Re: a MoM question
Hello and thanks for your comments, friends!
1) I can't prove my approach until the matrix [R] will be "healthy" for the theory, i.e. positive semidefinite (with all eigenvalues >=0) - as written in part III of Harrington's paper 011399990.pdf (in previous post), where the eigenvalues <0 are caused by numerical inaccuracies of MoM method
Testing with sphere is a very good point, as we know spherical modes analytically!!
2) I'm quite sure that my [Z] matrices are O.K, Makarov's code is working well with clasiccal MoM postprocessing step like [Z][J]=[E] (1), where E is driving vector (like voltage gap feeding the dipole).
BTW, by having eq. (1) we are missing information of the modes, because the whole MoM solution is superposition of "the characteristic modes".
I've choosed plate just because it's a simple geometry with considerable numbers of mesh nodes and the process in not so time consuming. I've also tested dipole, but the results were wrong too.
3) The separation of Z into R and X is described in another Harrington's paper attached here (it has the same name as the paper in my previous post, but they're different).
The generalized eigenvalue problem is a very common, for example if you solve structural mechanics problem with FEM (like resonant frequencies of a bridge), you finally get the equation:
[L] U_n = Lambda_n [M] U_n, where they call [L]
stiffness matrix and [M]
mass matrix; that's basically the same equation like in our theory!
4) Theory of char. modes has been developed in 70's and then forgotten (now being reborned by people from UPV).
My opinion why is because of boom of very sophisticated and powerfull EM packages (like IE3D, Sonnet, FEKO to name a few). This allows almost everybody to simulate a lot of structures with a little of physical knowledge. You can see plenty of papers "I've drawed patch antenna, cutted some slots and it has this and this behavior, bandwith etc.." But why it behaves so??
A lot of natural phenomenons have a resonant background (eigenmodes), but in result we see a behavior with feeding mechanism included (like bridge feeded by wind). People usually do not think of decomposing of the brigde movement to basic vibration states, it just vibrates (in antenna words vibration states are basic current flows like dipole resonances)
Theory of char. modes is something like an extension to well known cavity model, where we solve Helmholtz equation for eigenmodes. But cavity model solves only for one field component and neglects internal coupling (having patch lying in XY plane, we consider only Ez component). Char. modes comes from full-wave MoM, where internal coupling is naturally included. We have also access to eigenvalues behavior with frequency (note that Z=Z(f) and of course Z is problem dependent) which is related to mode bandwith.
You can see a lot of examples in attached presentations from UPV (Part 1 is theory, Part 3 some nice applications - I like the circularly polarized patch at slides 16+).
My interest are compact planar antennas, that's why I'm interested in this theory (Very often cavity model fails there or is highly inacurrate mainly because of ommiting the internal coupling).
I hope that my reasons are sufficient
PS: I don't know why this theory isn't included in current MoM packages, I think FEKO, IE3D, Sonnet, Supernec etc. are good candidates
Main programming effort on MoM is to create the Z matrix so I guess the rest would not be major problem for the developers...
Thanks for your time!
Cheers,
eirp