General Linear Map Description

The general linear map description provides the minimum information set needed to evaluate and manipulate a field. It is based on the "Generalized FEM" theory taken from Warren Hedley's Masters thesis, available from here. The components of the general linear map field description are as follows.

• Element Bases - definitions of the set of basis functions used for field interpolation within each element (see Specifying Element Bases.
• Ensemble Field Parameters - The values of the ensemble field parameters (a single vector).
• Linear Map - the matrix used for transforming ensemble field parameters to element field parameters. It may be most efficient to store this as a single sparse matrix, or in column groups for a given element and group of ensemble field parameters.

A Simple Example

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

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

  <map-information>
    <ensemble-field-parameters>
      <vector>
        <cn>5.0</cn> <cn>3.0</cn> <cn>4.0</cn>
        <cn>4.0</cn> <cn>5.0</cn> <cn>3.0</cn>
      </vector>
    </ensemble-field-parameters>

    <ensemble-element-field-parameter-map>
      <matrix>
        <matrixrow>
          <cn> 1.0</cn><cn> 0.0</cn><cn> 0.0</cn>
          <cn> 0.0</cn><cn> 0.0</cn><cn> 0.0</cn>
        </matrixrow>
        <matrixrow>
          <cn>-1.0</cn><cn> 1.0</cn><cn> 0.0</cn>
          <cn> 0.0</cn><cn> 0.0</cn><cn> 0.0</cn>
        </matrixrow>
        <matrixrow>
          <cn>-1.0</cn><cn> 0.0</cn><cn> 0.0</cn>
          <cn> 1.0</cn><cn> 0.0</cn><cn> 0.0</cn>
        </matrixrow>
        <matrixrow>
          <cn> 1.0</cn><cn>-1.0</cn><cn> 0.0</cn>
          <cn>-1.0</cn><cn> 1.0</cn><cn> 0.0</cn>
        </matrixrow>
        <matrixrow>
          <cn> 0.0</cn><cn> 1.0</cn><cn> 0.0</cn>
          <cn> 0.0</cn><cn> 0.0</cn><cn> 0.0</cn>
        </matrixrow>
        <matrixrow>
          <cn> 0.0</cn><cn>-1.0</cn><cn> 1.0</cn>
          <cn> 0.0</cn><cn> 0.0</cn><cn> 0.0</cn>
        </matrixrow>
        <matrixrow>
          <cn> 0.0</cn><cn>-1.0</cn><cn> 0.0</cn>
          <cn> 0.0</cn><cn> 1.0</cn><cn> 0.0</cn>
        </matrixrow>
        <matrixrow>
          <cn> 0.0</cn><cn> 1.0</cn><cn>-1.0</cn>
          <cn> 0.0</cn><cn>-1.0</cn><cn> 1.0</cn>
        </matrixrow>
      </matrix>
    </ensemble-element-field-parameter-map>
  </map-information>

</field>