of n nth-order basis functions required for linear interpolation across an element given n field values specified at equi-spaced local coordinate locations between 0.0 and 1.0.
  • "monomial_n" : the set of n monomial basis functions (a monomial basis of degree P consists of the P+1 powers of x from 0 to P). Note that the element field parameters combined with a monomial basis have no physical significance.
  • "cubic_Hermite" : the set of four third-order basis functions required for cubic interpolation across an element given field values specified at local coordinates 0.0 and 1.0, and field derivatives at the same points (in that order).

Defining New Basis Functions

As the need arises, a scheme for extending basis function definitions may be introduced whereby basis functions are expressed as linear combinations of monomial basis functions. For instance, a set of one-dimensional cubic Bezier basis functions where the element field parameters are the field values at x = 0 and x = 1 and guide points at x = 1/3 and x = 2/3 could be defined in FieldML as shown in Figure 2. Note that these could just as easily be defined one at a time, and referenced inside a <basis_function_set> element.


<define>
  <basis_function_set name="cubic_Bezier">
    <basis_function>
      <vector xmlns="http://www.w3.org/1998/Math/MathML">
        <cn> 1</cn> <cn>-3</cn> <cn> 3</cn> <cn>-1</cn>
      </vector>
    </basis_function>
    <basis_function>
      <vector xmlns="http://www.w3.org/1998/Math/MathML">
        <cn> 0</cn> <cn> 3</cn> <cn>-6</cn> <cn> 3</cn>
      </vector>
    </basis_function>
    <basis_function>
      <vector xmlns="http://www.w3.org/1998/Math/MathML">
        <cn> 0</cn> <cn> 0</cn> <cn> 3</cn> <cn>-3</cn>
      </vector>
    </basis_function>
    <basis_function>
      <vector xmlns="http://www.w3.org/1998/Math/MathML">
        <cn> 0</cn> <cn> 0</cn> <cn> 0</cn> <cn> 1</cn>
      </vector>
    </basis_function>
  </basis_function_set>
</define>

Figure 2 The <define> element can be used to define a basis function or basis function set in terms of monomial coefficients.

This functionality requires the content model of the <basis_function> element to be extended from plain text to include an XML infoset matching the vector production. (OK - so I haven't talked about this yet. Here goes:) Currently, all matrices and vectors have been marked up using MathML, but this isn't concise, and FieldML applications may want to define their own formats for these basic types, for instance a comma-separated list, with XML elements placed in an application specific namespace. The RegionML specification will define the expected information content of elements that are created to replace MathML's matrix and vector notation, and these definitions will be known as the matrix and vector productions.

Nodal Field Description

This format is based on the age-old way of doing things in Engineering Science and is basically and XML-isation of CMGUI's .exelem and .exnode file formats. The files currently necessary to describe a one-element cube can be found here: the .exelem file contains the mesh, element bases and map (implied) information, and the .exnode file contains the ensemble field parameters for the problem.

With the separation of mesh and field information in RegionML, the nodal field description reduces to:

  • Element Bases - definitions of the set of basis functions used for field interpolation within each element (see above).
  • Node Vector - the ensemble field parameters in the traditional model description are the field values (and possibly derivatives) at the nodes.
  • Node Indexing - a list containing the indices of the nodes in the global node vector that correspond to the local nodes for each element. Currently the order is implied, based on the local coordinates of the element.


<field name="temperature" num_components="1">

  <define>
    <basis_function_set name="one_D_cubic_hermite">
      <basis_function>cubic_Hermite</basis_function>
    </basis_function_set>
  </define>

  <field_component name="temperature">

    <nodal_field_description>

      <node_list>
        <field_ordering>
          <nl_field_value />
          <nl_field_derivative type="d/ds1" />
          <nl_field_value />
          <nl_field_derivative type="d/ds1" />
        </field_ordering>
        <field_node name="1">
          <vector><cn>0.0</cn><cn>1.0</cn></vector>
        </field_node>
        <field_node name="2">
          <vector><cn>0.0</cn><cn>1.0</cn></vector>
        </field_node>
      </node_list>

      <element_list>
        <element mesh_index="1">
          <basis_function_set_ref name_ref="one_D_cubic_hermite" />
          <field_node_map>
            <field_node_ref name_ref="1" />
            <field_node_ref name_ref="2" />
          </field_node_map>
          <scale_factors> ... </scale_factors>
          <scale_matrix> ... </scale_matrix>
        </element>
      </element_list>
    </nodal_field_description>

  </field_component>
</field>

Figure 3 Nodal description method example.


Last modified : Wed Apr 12 11:36:05 GMT+14:00 2000

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