Calculation of Eigenmodes in Superconducting Cavities

Calculation of Eigenmodes

in Superconducting Cavities

W. Ackermann, C. Liu, W.F.O. Müller, T. Weiland

Institut für Theorie Elektromagnetischer Felder, Technische Universität Darmstadt

Status Meeting December 17, 2012 DESY, Hamburg

Outline

▪ Motivation

▪ Computational model

- Problem formulation in 3-D

- Problem formulation in 2-D (boundary condition)

▪ Numerical examples

- 1.3 GHz structure, single cavity

- 3.9 GHz structure, string of four cavities (preliminary,without bellows)

▪ Summary / Outlook

Outline

▪ Motivation

- 3.9 GHz structure, string of four cavities (preliminary, without bellows)

Motivation

▪Particle accelerators

- FLASH at DESY, Hamburg

http://www.desy.de

TESLA 1.3 GHz

TESLA 3.9 GHz

RF Gun

Compressor

Diagnostics Accelerating Structures Collimator Undulators

Motivation

▪ Linac: Cavities

- Photograph

- Numerical model http://newsline.linearcollider.org

CST Studio Suite 2012

upstream downstream

Motivation

▪Superconducting resonator 9-cell cavity

Downstream

higher order mode coupler

coupler

Upstream

Motivation

▪Superconducting resonator 9-cell cavity

Downstream

coupler

Upstream

Variation:

Penetration depth

Variation:

Coupler orientation

Outline

▪ Motivation

- 3.9 GHz structure, string of four cavities (without bellows)

Computational Model

▪Problem formulation

- Time domain

• Efficient algorithms available (explicit method, coupled S-parameter)

• Field excitation to discriminate unknown mode polarizations

- Frequency domain (driven problem)

• Efficient algorithms available (coupled S-parameter)

• Field excitation to discriminate unknown mode polarizations

- Frequency domain (eigenmode formulation)

• Expensive algorithms

• Field distributions available including mode polarization

• Localized mode patterns with weak coupling naturally included

Computational Model

- Local Ritz approach

continuous eigenvalue problem

+ boundary conditions

vectorial function

global index

number of DOFs

scalar coefficient

discrete eigenvalue problem

Galerkin

Computational Model

▪Eigenvalue formulation

- Fundamental equation

- Matrix properties

- Fundamental properties

Notation:

A - stiffness matrix

B - mass matrix

C - damping matrix

for proper chosen scalar and vector basis functions

or static dynamic

Computational Model

▪Fundamental properties

- Number of eigenvalues

- Orthogonality relation

Notation:

A - stiffness matrix

B - mass matrix

C - damping matrix Matrix B nonsingular:

• matrix polynomial is regular

• 2n finite eigenvalues

If the vectors and are no longer B-orthogonal:

Computational Model

▪Geometrical model

9-cell cavity

Downstream

coupler

Input coupler

Upstream

coupler

High precision

cavity simulations

for closed structures

Outline

▪ Motivation

Computational Model

▪Port boundary condition

Port face, fundamental coupler

Computational Model

▪Port boundary condition

Port face, HOM coupler

Computational Model

vectorial function

global index

number of DOFs

scalar coefficient

Port face

Mixed 2-D vector and scalar basis

Computational Model

continuous eigenvalue problem, loss-free

+ boundary conditions

vectorial function

global index

number of DOFs

scalar coefficient

discrete eigenvalue problem

Galerkin

0 5 10 15 200

0 2 4 6 8 100

Computational Model

▪Wave propagation in the applied coaxial lines

- Main coupler

- HOM coupler

12.5 mm

60.0 mm

3.4 mm

16.0 mm

TE11 TE21

f0 = 1.3 GHz

Dispersion relation

propagation

damping

Computational Model

- Determine propagation constant for a fixed frequency

algebraic eigenvalue problem

eigenvector

eigenvalue

Computational Model

- Determine propagation constant for a fixed frequency

algebraic eigenvalue problem

eigenvector

eigenvalue

Mode 1 Mode 2 Mode 3 Mode 4 …

Computational Model

▪Memory requirement for waveguide formulations

- Ports with 2-D eigenmodes

- Impedance boundary condition

Port mode series expansion

Projection of the port fields on the set of 3-D basis

functions results in a dense matrix block

Same population pattern as PMC (natural boundary condition)

Computational Model

▪Memory requirement for waveguide formulations

- Example 1: 1.119.219 tetrahedrons

- Example 2: 1.218.296 tetrahedrons

- General rule:

b = mean bandwidth of sparse system matrix

h = mean grid step size

TESLA 1.3 GHz

Refined port mesh area

NNZ = number of non-zero elements

Computational Model

▪Waveguide ports implementation options

- Standard approach:

Incorporate dense matrix blocks into sparse system matrix

- Memory-efficient approach:

Utilize that system matrix is dense but of low rank

Port mode series expansion

Dyadic product of modified port mode vectors

Outline

▪ Motivation

▪Problem definition

- Geometry

- Task

Numerical Examples

TESLA 9-cell cavity

boundary condition

Search for the field distribution, resonance frequency and quality factor

boundary

conditions

Port boundary conditions

Numerical Examples

▪Computational model

9-cell cavity

coupler

Upstream HOM coupler

Distribute

computational load

on multiple processes

Numerical Examples

▪Simulation results

- Accelerating mode (monopole #9)

- Higher-order mode (dipole #37)

Numerical Examples

▪Simulation results

Accelerating mode

(monopole #9)

Higher-order mode

(dipole #37)

Beam tube

HOM coupler

Coaxial

input coupler

Coaxial line

Beam tube

HOM coupler

Coaxial

input coupler

Numerical Examples

500100020005000

Monopole

Dipole

Quadrupole

Sextupole

2.46 2.48 2.50 2.52 2.54 2.56 2.58 2.60

▪Controlling the Jacobi-Davidson eigenvalue solver

- Evaluation in the complex frequency plane

- Select best suited eigenvalues in circular region around

user-specified complex target

Real and imaginary part of the complex frequency

Numerical Examples

▪Quality factor versus frequency

Calculations kindly performed by Cong Liu

Numerical Examples

▪Field distribution of selected mode

- First/second dipole passband (mode 17)

Outline

▪ Motivation

Numerical Examples (preliminary)

▪3.9 GHz structure (3rd harmonic cavity)

- String of four cavities

- Field distribution for selected modes

4 main coupler, 8 HOM coupler and 2 beam pipes = 14 ports

Bellows omitted for the time being

1.040.212 tetrahedrons

6.349.532 complex DOF

Summary / Outlook

▪Summary:

Request for precise modeling of electromagnetic fields within

resonant structures including small geometric details

- Geometric modeling with curved tetrahedral elements

- Port boundary conditions with curved triangles

- Memory-efficient implementation to evaluate the port fields

now available

▪Outlook:

- Application to 1.3 GHz and 3.9 GHz structure and strings

Calculation of Eigenmodes in Superconducting Cavities

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