danilorj said:
So are you saying that the coefficients of the matrix are the same for the charged plate and the charged disk ? I think they cannot be the same, the integral is different. I don't know what to do with the disk. I have to slice it in some parts? Could you please just show me how would it be the integral over the disk's surface? The intervals?
You do not integrate over the entire surface. The point of the moment method is that you find a way to discretize the problem of continuous operators acting on a continuous space. This is done by approximating your space using a finite set of basis functions. With a surface integral problem, this is done by defining local basis functions over a mesh that approximates the surface of the problem. The easiest way to do this is to use triangular basis functions. Each integration in the moment method integrates over a testing and source triangle on the surface. Because we can parameterize the surface integral over a triangle, as shown in Gibson's text (I know he discusses this in the later chapters of the EFIE method of moments problems in 3D), the integration is more or less the same. The only thing that changes is the position, size, and orientation of our triangles in space due to the geometry of the problem. Still, once you know how to integrate over one triangle, you know how to integrate over all triangles.
Gibson does this explicitly in his rectangular sheet problem. He subdivides the sheet into rectangular patches. Technically, his moment method has two basis functions. The testing basis functions are pulse basis functions, constant values over the area of the patch. His source basis functions are just delta functions at the patch's centroid. That's ok for a rough approximation and it makes his b vector much simpler. If he had chosen a pulse basis function he would have to integrate across the source patch in the Z matrix and b vector. Trivial for the b vector since you would just scale it by the area of the patch but more work for the Z matrix.
Either way, you can see that he quickly arrives at an approximation for the self-term integrals and for integrals of non-overlapping patches. This allows for trivial substitution for filling the impedance matrix.
So for the disc, do the same thing except use triangular patches and work the integrals out for those. This is not too much work because he should have worked out the 1/r integration in closed form for you in later chapters when he works out the singularity subtraction in the 3D EFIE MOM chapter. So you can use his technique of a rough estimate of a one point integration for the off diagonal terms and use his closed form 1/r integral for the diagonal terms. If you want to be more exact, you could use more integration points on the triangles. This can be done using Gaussian quadrature and integrating over a triangular patch using Gaussian quadrature is a very well researched problem, in fact it should be explained in Gibson's book too. You could also add on a pulse basis function for the source integration (which he has implicitly done as a delta function integration I believe). This would require you to have a double surface integral in the Z matrix like in later problems and you would modify your b matrix since you would have to integrate the voltage over the appropriate source patch (which is a constant in his example so it would be trivially scaled by the patch's area like I stated above).
