Advancing Computational Science Research for Accelerator Design and Optimization Accelerator Science and Technology - SLAC, LBNL, LLNL, SNL, UT Austin, Columbia, RPI, Stanford, UC Davis, U Wisconsin

The SciDAC Accelerator project at SLAC is committed to advancing computational science research which is absolutely essential to the success of LARGE scale electromagnetic simulations that are necessary for the design and optimization of DOE accelerators such as the International Linear Collider (ILC). In order to be able to model an entire, realistic ILC cryomodule, petaflops-scale computing resources will be needed as well as advances in shape determination, nonlinear eigensolvers and adaptive refinement.

Forward Problem: solved by Omega3P to get state variables (i , Hi)

Adjoint Equations: (for each frequency to fit) Error Indicator

Relative error of E field

Relative error of H field

Gradient of energy densityIs used as error indicator for a cylindrical cavity. Error indicator is compared with relative field errors on a line close to the curved surface.

Nonlinear Eigensolvers (SLAC, TOPS/LBNL, SAPP/Stanford, UC Davis))

Shape Determination(SLAC, TOPS/UT, LBL, Columbia, TSTT/SNL, LLNL)

Adaptive Refinement (SLAC, TSTT/RPI))

Cavity Deformations on HOMs (compared to ideal cavity) Mode frequency shifted by MHz

Modes split 100’s instead of 10’s kHz

Qext’s scattered towards high side - may lead to dangerous modes

Shape Determination

Use measurements from TESLA cavity data bank as input (field frequencies and amplitudes) Solve an inverse problem to find cavity deformations using a PDE constrained optimization method:

F i fi d1,...,dm fi* 2

i1

n

ij(| Ej

(ri ) E ij

|)j1

p

i1

m

Minimize

Subject to

A reduce space method is used to solve the optimization problem where design sensitivities were computed using a continuous adjoint approach

Proof of Principle Experiment:

As a test case, we randomly generate a set of deformations (dr, dz, dt) on the ideal TESLA cavity and computed the deformed cavity frequencies and fields. Using these computed results, we apply our shape determination tool to solve the inverse problem and fully recover the set of deformations :

9-cell model cavity

A new nonlinear Jacobi-Davidson solver has been successfully implemented in the parallel FEM eigensolver Omega3P for finding the Higher-Order-Modes (HOMs) in various ILC cavity designs ity.

Maxwell’s equation in Frequency domain for a wavguide-loaded cavity:

(1) Select an initial space basis Q

(2) Get approximate eigen-pair (m, v), choose (3) Expand the space by:

- approximately solving correction equation

- Orthogonalize q=q-QQHq, q /= |q|, and Q=[Q q]

(4) Solve projected NEP QH T() Q y = 0

- Get Ritz pair m and v = Qy

(5) Test whether (m, v) converges. If not, goto step 3

Nonlinear Jacobi-Davidson Algorithm

.Number of Iterations

Residua

lConvergence of 7 eigenpairs in ILC cavity using nonlinear Jacobi-Davidson solver

Mode Analysis of the ILC Cryomodule

The damped modes in a wave-guide loaded cavity such as the ILC cryomodule require the solutions to a nonlinear eigenvalue problem resulting from the formulation of Maxwell’s equations in the frequency domain that includes the boundary conditions at the waveguide terminations. The Qext which measures the damping is given by the ½ the ratio of the real to imaginary part of the complex eigenfrequency.

Waveguide cutoffs

4-cavity Structure: A nonlinear eigensystem with more than 15 million of Degrees of Freedoms(DOF) was solved within 10 hours for 2 modes on 768 CPUs with 276 GB memory on Seaborg as a first step towards modeling an entire cryomodule.

Parallel “h-p” Adaptive Refinement

In simulating LARGE, COMPLEX geometries such as the ILC cryomodule, adaptive refinement can optimize computing resources while providing higher accuracy with faster convergence.

In SciDAC-1, progress was made in adaptive mesh (h) refinement for the parallel FEM eigensolver Omega3P. Working towards the goal of parallel “h-p” adaptive refinement, efforts are ongoing in the development of:

(1) New local error indicator – A posterior error indicator based the energy density gradient in domains with higher-order surface geometry which computes the local variation of both the electric and magnetic field components:

Test Case

(2) Parallel AMR (h) – A complete parallel AMR loop has been implemented in Omega3P and successfully executed.

DESY TTF CRYOMODULE #3

Test Case Application to the pi mode in the fully 3D ILC TDR cavity with tuning of the convergence parameters is in progress

The method will be applied to the TESLA cavities to obtain the TRUE cavity shape that corresponds to the measured frequencies and Qext’s.