EM Computations with Embedded Boundaries (cut cells) C. Nieter †, J.R. Cary †* (presenter), G.R....
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Transcript of EM Computations with Embedded Boundaries (cut cells) C. Nieter †, J.R. Cary †* (presenter), G.R....
EM Computations with Embedded Boundaries
(cut cells)
C. Nieter†, J.R. Cary†* (presenter), G.R. Werner*, D.N. Smithe†, P.H. Stoltz†, S. Ovtchinnikov†
Tech-X Corporation, U. ColoradoCOMPASS Collaboration Meeting, Sep. 17-18, 2007
We also acknowledge assistance from the rest of the VORPAL team: G. I. Bell, D. L. Bruhwiler, R. S. Busby, J. Carlsson, B. M. Cowan, D. A. Dimitrov,
A. Hakim, P. Messmer, P. J. Mullowney, K. Paul, S. W. Sides, N. D. Sizemore, S. A. Veitzer, D. J. Wade-Stein, N. Xiang, W. Ye.
Work supported byOffices of FES, HEP, and NP of the Department of Energy, the SciDAC
program, AFOSR, JTO, Office of the Secretary of Defense, and the SBIR programs of the Department of Energy and Department of Defense
Tech-X Corporation/COMPASS 2
Outline
•Embedded boundaries: theory and use•Frequency extraction•Richardson extrapolation use and results•Areas for collaboration
Tech-X Corporation/COMPASS 3
Finite difference time domain (FDTD) based on accurate derivatives
• Simple• Fast
–No matrix inversions• Manifestly stable
–Symmetric update matrix• Works well with particles
–The choice of PIC codes• Parallelizes well
–Only boundary information exchanged between domains
Ez
yj yj+1yx
z BxEz
Ez
Ey
Ey
By
Ex
Bz
VORPAL BG/L Speedup
Tech-X Corporation/COMPASS 4
Error in frequency for sphere
y = 0.1781x -0.875
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+01 1.00E+02 1.00E+03 1.00E+04N
Error
Stair-step
Power (Stair-step)
But historically, FDTD failed to do well with curved surfaces
• N (L/x) cells in each direction• Error of (x/L)3 at each surface cell• O(N2) cells on surface• Error = N2(x/L)3 = O(1/N)
120x24x24 = 71,424 cells= 215,000 degrees of freedom
Modest problems require 1012 cells for 10-5 error
Tech-X Corporation/COMPASS 5
Embedded boundaries give locally first order error
• AKA cut-cell or conformal
• Dey-Mittra (NOT Yu-Mittra)
• Justified as–Update flux by line
integral–Divide by area to get
change of B
Tech-X Corporation/COMPASS 6
Embedded boundaries have global second-order error
• Extensive numerical validation
• Computations now doable
• Mesh generation parallelizes well
• 10-5 error with 100 cells per direction
• 106 cells usually suffices for simple structures
• Variable mesh will reduce further
Tech-X Corporation/COMPASS 7
Embedded boundaries have some "issues"
• No real derivation in literature– Wrong centering– No cut cells for B– Lack of understanding prevents
development of higher-order method• Smaller cells decrease the maximum
stable time step– Matrix elements ~ inverse triangle size– Must discard tiny cells– Results in
• O(x) scaling at small x• Smaller time step for stability• Locally trapped high frequency modes• Interferes with Richardson scaling
€
˙ B z =1
Axyl iEi
x−y edges
∑
€
Matrix coefs =l i
Axy
li Axy
C. Nieter, J.R. Cary, G.R. Werner, D.N. Smithe, P.H. Stoltz, Application of Dey-Mittra conformal boundary algorithm to 3D electromagnetic modeling, preprint, 2007.
Tech-X Corporation/COMPASS 8
Frequencies obtained from subspace diagonalization
• We can beat Heisenberg!• Ring up finite bandwidth, compute
time series in subspace• Diagonalize subspace• Multiple simulations if near
degeneracies
G.R. Werner and J.R. Cary, Extracting Degenerate Modes and Frequencies from Time Domain Simulations, J. Comp. Phys., submitted (2007).
Tech-X Corporation/COMPASS 9
Application to nearly degenerate square shows accurate degeneracy extraction
• Lx = 1m, Ly = 1.00001 m• Simulation set up to
capture modes in +- 10% band
• Expected number of modes from density of states
• Four-fold near-degeneracy, so five simulations
• Frequencies obtained to parts in 107-10-9
• Computation error dominates over extraction error
Tech-X Corporation/COMPASS 10
Method now extended to complex frequencies
• Can get Q measurements again from 10s of oscillations
• Applied to simple cavities only so far
Tech-X Corporation/COMPASS 11
Richardson extrapolation gets accuracy to next order
• Fit frequency:• Solve for and 0 from two
measurements• Requires smooth variation: similarity
€
0 =ωi +αΔxi2
Elliptic cavity, direct Elliptic cavity, extrapolated
Tech-X Corporation/COMPASS 12
Results: crab cavity frequencies to 50 kHz
• 13 cell crab cavity• Varying resolution
–192x40x40–…–752x144x144
• Fit different ways– last two points– last three points–keep or not other
polynomial terms– results differ by less than
50 kHz
Frequency versus resolution, 0.05cm indent
y = 1.6402E+15x2 - 2.0490E+11x + 3.9030E+09
R2 = 9.9677E-01
y = 8.6832E+14x2 - 1.8779E+11x + 3.9019E+09
R2 = 9.9555E-013.897E+09
3.898E+09
3.899E+09
3.900E+09
3.901E+09
3.902E+09
3.903E+09
0.00E+000 1.00E-005 2.00E-005 3.00E-005
1/NX^2
Frequency (Hz)
Tech-X Corporation/COMPASS 13
Results differ from previous in frequencies
• Observing 3 MHz difference Frequency vs. Indentation, Comparison
752 x 144 x 144 cell
3.890E+09
3.900E+09
3.910E+09
3.920E+09
3.930E+09
3.940E+09
0.050 0.100 0.150 0.200 0.250Cavity Indentation (m)
Frequency (Hz)
Upper
Lower
MAFIA_U*MAFIA_L*
Tech-X Corporation/COMPASS 14
Separation vs. Indentation, Extrapolated
0.000E+00
1.000E+07
2.000E+07
3.000E+07
0.050 0.100 0.150 0.200 0.250
Cavity Indentation (cm)
Frequency (Hz)
Difference
MAFIA_U diff
Differences also seen in the splitting
• VORPAL sees typically 0.5-1MHz lower split
Tech-X Corporation/COMPASS 15
Verification shows sampling okay, but pipe length effects present
• Effects of modified end groups?• Need outgoing wave for pipe ends?
Upper Frequency vs. Beam Pipe Length
3.9052E+09
3.9056E+09
3.9060E+09
3.9064E+09
3.9068E+09
3.9072E+09
3.9076E+09
0.040 0.060 0.080 0.100 0.120
PIPELEN (m)
Frequency (Hz)
Upper
Lower
3.9000E+09
3.9020E+09
3.9040E+09
3.9060E+09
0 4 8 12 16
Number of Sampling Points
Frequency (Hz)
Tech-X Corporation/COMPASS 16
Doing few calculations on Tesla, multipactoring
• Jlab multipactoring• Tesla cavities
Tech-X Corporation/COMPASS 17
There are potentially fruitful collaborations
• Higher-order embedded boundaries • Eliminate stable time step reduction• Particle motion near boundaries• Visualization
Tech-X Corporation/COMPASS 18
Higher-order embedded boundaries would make a large impact
• Boundary error same as interior–Boundary error is O(x), gives O(x2) globally– Interior error is O(x2)
• With Richardson extrapolation–Boundary error is O(x2), gives O(x3) globally– Interior error is O(x4)
• Boundary error is limiting with extrapolation• Improved boundary error will lead to overall
error of O(x4)!• We now have a derivation of Dey-Mittra• Have higher-order algorithm, but
–Very complex–Not manifestly symmetric
Tech-X Corporation/COMPASS 19
Elimination of time-step reduction improves modeling
• Reduces work by a factor of 4-10• Eliminates spurious trapped high-frequency
modes (important for multipactoring studies)• I. A. Zagorodnov, R. Schuhmann, T. Weiland, [A
uniformly stable conformal FDTD-method in Cartesian grids, Int. J. Numer. Model., 16, 127 (2003)] has heuristic approach based on area borrowing.
• Can one prove the above?• Understand how to have minimal impact?• How is symmetry imposed?
Tech-X Corporation/COMPASS 20
Particle dynamics near boundaries critical for accurate modeling
• Charge conservation near boundaries critical to avoid nonphysical charge buildup
• What does one do with dynamics? Without some care, we have observed self forces and excess heating.
• We are approaching heuristically: copy over
• Does this avoid self forces?
Tech-X Corporation/COMPASS 21
Visualization and code comparision
• Visualization what one is solving is a great aid• For verification, would like to have easier ways to
compare results–Exchange standards for data, geometries
• Ultimately, solve different problems and have increased productivity
Tech-X Corporation/COMPASS 22
Summary and conclusions
• FDTD has made a number of advances in EM with embedded boundaries. Now have accurate, charge conserving solutions
• Potential collaborations with physicists–Data exchange–Formats
• Potential collaborations with applied mathematicians–Higher order embedded boundaries–Elimination of time-step reduction–Dynamics of particles near boundaries–Visualization, data formats