Page 278 - Process Modelling and Simulation With Finite Element Methods

Coupling Variables Revisited 265

Projection Coupling Variables and Line Integrals

The projection coupling variable performs a line integral across a 2-D domain
according to a specified coordinate dependent transformation, i.e. a path integral.
That this is a useful concept is attested to by its use in formulating quantum
electrodynamics [14]. In its simplest form, the path is taken as one of the
coordinates (a simple grid line) and thus achieves a reduction in the order of the
domain or variable dependence:

(7.7)

I(x) is the coupling variable, which must be defined on a domain D1 of
dimension one less than D2= (xl,x2)x(yl(x),y2(x)) in the case shown above. I:
D2 + D,. A more complicated projection can be achieved by local mesh
transformation using either the space coordinates (dependent variables x, y, z . . .)
or local mesh parameters, e.g. s, sl, or s2, which are then used to make a new
source mesh either for interpolation or directing the curves on which the
line/projection integrals are to be computed. For example

where the curve C is parametrized by n. So there is one such line integral for
each point x in the destination domain. Generally, a projection coupling variable
is one order of dimensionality lower than the source domain and therefore must
be defined on a new domain, perhaps created explicity to receive the coupling
variable as its destination domain. Inherently, a projection coupling requires two
distinct domains (though the destination might be a boundary of the source
domain) and thus must be planned from the start as at least a two domain (and
potentially two geometry) model.
The coupling variable I(x) contains more information than one line integral.
So if you are interested in a particular value of the line integral, then you need
only click on the point in the destination domain on the post plot of the coupling
variable, and the message window will display the interpolated value at that
point. Alternatively, you can export the FEM structure and use postinterp to
provide numerical value.

Example: Lidar positioning and sizing of a dispersing pollutant cloud

Lidar works on the same principle as several other optical devices, for instance
spectrometry and spectroscopy, where light received of a given wavelength is of