Analysis of vector fields, fourier and harmonics

[SirAskalot]

Hi

I am working on a optimization problem involving vector fields. In order to define a objective function I need a measure (scalar quantity) of some properties of the vector field. The vector field comes from a finite element analysis, that is the vector field is calculated on a discretized domain. If possible I want to exclude the time domain.

The properties I am interested in are the path/direction of the vector field. Due to the geometry and material properties the vector field (or flux) travels in an unwanted path.
Until now I have used a 1-dimensional measure( flux through a surface), calculated the Fourier transform and total harmonic distortion to describe the problem. But I feel this is not a good enough description of the behavior.

I have tried searching for a better solution but having difficulties finding the proper solution. From all the math class I have had up until now I don't recall a method having such a function. I have thought of a 2D Fourier transform but i lack the knowledge of analyzing the results. In essence I need a scalar quantity in the end. I don't know if one can describe the vector field by harmonic content as with the 1D case.

I don't need a actual unit on the measure, but a quantity who describe the field and how the field changes when the geometry of the domain is changed.

I would be very happy if any of you guys and girls could guide me in the right direction and suggest some methods I can look into.

Thanks,
SirAskalot

(if this thread fits better in another sub-category, let me know)

 

[joeblow]

TBH, I'm not entirely sure what you're asking. What follows is an attempt to answer what I *think* you're asking.

A path integral of a force field represents work (i component) and flux (j component). If you only have flux, you can only describe half the force field.

 

[SirAskalot]

Maybe my inquiry was unclear. I should have attached a figure showing my intention. Unfortunately I don't have access to one at the moment, maybe i'll make one tomorrow.

A line integral would be a possibility and I have thought of doing such, but I am unsure on what line/path/geometry to choose. Maybe along some streamline, but then again choosing a similar streamline in the next trial would be hard.

What I was hoping for was a method utilizing the complete vector space. Not depending on the geometry of the domain.

joeblow: Thanks for the input, maybe you got some more if I describe the problem a little better.

If any of you got another ideas I would be very happy, anything goes.