22 July 2008 John Hart Toroid Field Parameterisation 1 Toroid Field Parameterisation An informal report to the RAL ATLAS meeting John Hart 22 July 2008

22 July 2008 John Hart Toroid Field Parameterisation 1

Toroid Field Parameterisation

An informal report to the RAL ATLAS meeting

John Hart

22 July 2008



What is the problem?

Outline of proposal

Implementation

The meshing algorithm

Present status

Next steps

Acknowledgements to Witold Kozanecki and

Laurent Chevalier



What is the problem?Initial estimate of storage space was ~I GB, but

can only afford ~200 MB.

Need to use linear interpolation for speed. Aim is to find field in 1-2 μs.

Require field accurate to 4 mT to avoid increasing error in muon momentum by more than 3%.


ATLAS magnet system


The barrel toroid


The endcap toroids



Outline of proposalDivide field into zones to avoid fine mesh over

large volume. (Topographical analogy.)

Mesh size determined by 2nd derivative of field Rapidly changing field ⇨ fine mesh

Subtract Biot-Savart field from nearby conductors to make field smoother.


Implementation

3 steps:Define zones and specify conductor segments

“manually” via control cards. Optimise by trial and error. Program can replicate in φ.

Toroid coil parameter stored to simplify specification of conductor segments.

Create mesh automatically to satisfy 4 mT criterion.

Calculate and store field at each node of the mesh (after subtracting Biot-Savart component).


Implementation

Some detailsUse cylindrical polar coordinates and define

zones with boundaries of constant r, φ and z.

Mesh r, φ and z coordinates “independently” to generate a “rectangular” grid enabling fast access to the field at any point.

Use a lookup vector to find r, φ and z node numbers – faster than binary search.

Need to optimise zone finding. (Some ideas.)

Most field updates will require only recalculation.



Apply 4 mT criterion to the field vector B.

Mesh size at any point estimated from 2nd order differences, taking maximum in (r,φ) for a given z, and so on.

Exclude points in sub-zones from calculation of maxima.

Enforce mesh points at sub-zone (but not sub-sub-zone) boundaries.

Start with a rectangular grid, but allow for iteration. Normally get approximate convergence in a few iterations.



Meshing is very sensitive to the size of sub-zones round the conductors.

Plots of the 2nd derivative of B help in fixing sub-zone dimensions.

May be able to relax meshing criterion slightly and still satisfy 4 mT criterion on average.

Alternative criterion based on B⊥ is possible and slightly reduces number of mesh points.

Can test field parameterisation by checking field at random points and by calculating ∫Bdl.


Present status

A flexible program exists for defining field zones, building meshes, calculating the field values and testing the accuracy of the field.

It works well for the central region (|z|<10.5 m) of the barrel toroid field in terms of storage space and accuracy, excluding the coils themselves.

Preliminary zones have been defined for the whole toroid field.


Next steps

Test meshing of more difficult zones near the corners of the coils and in the ECT region

Provide prototype for tests with muon tracking

Optimise for speed and test timing

Try storing field as 2 byte integer

Study zones closer to conductor

Quadratic interpolation in a few difficult regions?

Mechanism for enforcing perfect continuity?

Documents

22 July 2008 John Hart Toroid Field Parameterisation 1 Toroid Field Parameterisation An informal report to the RAL ATLAS meeting John Hart 22 July 2008