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Three-dimensionalRobust Solver for

Parabolic EquationLanfa Wang

5.18.2011Proposal in LCLS effort meeting

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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 2Dsolver (3D problem, z is treated like time)

We propose to develop fast 3D parabolic solver

for general cross-section of the beam pipe.

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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

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Genesis (boundary approximation)

Set the field ZERO out the domain of interest

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CSRCSR ( 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 104401Demin, in preparation

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Impedance calculation Gennady Stupakov,New Journal of Physics 8

(2006) 280(mathematica code)

Axis ymmetric geometry

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GENERALITY

IF We neglect the 1stterm

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Various Solver we have developed

Solver for all modes inDisk-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 forPoisson Equation (2D,3D), PRSTAB 5, 124402 (2002)

Adaptive impedance Analysis of grooved surface (THPAS067 ,PAC07)

Two-dimensional FEM Code for Impedance Calculation (IPAC'10)

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Fields in Disk-loaded Structures

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Advantages of FEMIrregular grids

Arbitrary geometryEasy to handle boundary

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Impedance of

Grooved surface

Shape A

Shape B

Shape C

Rounded Tip

(b)

(THPAS067 ,PAC07)

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Advantages of FEMIrregular grids

Arbitrary geometryEasy 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

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Arbitrary geometry of beam pipe

Any shape of beam

Mesh of chamber & beam

2D b li l f

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2D parabolic solver for

Impedance calculation

L. Wang, L. Lee, G. Stupakov,fast2D 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,ImZ(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(mm)

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,

ImZ(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(cm)

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HIGHER ORDER ELEMENTS

Tetrahedron elements

1

9

8

7

10

2

5

6

3

4

10 nodes, quadratic:

1

13

12

7

15

2

9

63

4

5

8

10

11

14

16

17

18

19

20

20 nodes, cubic:

z

y

i

j

l

k

1 =

4 =

2 =

3 =

=0

=1

=1

=constant

P

Q

4 nodes, linear: