1. 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

2. John Hart Toroid Field Parameterisation 2 Toroid Field Parameterisation What is the problem? Outline of proposal Implementation The meshing algorithm Present status Next steps Acknowledgements to Witold Kozanecki and Laurent Chevalier

3. 22 July 2008 John Hart Toroid Field Parameterisation 3 Toroid Field Parameterisation 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%.

4. 22 July 2008 John Hart Toroid Field Parameterisation 4 ATLAS magnet system

5. 22 July 2008 John Hart Toroid Field Parameterisation 5 The barrel toroid

6. 22 July 2008 John Hart Toroid Field Parameterisation 6 The endcap toroids

7. 22 July 2008 John Hart Toroid Field Parameterisation 7 Toroid Field Parameterisation Outline of proposal Divide field into zones to avoid fine mesh over large volume. (Topographical analogy.) Mesh size determined by 2 nd derivative of field  Rapidly changing field ⇨ fine mesh Subtract Biot-Savart field from nearby conductors to make field smoother.

8. 22 July 2008 John Hart Toroid Field Parameterisation 8 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).

9. 22 July 2008 John Hart Toroid Field Parameterisation 9 Implementation Some details Use 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.

10. 22 July 2008 John Hart Toroid Field Parameterisation 10 The meshing algorithm Apply 4 mT criterion to the field vector B. Mesh size at any point estimated from 2 nd 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.

11. 22 July 2008 John Hart Toroid Field Parameterisation 11 The meshing algorithm Meshing is very sensitive to the size of sub-zones round the conductors. Plots of the 2 nd 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.

12. 22 July 2008 John Hart Toroid Field Parameterisation 12 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.

13. 22 July 2008 John Hart Toroid Field Parameterisation 13 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?