1. Three-dimensional Robust Solver for Parabolic Equation Lanfa Wang 5.18.2011 Proposal in LCLS effort meeting

2. Motivation Parabolic equation has been solved in FEL, CSR, and Impedance calculations, etc. (Important for LCLS and LCLSII, etc). The present codes(solver) are limited for simple cases (geometry), or/and slow, and kind of 2D solver (3D problem, z is treated like time) We propose to develop fast 3D parabolic solver for general cross-section of the beam pipe.

3. FEL (for example, Genesis by sven reiche) FEL Modeling challenges : EE-HG (D. Xiang and G. Stupakov, PR STAB 12, 030702 (2009) Large number of particles, CSR in Chicane New numerical methods have to be applied to solve field equation

4. Genesis (boundary approximation) Set the field ZERO out the domain of interest

5. CSR CSR ( for example, CSR in bend magnet (Tomonori Agoh, Phys. Rev. ST Accel. Beams 7, 054403 (2004)) All this type of codes can only for rectangular cross-section! Agoh, PRSTAB 054403 Gennady, PRSTAB 104401 Demin, in preparation

6. Impedance calculation Gennady Stupakov, New Journal of Physics 8 (2006) 280(mathematica code ) Axis ymmetric geometry

7. GENERALITY IF We neglect the 1 st term

8. Various Solver we have developed  Solver for all modes in Disk-loaded Structures, NIMA, Vol. 481, 95(2002). (Traveling wave, all mode, meshless method)  Solver for microwave element and accelerating structure High Energy Physics &Nuclear Physics, 25 (2001)(2D)  Solver for Poisson Equation (2D,3D), PRSTAB 5, 124402 (2002)  Adaptive impedance Analysis of grooved surface (THPAS067,PAC07)  Two-dimensional FEM Code for Impedance Calculation (IPAC'10)

9. Fields in Disk-loaded Structures

10. Advantages of FEM Irregular grids  Arbitrary geometry  Easy to handle boundary

11. Impedance of Grooved surface (THPAS067,PAC07)

12. Advantages of FEM Irregular grids  Arbitrary geometry  Easy to handle boundary  Small beam in a large domain (FEL in undulator)  CPU (fast)  Accuracy(higher order element, adaptive mesh, etc) Disadvantage & Challenge: Complexity in coding (irregular grid, arbitrary geometry, 3D…) Time tables of milestones: (hard to predict) (1) coding---6 months (2)benchmark, application. Deliverables : SLAC-pub, and maybe Journal paper

13. Arbitrary geometry of beam pipe Any shape of beam Mesh of chamber & beam

14. 2D parabolic solver for Impedance calculation L. Wang, L. Lee, G. Stupakov, fast 2D solver (IPAC10)

15. HIGHER ORDER ELEMENTS Tetrahedron elements 10 nodes, quadratic: 20 nodes, cubic: 4 nodes, linear: