Implement envelope tracker

[austin-hoover]

This PR implements an envelope/centroid tracker (Issue #51). We track the centroid $\mathbf{\mu} = \langle \mathbf{x} \rangle$ and covariance matrix $\mathbf{\Sigma} = \langle (\mathbf{x} - \mathbf{\mu}) (\mathbf{x} - \mathbf{\mu})^T \rangle$, where $\mathbf{x} = [x, x', y, y', z, \Delta E ]^T$ is the 6D phase space vector.

Evolution equations:

$$ \begin{align} \mathbf{x} &\rightarrow \mathbf{M} \mathbf{x} + \mathbf{u} , \\ \mathbf{\mu} &\rightarrow \mathbf{M} \mathbf{\mu} + \mathbf{u} , \\ \mathbf{\Sigma} &\rightarrow \mathbf{M} \mathbf{\Sigma} \mathbf{M}^T . \end{align} $$

Accelerator nodes and child nodes are mapped to transfer matrices at the Python level. An error is raised if the node is not recognized. Space charge is handled by assuming a uniform charge density within an ellipsoid in the $x$-$y$-$z$ plane (for 3D space charge) or within an ellipse in the $x$-$y$ plane (for 2D space charge).

[austin-hoover]

The SNS linac benchmark uses the SpaceChargeCalcUniformElllipse solver with one ellipsoid and the MatrixRFGap model for rf cavities. The tracking shows dependence on the initial distribution; the closest result is for the 6D KV distribution.

python test_sns_linac.py --seq-stop CCL1 --sc 0 --dist kv

python test_sns_linac.py --seq-stop CCL1 --sc 1 --dist kv

python test_sns_linac.py --seq-stop CCL1 --sc 1 --dist waterbag

python test_sns_linac.py --seq-stop CCL1 --sc 1 --dist gauss

[woodtp]

Doesn't really matter, but just an FYI: instead of storing an int, if you want an on/off switch you can do something like

parser.add_argument("--sc", action="store_true")

Which will toggle space charge by just calling the script with that flag

script.py --sc

[austin-hoover]

Notes from meeting:

• Do not merge yet.
• Try implementing trackEnvelope functions for each node rather than current implementation.
• Add option to use matrix fitting for unknown or nonlinear elements.
• Look into SNS linac example a bit more. I think it makes sense but want to check against OpenXAL implementation.