Solving Load Density Problem Using Moment Method (MOM)

[danilorj]

I'm trying to solve a problem that came to me using a numerical method.
I'm going to put it here.
Using the moment method (MOM) find the load density at a conductor disc of radius a due to a punctual load q placed at a distance l of the disc.
The analytic answer I'm going to put here. But numerically how do I solve this. Does someone has ever done something like this once. I'm trying to do using MATLAB. If someone has the code i would appreciate too much.
Thanks.

 

[Born2bwire]

What are the equations that you need to solve? The moment method is just a general method of changing a continuous problem into a discretized linear algebra problem.

 

[danilorj]

I'm trying to solve a equation like this: [p] = A^-1B, where A is the matrix of the potentials and p is the matrix of the densities and B= 1/(4piE0)*(1,1,1...1)^T.

 

[Born2bwire]

I'm trying to solve a equation like this: [p] = A^-1B, where A is the matrix of the potentials and p is the matrix of the densities and B= 1/(4piE0)*(1,1,1...1)^T.

So in Matlab just do p = A\B; or something like that. Matlab uses LAPACK to solve linear equation problems.

 

[danilorj]

But how do I find the Matrix A?

 

[danilorj]

You can see what I'm saying at the book of Sadiku Eletromagnetics 3th edition at the 15 chapter, numeric methods. There the book solves a problem similar to this but for a thread, and then he builds the matrix A. But the big problem that the book does not explain the right way is how to find the elements of the principle diagonal. I just need the elements of the matrix A for the problem I want to solve, the disc.

 

[Born2bwire]

Like I asked earlier then, what are your equations that you need to solve?

 

[danilorj]

I'm going to explain it better. Look at this link here. http://books.google.com.br/books?id...K#v=onepage&q=moment method potential&f=false.
That is from a book. Take a look at the topic 3.1.2 Charged Plate. See that the book calculates the terms Zmn for m=n and for m<>n, where m and n are the lines an columns of the matrix. I just want to know how to calculate this terms for a disc charged.

 

[Born2bwire]

I'm going to explain it better. Look at this link here. http://books.google.com.br/books?id...K#v=onepage&q=moment method potential&f=false.
That is from a book. Take a look at the topic 3.1.2 Charged Plate. See that the book calculates the terms Zmn for m=n and for m<>n, where m and n are the lines an columns of the matrix. I just want to know how to calculate this terms for a disc charged.

I am more than familiar to Gibson's text. In that text and in Harrington's they both tackle the case of charged sheets and applying the method of moments to electrostatic problems. What is your difficulty with applying this to your problem?

 

[danilorj]

I already said. I want to find the coefficients of the matrix, as the book does, but it does for a charged plate. I want for charged disc. Now did you understand the situation?

 

[Born2bwire]

I already said. I want to find the coefficients of the matrix, as the book does, but it does for a charged plate. I want for charged disc. Now did you understand the situation?

No. What is the difficulty that you are having? Gibson and Harrington work through the problem explicitly in their texts. What are you having trouble with when it comes to doing it with a circular disc? All it is a different geometry but for the purposes of a moment method it does not change the problem at all with the exception of the mesh.

 

[danilorj]

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?

 

[Born2bwire]

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).

 

[danilorj]

Do you have an idea about the Matlab's code to perform this?

 

[Born2bwire]

For a 2D object you can have Matlab generate the mesh then you can apply the mesh information to organize your impedance matrix. Then iterate through and fill the matrix for Z and b and then just use the Matlab operations to solve the linear system.

 

[danilorj]

Let's suppose we have a disk which is charged to a constant potential V0, I just want the steps to find the surface charge density of the disk numerically. Only this.
If it's possible, could you send to me an essay of matlab's code to perform this?
Thanks.

 

[Born2bwire]

I don't have Matlab codes for this, I do not code in Matlab nor do I develop electrostatic software. You should have more than enough to do this from what I have given you, all of which should be covered in Gibson's book.

Gibson gives you a way of solving numerically for a plate of a given voltage. All you need to change is instead of using a rectangular mesh, use a triangular mesh. The self-integral over a triangular mesh of the function 1/r is already given to you in closed form in Gibson as well. I know for a fact that Matlab has mesh generating functions. You can use those to automatically generate your mesh.

However, if the voltage is zero, the charge distribution on the plate is zero as well unless there are sources involved.