# Presentation: 279693

## Analyzing Current Distributions in Rutherford Cables: Fast-Multipole-Method Version of the Network Model

- Jason McDonald
- Presented at:
- Workshop on Instabilities in Nb3Sn Strands, Cables and Magnets
- Fermilab, April 28-30, 2004

## Ultimate Goal

- I am interested in developing a combined network model (electromagnetic/thermal) for studying stability and quench dynamics in Rutherford cables
- I want the geometry of the Rutherford cable to be adequately represented (ie. Many strands, actual transposition pattern, etc.), and the method to be accurate and computationally feasible:
- It should be capable of obtaining high accuracy           (error << 1%) so we can feel confident about the results
- The storage requirements and operation count should scale linearly with the number of unknown currents (ie. If you double the cable length the computation time should double)

## Network Model of Rutherford Cable(A. Verweij, PhD Thesis)

- There are       strands in the cable
- There are         bands per pitch length, and         bands per cable length
- Each band contains              strand sections,        adjacent contact resistances, and           cross-over contact resistances, for a total of                                          	unknown currents
- The equations determining the unknown currents are derived by applying Kirchoff’s laws

## Evaluating Strand Voltages

- The equations from the circuit analysis involve voltage drops along each strand section.  These voltages involve both resistive and inductive parts:
- The inductive part causes problems because it destroys the matrix sparsity and very quickly becomes too expensive to store and to compute with:
- As a rule, if you have to invert a full matrix your algorithm is useless
- Your only chance is to use an iterative method, and even then the matrix-vector products eventually end up killing you, because they grow with quadratic complexity

## Fast Algorithm for Inductive-Voltage Calculations

- What is needed is an accurate method of approximating the matrix-vector multiplications in the inductive calculation, which is ‘cheap’ with respect to storage and computation
- The easiest way is to completely neglect all the strand sections beyond a certain distance away, thus creating a banded matrix; but this is impractical because of the long-range ( ~ 1 / *r* ) nature of the inductive interaction: you would need to take a huge number of neighbors to get decent accuracy
- A better way is to directly evaluate all the interactions within a small local region, and then approximate the influence of all distant sources by a short series; this is called the Fast Multipole Method or FMM (Rokhlin, Greengard)

## FMM applied to a Rutherford cable

- The bands are grouped into regions; each region containing       a certain number of bands (I use 8 bands per region)
- A truncated expansion is constructed for each region, which represents the electric field induced by distant sources (strand sections); this series is formed by applying ‘gymnastics’ to the well-known multipole expansion of the Coulomb potential      (J. D. Jackson, *Electrodynamics*).  The multipole expansion allows you to compress the information about the distant sources into a compact representation
- The inductive voltage of a strand section is then computed in two stages:
- Evaluate all pair-wise interactions in the same region and nearest-neighbor regions using the known analytical expressions [F. Grover, *Inductance Calculations*]
- Evaluate the multipole expansion at the location of the strand

## Results of FMM for a 40-strand cable

*  *
-          *L**cab *(m)           RMS Error (%)                CPU Time (sec)         Time Ratio

|     0.704  |           7.552x10-3  | FMM               3.004 
Direct            18.807 |           6.3 |
| --- | --- | --- | --- |
|     1.408
 |           7.972x10-3  | FMM               6.059 
Direct            74.988  |         12.4 |
|     2.816 |           8.194x10-3  | FMM             12.208 
Direct          302.355  |         24.8 |
|     5.632 |           8.320x10-3  | FMM             24.255 
Direct        1201.287  |         49.5 |
|   11.264 |          8.404x10-3 
 | FMM             48.670 
Direct        4774.285  |         98.1 |

## Implementing the FMM in the Network Model

-  The linear-equation system is created from Kirchoff’s laws
-  The inductive voltages are moved to the right-hand-side of the equation system, leaving a ‘very-sparse’ matrix which can be factored with linear complexity (ie. You double the length, the computation time doubles)
-  An iterative method is used for the nonlinear system, and the inductive voltages are updated at each iteration using the FMM
-  Interestingly, once the FMM is implemented the sparse-matrix factorization becomes, by far, the most time-consuming part of the calculation

## Testing the algorithm

- In addition to testing the FMM against the direct matrix-vector multiplication, the resulting network model was also compared to known results, for example:
-  Current distributions in a uniform cable placed in a uniform, ramped field
-  Power dissipation in cables of finite length
-  etc.

## Ramped transport current in a short cable with soldered ends

- Superconducting fraction = 50%

## Strand numbering used in figures

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## Longer cable

- Superconducting fraction = 50%

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## Summary

-  The Fast-Multipole-Method has been implemented in a network model for analyzing the transient electromagnetic response of Rutherford-type cables
-  The storage requirements and simulation time of the resulting algorithm scale linearly with respect to cable length, therefore, the algorithm can be termed ‘fast’
-  As an initial application, a short-sample 28-strand cable has been modeled to determine the effect of soldered connections on the current distributions.  The simulations assumed zero applied field and various ramprates for the transport current
-  The simulations predict current imbalances but the imbalances seem to be insignificant and can not explain the large critical-current degradations found experimentally.  The imbalances appear to be a self-field version of boundary-induced coupling currents, because their presence is definitely caused by the discontinuity is contact resistance and by the changing self field
-  The next step is to implement thermal analysis solver and to couple it to the existing electromagnetic solver.  Once this is done, more interesting analyses can be performed