Three-dimensional Three-dimensional Robust Solver for Robust Solver for

Parabolic EquationParabolic EquationLanfa Wang

5.18.2011

Proposal in LCLS effort meeting

MotivationMotivation 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.

FEL (for example, Genesis by sven reiche)

FELFEL

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

Genesis (boundary approximation) Genesis (boundary approximation)

Set the field ZERO out the domain of interest

CSRCSRCSR ( 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

Impedance calculation Impedance calculation Gennady Stupakov, New Journal of Physics 8 (2006) 280(mathematica code )

Axis ymmetric geometry

GENERALITYGENERALITY

IF We neglect the 1st term

Various Solver we have developedVarious Solver we have developedSolver 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)

Fields in Disk-loaded StructuresFields in Disk-loaded Structures

Advantages of FEMAdvantages of FEMIrregular grids

Arbitrary geometryEasy to handle boundary

Impedance ofImpedance ofGrooved surfaceGrooved surface

Shape A

Shape B

Shape C

Rounded Tip

(b)

(THPAS067 ,PAC07)

Advantages of FEMAdvantages of FEMIrregular grids

Arbitrary geometryEasy to handle boundarySmall beam in a large domain (FEL in undulator)CPU (fast)Accuracy(higher order element, adaptive mesh, etc)

Disadvantage & Challenge:Disadvantage & Challenge: Complexity in coding (irregular grid, arbitrary geometry, 3D…)Time tables of milestones: (hard to predict) Time tables of milestones: (hard to predict) (1) coding---6 months (2)benchmark, application.

Deliverables :Deliverables : SLAC-pub, and maybe Journal paper

•Arbitrary geometry of beam pipe

•Any shape of beam

Mesh of chamber & beamMesh of chamber & beam

2D parabolic solver for 2D parabolic solver for Impedance calculation Impedance calculation

L. Wang, L. Lee, G. Stupakov, fast 2D solver (IPAC10)

0 200 400 600 800 1000-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

f (GHz)

ReZ

, Im

Z (

k)

Real, ECHO2Imaginary, ECHO2Real, FEM codeImaginary, FEM code

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

z (mm)

r (m

m)

0 200 400 600 800 1000-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

f (GHz)

ReZ

, Im

Z (

k)

Real, ECHO2-Imaginary, ECHO2

dot-lines: FEM code

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

z (cm)

r (c

m)

HIGHER ORDER ELEMENTSHIGHER ORDER ELEMENTS

Tetrahedron elements

1

9

8

7 10

2

5

6

3

4

10 nodes, quadratic:

1

13 12

7

15

2

9

6 3

4

5

8

10

11

14

16

17

18

195

20

20 nodes, cubic:

z

x

y

i

j

l

k

1 =

4 =

2 =

3 =

=0

=1

=1

=constant

P

Q

4 nodes, linear: