Natural Field Description

The natural field description provides a concise way of describing the components of the general linear map description, without explicitly calculating or specifying the map. Software may interpret this description to generate the general linear map information corresponding to this description. The information set required in the natural field descpription is :

• Element Bases - definitions of the set of basis functions used for field interpolation within each element (see Specifying Element Bases.
• Ensemble Parameters - the physical representation and values of the ensemble parameters that will be used to control the field. The representations are used to express the ensemble field parameters as a linear combination of the element field parameters, and will normally take the form of field values or derivatives at a given set of local coordinates in some element.
• Continuity Constraints - Constraints on the field along inter-element boundaries. Typically, these will specify C¹ continuity or higher, but could also include constant terms in order to define step changes. [to do : more]

If the ensemble field parameters and continuity constraints can be formulated as linear combinations of the element field parameters, then a linear map can be calculated as discussed in the "Generalized FEM" theory accessible from here..

A Simple Example

<field
    name                = "temperature"
    description-method  = "natural"
    type                = "data">

  <element-bases>
    <use-bases>monomial-1, monomial-1</use-bases>
  </element-bases>

  <natural-information>
    <global-parameters>
      <gp-field-value
          local-coordinates  = "0.0, 0.0"
          in-element         = "1">5.0</gp-field-value>
      <gp-field-value
          local-coordinates  = "1.0, 0.0"
          in-element         = "1">3.0</gp-field-value>
      <gp-field-value
          local-coordinates  = "1.0, 0.0"
          in-element         = "2">4.0</gp-field-value>
      <gp-field-value
          local-coordinates  = "0.0, 1.0"
          in-element         = "1">4.0</gp-field-value>
      <gp-field-value
          local-coordinates  = "1.0, 1.0"
          in-element         = "1">5.0</gp-field-value>
      <gp-field-value
          local-coordinates  = "1.0, 1.0"
          in-element         = "2">3.0</gp-field-value>
    </global-parameters>

    <constraints>
      <-- This will be applied across the whole mesh -->
      <field-continuity
          order  = "C0" />
    </constraints>

  </natural-information>

</field>