Three-dimensional
Robust Solver for
Parabolic Equation
Lanfa Wang
5.18.2011
Proposal in LCLS effort meeting
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.
FEL
FEL (for example, Genesis by sven reiche)
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)
Set the field ZERO out the domain of interest
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
Impedance calculation

Gennady Stupakov, New Journal of Physics 8
(2006) 280(mathematica code )
Axis ymmetric geometry
GENERALITY
IF We neglect the 1st term
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)
Fields in Disk-loaded Structures
Advantages of FEM
Irregular grids
Arbitrary geometry
Easy to handle boundary
Impedance of
Grooved surface
(THPAS067 ,PAC07)
Shape A
Shape B
(b)
Shape C
Rounded Tip
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
Mesh of chamber & beam
•Arbitrary geometry of beam pipe
•Any shape of beam
2D parabolic solver for
Impedance calculation

L. Wang, L. Lee, G. Stupakov, fast 2D solver (IPAC10)
0.5
2.5
0.4
1.5
r (cm)
r (mm)
2
1
0.3
0.2
0.5
0.1
0
0
10
20
30
40
50
60
0
0
z (mm)
2
4
6
8
10
z (cm)
0.14
0.07
0.05
dot-lines: FEM code
0.1
0.04
0.03
0.02
0.01
0.08
0.06
0.04
0.02
0
0
-0.01
0
Real, ECHO2
-Imaginary, ECHO2
0.12
ReZ, ImZ (k )
ReZ, ImZ (k )
0.06
Real, ECHO2
Imaginary, ECHO2
Real, FEM code
Imaginary, FEM code
200
400
600
f (GHz)
800
1000
-0.02
0
200
400
600
f (GHz)
800
1000
x
HIGHER ORDER ELEMENTS

Tetrahedron elements
4=
20 nodes, cubic:
10 nodes, quadratic:
4 nodes, linear:
4
4
l
=constant
3= k
1=
i
Q
=1
16
11
2
7
7
10
1
2= j
=1
17
3
9
10
5
6
2
8
19
5 18 15
1
5
P
y
13
20
8
=0
z
14
12
9
6
3