Field Definition

Poul asks : "What is the minimal set of information needed to define a field? Why not just store that?"

Warren says : "That's friggin' ridiculous ! That's the stupidest thing I've ever heard in my life !"

As Warren writes these pages, some artistic license may have been taken in the accurate transcription of this conversation. There may even be a slight bias to the content of these pages - unbelievable! On to more serious things ...

The Minimal Information Set

As Poul points out, it is sufficient to store a linear map, a set of basis functions (including information on element coordinate bounds), and a vector of ensemble field parameters to fully describe a field. Given this information it is a simple matter (and quick) to produce interpolated points on the field and to solve most classes of FE problems dealt with in this department. The field description is reasonably concise and any duplication and hence inconsistency in the data is avoided.

As David pointed out, given only the minimal information set, it is virtually impossible to determine what an ensemble field parameter physically represents (although this can probably be determined from inspection of the ensemble basis functions). It is also difficult to reconstruct the continuity constraints on the field. Both of these problems make it impossible to change the structure of the mesh (eg. mesh refinement) for a field, given only the minimal information set.

This provides some compelling reasons not to rely on the MIS for field specification. However it can not be discarded altogether because it allows the user to enter problems into solution software that can not be easily specified using a typical graphical interface (eg. Warren's C¹ continuous ventricle model), as linear maps may easily be created using programs external to the solution/rendering software. The specification of a matrix in content markup using MathML 2.0 looks like this :

<matrix>
  <matrixrow>
    <cn> 0 </cn> <cn> 1 </cn> <cn> 0 </cn>
  </matrixrow>
  <matrixrow>
    <cn> 0 </cn> <cn> 0 </cn> <cn> 1 </cn>
  </matrixrow>
  <matrixrow>
    <cn> 1 </cn> <cn> 0 </cn> <cn> 0 </cn>
  </matrixrow>
</matrix>

Although not exactly concise, specialised XML compression tools such as XMill (which I haven't tested yet) should solve this problem. If matrices and vectors can be simply described using MathML, there only remains the problem of basis function definition.

More Verbose Methods

The MIS provides a compact way to provide a field specification to solution software, but a more verbose method may be needed to provide all of the information necessary to manipulate a field in a useful manner. At the opposite end of the linear map generation process is specification of ensemble field parameters and continuity constraints as linear combinations of element field parameters. Extending this further, for most simple problems the physical meaning of the ensemble field parameters and a set of general continuity conditions (the current situation in CMISS essentially) would provide enough information to generate the linear map.