Possible wiffle-ball analytical solution

Discuss how polywell fusion works; share theoretical questions and answers.

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Indrek
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Post by Indrek »

I finally got around to putting all the equations, images and code into a single fancy PDF:

http://www.mare.ee/indrek/ephi/images.pdf

Btw Art what would be your response to Icarus' objections?

- Indrek

Art Carlson
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Re: Mid-point singularities

Post by Art Carlson »

icarus wrote:kcdodd has pointed out the mid-point singularities earlier in the thread. These mid-point singularities are very, very interesting and not what anyone was expecting.

On the surface of the sphere, they appear to be saddle points in terms of field line topology but minimums of field strength. Moving radially away from the sphere in the Meridian45 plane, the field lines converge to the mid-point singularity. I'm not even sure if these singularities have a mathematical classification. A saddle point on a spherical surface but with a radial convergence.

I think for this topology wiffle-ball (sphere) we can say then there are NO line cusps but there are these weird mid-point singularities and the corner cusps.
With equal conviction we can say that the corners, whatever they are, are not point cusps. Whatever the scaling of the losses is, Bussard's leap to conclude that all the cusps behave like point cusps was unjustified.
icarus wrote:8 corner cusps, 12 mid-point singularities and 6 face-centered cusps., no lines is my take on it.

Now my question has to be, if we are getting results that no-one expects just looking at basic topologies, why would we waste time doing detailed electronic analysis around "cusps" that may or may not exist and may or may not look like we are showing??

ART: In all fairness, I think doing 2 and 3 on your shopping list is a waste of time at this stage until the actual topology of the plasma can be nailed down.
I am all for getting the magnetic topology straight before doing too much else (like designing a reactor). The wierdness along the edges (and therefore at the corners) may be an artifact, e.g. resulting from our approximation of the plasma as a superconducting sphere.
But I think my item 3 is still worthwhile.
3. The third thing that would interest me is what happens to the particles that pass through a cusp. I'm afraid this will require addition of electric charge/potential associated with both the central sphere and the coils. (Might be tricky to get this exactly right, but a decent approximation should not be too hard.) I am interested in the question of whether single particles (collective effects will come later) can really be reflected (recirculated) back into a cusp. I suspect that a good fraction of them won't make it.
The strange field topology is all between the plasma and the coils. Where ever the particles come out along the cube edges (everywhere, or mostly at the corners), once they are outside the radius of the magrid, the field topology is straightforward, and tracing particle trajectories should be able to answer the question of whether orbit effects alone are enough to prevent or at least degrade recirculation.

kcdodd
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Post by kcdodd »

Yes, we should instead leap to conclude they all behave as line cusps. ;) Hehe, I'm just joshing ya. And yeah it's more of a why not simulate. Learn something no matter what.
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Art Carlson
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Re: Mid-point singularities

Post by Art Carlson »

Art Carlson wrote: But I think my item 3 is still worthwhile.
3. The third thing that would interest me is what happens to the particles that pass through a cusp. I'm afraid this will require addition of electric charge/potential associated with both the central sphere and the coils. (Might be tricky to get this exactly right, but a decent approximation should not be too hard.) I am interested in the question of whether single particles (collective effects will come later) can really be reflected (recirculated) back into a cusp. I suspect that a good fraction of them won't make it.
The strange field topology is all between the plasma and the coils. Where ever the particles come out along the cube edges (everywhere, or mostly at the corners), once they are outside the radius of the magrid, the field topology is straightforward, and tracing particle trajectories should be able to answer the question of whether orbit effects alone are enough to prevent or at least degrade recirculation.
I'm probably screwing up here. The fact that the magnetic topology is boring outside the cusps also means that the particles will be able to get back through the cusp after they bounce. The tricky part is what happens when they hit the rapid field changes that break adiabaticity. Will they get back into the high beta plasma region or will they get stuck on a flux tube farther out? To figure that out, you need to know the field topology near the plasma surface, and that is what we're not sure about.

What's the alternative? Instead of calculating particle trajectories with the spherical superconductor, we could try to improve the plasma model first. What's the next easiest model? Maybe a superconducting bag with a fixed volume (or pressure), which would squeeze into the cusps a bit. I think this is well-defined, but I don't know how I would calculate it. Either parameterize the surface position and then adjust it until I have pressure balance everywhere. Or represent the plasma as the six coils we have now plus a multipole expansion, adjusting the terms until they minimize the pressure variations on some flux surface. It doesn't sound easy either way.

Solo
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Post by Solo »

I've got a half-baked idea. I wonder if you could start by assuming the electron sheath we expect, and try to see whether it made sense. I'm not sure how you'd go about modeling it. I'm thinking of using Indrek's image-coil field (just assume zero field inside the sphere) and E-field data (from the grid only). Then do a PIC model, keep track of the electron density on the mesh and use it to find the E-field (no ions modelled, but assume electrons inside the null sphere don't contribute to charge). Also assume monoenergetic electrons, no collisions. This ought to produce a potential well. Maybe that will tell whether the well shape is going to allow convergence.

Maybe then use the new E-field data and either the unmodified B-field or the B-field with the null sphere, and run the code 'the other way' to find the B-field produced by the electrons. Maybe with a few iterations like this we might converge on a solution?

kcdodd
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Post by kcdodd »

Actually I think the bag idea might not be that difficult. I'm pretty sure the only real currents will be surface currents of whatever "bag shape" you have. So, you make a discrete mesh with each edge of the mesh having a current. The B-field from the real coils are constant so just calculate them at all the surface points. Then calculate a contribution b-field from every surface current element leaving the current values as unknowns. Then solve by linear algebra for currents creating fields at all points so that the flux is zero.

Now, the trick is still to figure out what the shape of your bag is supposed to be in the first place.
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Art Carlson
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Post by Art Carlson »

kcdodd wrote:Actually I think the bag idea might not be that difficult. I'm pretty sure the only real currents will be surface currents of whatever "bag shape" you have. So, you make a discrete mesh with each edge of the mesh having a current. The B-field from the real coils are constant so just calculate them at all the surface points. Then calculate a contribution b-field from every surface current element leaving the current values as unknowns. Then solve by linear algebra for currents creating fields at all points so that the flux is zero.

Now, the trick is still to figure out what the shape of your bag is supposed to be in the first place.
There are two parts to the problem. (1) You have to make sure the field is zero inside the bag. (2) You have to make sure the bag is in pressure balance with the field. If you use an interative approach (employing some linear algebra if that makes it easier), you should be able to alternate these steps. First adjust the currents to make the interior field smaller. With these currents, adjust the radius of the surface to make the force imbalance smaller. With the new shape, go back and calculate new currents. And on and on.

I would suggest expressing the currents not as the currents along the edges of the mesh, because then you have to worry about divergence of the current. I would do the calculation in terms of current loops in the mesh elements. Divergence-free currents are guaranteed, the current in any edge (if you need it) is just the difference of the currents in the two adjacent loops, and the incremental change in the current can be simply determined from the normal component of the B field and the area of the loop.

To adjust the radius of a vertex point (i.e., the shape of the bag), I would calculate the force (current X field X length) on each edge leading to the vertex and move the point out a bit if the net force is outward and in a bit if the net force in inward. If you keep track of the force from the previous step, you can also jump to the position where the extrapolated force is zero.

This is gonna take some work, but the algorithm seems fairly clear and I would expect it to converge rapidly.

kcdodd
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Post by kcdodd »

We have the force from the magnetic field pushing the surface inward since we are diminishing the interior field. What force do we use to push out? nKT*Area?
Carter

rnebel
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Post by rnebel »

Art:

Just a thought, but I suspect that you can answer your question about electrons trapping on a flux tube analytically. Write the equation of motion of the electron in either LaGrangian or Hamiltonian form. If the vector potential of the magnetic field has any symmetry, then you should get a conserved canonical momentum. This, combined with total energy conservation, should tell you where the particle can go.

icarus
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Post by icarus »

I would suggest expressing the currents not as the currents along the edges of the mesh, because then you have to worry about divergence of the current. I would do the calculation in terms of current loops in the mesh elements. Divergence-free currents are guaranteed, the current in any edge (if you need it) is just the difference of the currents in the two adjacent loops, and the incremental change in the current can be simply determined from the normal component of the B field and the area of the loop.

To adjust the radius of a vertex point (i.e., the shape of the bag), I would calculate the force (current X field X length) on each edge leading to the vertex and move the point out a bit if the net force is outward and in a bit if the net force in inward. If you keep track of the force from the previous step, you can also jump to the position where the extrapolated force is zero.
Art: I think the method you describe here is already well-known and used in hydrodynamics, it is essentially the magneto-static analogy for the "vortex panel methods" used to calculate forces (lift, drag) on aircraft wings and other bodies subject to hydrodynamic forces. The twist would be to have a flexible membrane surface as your immersed body, so probably getting very much like a yacht sail optimisation design program for a specified pressure distribution, (actually it might be a more numerically stable problem to solve because the boundary is a priori a simple, closed surface).

They use a mesh of panels, each one being made up of vortex loops, entirely covering the surface and solving by inverting the matrix of the coefficients of all the panels, just as you describe. Very quick, very effective first order design engineering method.

Art Carlson
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Post by Art Carlson »

@Carter: That's right. Since we want a high beta plasma, there is no field and the pressure must be constant. (As a first step anyway. I don't know if Rick is going to insist on a tensor pressure at some point.) The shape is most easily defined by the positions of the verticex points, the magnetic force is most easily defined for an edge line segment, and the fluid pressure force is most easily defined for the area of a face. No big problem, but we need to find a consistent way to relate these all to each other.

@Rick: Good idea. Schmidt (p.42) does a calculation like this, but I need to sit down and think a spell before I understand it.

@icarus: If the idea is any good, then it is bound to be at least 50 years old. The question is whether it is easier to re-invent the wheel or to read books until you find a description of the wheel. For better or worse, I usually take the first route.

tombo
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Post by tombo »

It's more fun to reinvent the wheel.
It's more productive to buy one off the shelf,

That is especially true when someone else has spent 50 years and who knows how many millions refining the idea and comparing the results to reality.
I personally don't trust fancy math until it is tested by someone else, preferably against reality.
-Tom Boydston-
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein

kcdodd
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Post by kcdodd »

I deleted my last post because I have since gotten it to work. I used actual current loops instead of edge currents and it works like jewel even for polywell fields.

My first test is a superconducting sphere, using the discrete method, in a uniform B-field pointing in z direction.

Original B-field at surface: http://www.andromedaspace.com/files/bag0001.png

B-field generate by the currents which were solved for: http://www.andromedaspace.com/files/bag0002.png

Total b-field of solution: http://www.andromedaspace.com/files/bag0003.png

element flux, or dot(B, normal). Basically how far off it was since it should be zero: http://www.andromedaspace.com/files/bag0004.png

The biggest element flux was around 10^-13, which doesn't seem too bad.

edit: that was actually from using edge currents, with loop currents its more like max 10^-15 but otherwise pretty much the same.

Ok, now for the polywell field.

Original B-field at surface: http://www.andromedaspace.com/files/poly0000.png

B-field generate by the currents which were solved for: http://www.andromedaspace.com/files/poly0004.png

Total b-field of solution: http://www.andromedaspace.com/files/poly0005.png
And b-field magnitude: http://www.andromedaspace.com/files/poly0006.png

and higher refinement: http://www.andromedaspace.com/files/polyh0001.png

element flux, or dot(B, normal). Basically how far off it was since it should be zero: http://www.andromedaspace.com/files/poly0007.png

biggest error flux is 10^-17.

Ok, so this is pretty much the same result icarus and indrek got from the image coils, so it does work so far.

The mesh is morphable so it can do Art Carlson's next idea to balance plasma and bfield pressure. But, still not totally sure how to do that I have to think on it a bit now that I made the first part work.
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drmike
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Post by drmike »

Nice work! Are the currents square (i.e. straight lines)? The symmetry looks like it.

Can you do that same plot for different radius levels? It would be interesting to see 1/4, 1/2 and 2/3 from the center out to the coils.

Looks like a fun task, keep up the good work!

kcdodd
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Post by kcdodd »

Are you asking about the element currents? They are circular loops of currents inscribed within the triangular elements. I tried line-segment current loops but they didn't work so well though I'm not really sure why, I didn't see anything wrong with the equation. The loops work well for finding the field at the test points but they are not perfect either because they leave gaps in the field between the circles.
Carter

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