State-of-the-art stellarator optimization code
This tutorial will walk the user through running VMEC with a fixed boundary condition. For this example the National Compact Stellarator Experiment (NCSX) configuration will be used. This machine is stellarator symmetric with a periodicity of three.
The inputnamelist (input.ncsx_c09r00_fixed) controls the execution of the VMEC code. The suffix of the input file will be appended to each of the output files as we will see after execution. The Fourier coefficient in this file have been generated through an optimization routine. In general, more simple initial conditions will suffice for the axis position and outer most flux surface.
&INDATA
!----- Runtime Parameters -----
DELT = 9.00E-01
NITER = 2500
NSTEP = 200
TCON0 = 2.00E+00
NS_ARRAY = 9 28 49
FTOL_ARRAY = 1.000000E-06 1.000000E-08 1.000000E-11
!----- Grid Parameters -----
LASYM = F
NFP = 3
MPOL = 11
NTOR = 6
NZETA = 32
PHIEDGE = 4.97070205837336E-01
!----- Free Boundary Parameters -----
LFREEB = F
MGRID_FILE = 'none'
EXTCUR = 0
NVACSKIP = 0
!----- Pressure Parameters -----
GAMMA = 0.000000E+00
BLOAT = 1.000000E+00
SPRES_PED = 1.00000000000000E+00
AM = 6.85517649352426E+04 -5.12027745123057E+03 -3.61510451745464E+04 -4.74263014113066E+05
1.78878195473870E+06 -3.21513828868170E+06 2.69041023837233E+06 -8.17049854168367E+05
0.00000000000000E+00 0.00000000000000E+00 0.00000000000000E+00
!----- Current/Iota Parameters -----
CURTOR = -1.78606250000000E+05
NCURR = 1
AI = 0.00000000000000E+00 0.00000000000000E+00 0.00000000000000E+00 0.00000000000000E+00
0.00000000000000E+00 0.00000000000000E+00 0.00000000000000E+00 0.00000000000000E+00
0.00000000000000E+00 0.00000000000000E+00 0.00000000000000E+00
AC = 8.18395699999999E+03 1.43603560000000E+06 -1.07407140000000E+07 7.44389200000000E+07
-3.22215650000000E+08 8.81050800000000E+08 -1.49389660000000E+09 1.52746800000000E+09
-8.67901590000000E+08 2.10351200000000E+08 0.00000000000000E+00
!----- Axis Parameters -----
RAXIS = 1.49569454253276E+00 1.05806400912320E-01 7.21255454715878E-03 -3.87402652289249E-04
-2.02425864534069E-04 -1.62602353744308E-04 -8.89569831063077E-06
ZAXIS = 0.00000000000000E+00 -5.19492027001782E-02 -3.18814224021375E-03 2.26199929262002E-04
1.28803681387330E-04 1.11266150452637E-06 1.13732703961869E-05
!----- Boundary Parameters -----
RBC(0,0) = 1.40941668895656E+00 ZBS(0,0) = 0.00000000000000E+00
RBC(1,0) = 2.79226697269945E-02 ZBS(1,0) = -1.92433268059631E-02
RBC(2,0) = -1.54739398509667E-03 ZBS(2,0) = 1.11459511078088E-02
RBC(3,0) = 2.90733840104882E-03 ZBS(3,0) = -3.97869471888770E-03
RBC(4,0) = -8.91322016448873E-04 ZBS(4,0) = 1.34394804673514E-03
RBC(5,0) = -7.81997770407839E-05 ZBS(5,0) = -1.57143910159387E-04
RBC(6,0) = 1.06129711928351E-04 ZBS(6,0) = 9.58024291307491E-05
RBC(-6,1) = 2.48228899767757E-05 ZBS(-6,1) = -2.28386224209054E-05
RBC(-5,1) = 8.23567667077671E-05 ZBS(-5,1) = 3.30176003890210E-04
RBC(-4,1) = -7.20220898033597E-04 ZBS(-4,1) = 1.28038328362904E-04
RBC(-3,1) = 2.76250777733235E-03 ZBS(-3,1) = 3.43199911886317E-04
RBC(-2,1) = -1.24883373588382E-02 ZBS(-2,1) = 6.12174680232785E-04
RBC(-1,1) = 1.52272804511910E-02 ZBS(-1,1) = -2.70066914159594E-02
RBC(0,1) = 2.89195233044040E-01 ZBS(0,1) = 4.50462554508443E-01
RBC(1,1) = -1.17988850341728E-01 ZBS(1,1) = 1.93490230971634E-01
RBC(2,1) = -3.84923299492945E-03 ZBS(2,1) = 5.72865331625290E-03
RBC(3,1) = -1.44452305429529E-03 ZBS(3,1) = 2.19788951889214E-03
RBC(4,1) = -2.11622985211109E-04 ZBS(4,1) = 1.31883972780290E-03
RBC(5,1) = 1.79091719677667E-04 ZBS(5,1) = -5.63363462408534E-04
RBC(6,1) = 1.31982402741742E-04 ZBS(6,1) = -9.31801467009349E-05
RBC(-6,2) = -2.40882614870476E-05 ZBS(-6,2) = -3.95416405717970E-05
RBC(-5,2) = -4.92449386382591E-05 ZBS(-5,2) = -3.25048356502217E-06
RBC(-4,2) = 1.50530476034115E-04 ZBS(-4,2) = 4.61522421935086E-05
RBC(-3,2) = -1.23084235126550E-03 ZBS(-3,2) = -3.40868203306282E-04
RBC(-2,2) = 2.01350576071929E-04 ZBS(-2,2) = -4.19781517712033E-03
RBC(-1,2) = 2.36777003797179E-03 ZBS(-1,2) = 1.98753868216412E-02
RBC(0,2) = 5.73443941583452E-02 ZBS(0,2) = 4.81527027892127E-03
RBC(1,2) = 6.89385874058265E-02 ZBS(1,2) = -9.28353553039424E-03
RBC(2,2) = 4.71996849673782E-02 ZBS(2,2) = -2.04292782322197E-02
RBC(3,2) = -5.50889052720066E-04 ZBS(3,2) = 8.81593501270446E-04
RBC(4,2) = 4.24491391207156E-04 ZBS(4,2) = -6.08871281835245E-04
RBC(5,2) = -2.07538883155595E-04 ZBS(5,2) = -3.88708113241096E-04
RBC(6,2) = -1.62304038006678E-04 ZBS(6,2) = 1.72340342752605E-04
RBC(-6,3) = -1.01105699684233E-04 ZBS(-6,3) = -6.16215454248342E-05
RBC(-5,3) = 5.15925605980565E-05 ZBS(-5,3) = 1.23419431936950E-04
RBC(-4,3) = -3.79290487874111E-05 ZBS(-4,3) = 3.98637008165582E-06
RBC(-3,3) = -2.96154201246223E-04 ZBS(-3,3) = -7.01248486620889E-04
RBC(-2,3) = 1.27628943631957E-03 ZBS(-2,3) = 3.19332333533202E-03
RBC(-1,3) = 3.12803506573940E-03 ZBS(-1,3) = -8.24657727838880E-03
RBC(0,3) = -1.34574092972690E-02 ZBS(0,3) = 5.05936199755365E-03
RBC(1,3) = -8.02339287294677E-03 ZBS(1,3) = -3.90421394288867E-03
RBC(2,3) = -1.68510947837154E-02 ZBS(2,3) = 3.75441853342170E-03
RBC(3,3) = -8.00581733372124E-03 ZBS(3,3) = 6.00542774606014E-03
RBC(4,3) = 1.80667899211621E-03 ZBS(4,3) = -4.16787432635077E-04
RBC(5,3) = 3.10773970094350E-05 ZBS(5,3) = 5.44335921432213E-05
RBC(6,3) = 8.32496816115997E-05 ZBS(6,3) = -4.15830451164888E-05
RBC(-6,4) = -1.19874891436340E-05 ZBS(-6,4) = 1.56845408711308E-05
RBC(-5,4) = 1.22793444338155E-04 ZBS(-5,4) = -3.97576733690054E-05
RBC(-4,4) = -1.30945484439682E-04 ZBS(-4,4) = -7.22429623460448E-05
RBC(-3,4) = -1.21368603604647E-04 ZBS(-3,4) = 3.52928331257216E-04
RBC(-2,4) = 1.00352526472782E-03 ZBS(-2,4) = -1.23710282249961E-04
RBC(-1,4) = -1.73680844498789E-03 ZBS(-1,4) = -1.50689928334813E-03
RBC(0,4) = 1.80149787198970E-03 ZBS(0,4) = 1.56109492686192E-03
RBC(1,4) = 3.82771889154294E-03 ZBS(1,4) = 3.80910842862487E-03
RBC(2,4) = 5.43835034437129E-03 ZBS(2,4) = 2.06275075117804E-03
RBC(3,4) = 8.39729828422411E-04 ZBS(3,4) = -1.54779126563731E-03
RBC(4,4) = 6.74263596810560E-04 ZBS(4,4) = -1.33149943553452E-03
RBC(5,4) = -6.98647584180715E-04 ZBS(5,4) = 3.81307095116973E-04
RBC(6,4) = 8.77670652920776E-05 ZBS(6,4) = -1.40433963574141E-05
RBC(-6,5) = 6.78635213884316E-06 ZBS(-6,5) = -1.22283666932084E-05
RBC(-5,5) = 3.87846546342867E-05 ZBS(-5,5) = 4.64829761643373E-05
RBC(-4,5) = -3.78300368387435E-05 ZBS(-4,5) = -7.03801581329045E-05
RBC(-3,5) = -1.21743926248229E-05 ZBS(-3,5) = 1.85735151533626E-04
RBC(-2,5) = -2.68229697014545E-04 ZBS(-2,5) = -9.33216243296025E-04
RBC(-1,5) = 1.19567316567517E-03 ZBS(-1,5) = 2.12648562837673E-03
RBC(0,5) = -7.12579133390599E-04 ZBS(0,5) = -1.97890515574565E-03
RBC(1,5) = 8.81127157923892E-04 ZBS(1,5) = 2.71321673191593E-03
RBC(2,5) = 9.67210453659238E-04 ZBS(2,5) = 8.74618447862515E-04
RBC(3,5) = 2.11794179698155E-04 ZBS(3,5) = 8.43817701627930E-04
RBC(4,5) = 1.29403911922840E-03 ZBS(4,5) = 6.51808476607835E-04
RBC(5,5) = -1.30477683585083E-04 ZBS(5,5) = 1.01349326961770E-04
RBC(6,5) = 1.86680624010370E-04 ZBS(6,5) = -2.13838628730300E-04
RBC(-6,6) = -4.08213549686361E-05 ZBS(-6,6) = -7.53394429655583E-06
RBC(-5,6) = 7.11305157811999E-05 ZBS(-5,6) = 2.54876062250879E-05
RBC(-4,6) = -1.33727065581923E-04 ZBS(-4,6) = -1.70180862196520E-05
RBC(-3,6) = 1.65191943182183E-06 ZBS(-3,6) = -1.31350577800873E-04
RBC(-2,6) = 2.19460449719541E-04 ZBS(-2,6) = 4.38914760402648E-04
RBC(-1,6) = 4.68618562605432E-04 ZBS(-1,6) = -4.44537659614533E-04
RBC(0,6) = -8.51896573200937E-04 ZBS(0,6) = 7.36122964253313E-04
RBC(1,6) = -5.26623264534578E-05 ZBS(1,6) = -1.12352425125337E-03
RBC(2,6) = -1.31954654361710E-04 ZBS(2,6) = -2.22905186553194E-03
RBC(3,6) = -8.91482312658694E-04 ZBS(3,6) = -2.11193996461398E-03
RBC(4,6) = -3.89733094884781E-04 ZBS(4,6) = -3.44184359663702E-04
RBC(5,6) = -2.74329775462215E-04 ZBS(5,6) = -5.06914660659672E-05
RBC(6,6) = 2.47385092660320E-04 ZBS(6,6) = 3.74971583066409E-05
RBC(-6,7) = 9.61516193308531E-06 ZBS(-6,7) = -3.66121037894761E-06
RBC(-5,7) = -2.51122684780459E-05 ZBS(-5,7) = 3.72828134065079E-05
RBC(-4,7) = 4.44568599556351E-05 ZBS(-4,7) = -8.74488353626824E-05
RBC(-3,7) = -1.42433799354752E-04 ZBS(-3,7) = 1.48694485468843E-04
RBC(-2,7) = 4.85802385952487E-04 ZBS(-2,7) = -2.27519962800893E-04
RBC(-1,7) = -9.00652688032426E-04 ZBS(-1,7) = 4.16601324903870E-04
RBC(0,7) = 9.59457670863182E-04 ZBS(0,7) = -3.25818663499641E-04
RBC(1,7) = -3.37159659594826E-04 ZBS(1,7) = -2.34240245561361E-04
RBC(2,7) = -4.64969900861713E-04 ZBS(2,7) = 4.87821281121050E-04
RBC(3,7) = -4.09185322970312E-04 ZBS(3,7) = 8.50140634573578E-04
RBC(4,7) = 5.32088748759921E-05 ZBS(4,7) = 5.93528572346752E-04
RBC(5,7) = -3.21692982976907E-04 ZBS(5,7) = -2.54775193277671E-04
RBC(6,7) = -4.82403633897412E-05 ZBS(6,7) = 1.41947169759239E-05
RBC(-6,8) = -2.23522770283961E-05 ZBS(-6,8) = -4.00911971000495E-06
RBC(-5,8) = 3.95696912099304E-05 ZBS(-5,8) = 1.34684147523625E-05
RBC(-4,8) = -6.50775924544567E-05 ZBS(-4,8) = -2.94168940555405E-05
RBC(-3,8) = 1.71610112932980E-04 ZBS(-3,8) = 2.17875987311858E-05
RBC(-2,8) = -3.45412623614909E-04 ZBS(-2,8) = -7.26482153663716E-05
RBC(-1,8) = 5.61089095467387E-04 ZBS(-1,8) = 2.51145295676537E-04
RBC(0,8) = -5.84359101746051E-04 ZBS(0,8) = -5.42465826224607E-04
RBC(1,8) = -6.16860761080513E-05 ZBS(1,8) = 3.93697603313273E-04
RBC(2,8) = 5.99275780897287E-04 ZBS(2,8) = 3.30798770955874E-04
RBC(3,8) = 5.68520162541870E-04 ZBS(3,8) = 5.47788467933391E-04
RBC(4,8) = 4.47404034542356E-04 ZBS(4,8) = 2.43547539548605E-04
RBC(5,8) = 2.76704814165950E-04 ZBS(5,8) = 9.15194583315619E-05
RBC(6,8) = 2.97621090888441E-04 ZBS(6,8) = 1.65605427353701E-04
RBC(-6,9) = -3.78145897931544E-06 ZBS(-6,9) = -1.85759364750771E-06
RBC(-5,9) = -1.57985677623482E-05 ZBS(-5,9) = 1.06970045147331E-05
RBC(-4,9) = 7.91641381274532E-05 ZBS(-4,9) = -1.16252171939772E-05
RBC(-3,9) = -1.97587428419659E-04 ZBS(-3,9) = -3.08457797690412E-05
RBC(-2,9) = 3.95855751452672E-04 ZBS(-2,9) = 2.03418980231168E-04
RBC(-1,9) = -5.41153103221438E-04 ZBS(-1,9) = -3.99552958537408E-04
RBC(0,9) = 4.98714381092541E-04 ZBS(0,9) = 4.32916759100965E-04
RBC(1,9) = -8.06048953531492E-05 ZBS(1,9) = -1.84722027458208E-04
RBC(2,9) = -8.67990109801738E-05 ZBS(2,9) = 2.52568631885491E-04
RBC(3,9) = 4.35340840113358E-04 ZBS(3,9) = -3.50159782847442E-04
RBC(4,9) = 2.33585243788111E-04 ZBS(4,9) = -7.06133299107118E-04
RBC(5,9) = -7.69581174305243E-06 ZBS(5,9) = -3.79072907220561E-04
RBC(6,9) = -2.85256407945938E-05 ZBS(6,9) = -4.49599333610498E-05
RBC(-6,10) = 1.20206720198758E-05 ZBS(-6,10) = 4.73580005255806E-06
RBC(-5,10) = 7.02670536357846E-06 ZBS(-5,10) = -6.99664911015022E-06
RBC(-4,10) = -2.76926398374910E-05 ZBS(-4,10) = -9.18014408856618E-06
RBC(-3,10) = 5.20223745639364E-05 ZBS(-3,10) = 6.80574180383753E-05
RBC(-2,10) = -7.88310431749746E-05 ZBS(-2,10) = -1.06370673487973E-04
RBC(-1,10) = 1.21755712542490E-05 ZBS(-1,10) = 1.22161894513591E-04
RBC(0,10) = 3.22193519645521E-05 ZBS(0,10) = -6.04052049877600E-05
RBC(1,10) = -1.08453911913102E-06 ZBS(1,10) = 8.60890353665843E-05
RBC(2,10) = 1.04051545504927E-04 ZBS(2,10) = -2.17661420286656E-04
RBC(3,10) = -5.21965328013036E-04 ZBS(3,10) = -2.67111216700977E-04
RBC(4,10) = -4.95991087393098E-04 ZBS(4,10) = 2.43875640076056E-05
RBC(5,10) = -1.94520415280627E-04 ZBS(5,10) = 1.55759001593971E-04
RBC(6,10) = -6.94143617569942E-05 ZBS(6,10) = 4.40565098025554E-05
/
&END
To simplify execution of the code,
the VMEC compilation scripts create a directory called ‘bin’ in
your home () directory. Symbolic links are then placed there
pointing to each of the compiled codes in their respective
‘Vrelease’ subdirectories. In practice, the screen output from
VMEC should be redirected to a log file and put in the background
(>& log.ncsx_c09r00_fixed &) but for this tutorial we will simply
instruct the user to run the code so they can see how VMEC executes.
This is done by passing the suffix of the input file to the VMEC
code through the command line.
>~/bin/xvmec2000 ncsx_c09r00_fixed
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
SEQ = 1 TIME SLICE 0.0000E+00
PROCESSING INPUT.ncsx_c09r00_fixed
THIS IS VMEC2000, A 3D EQUILIBRIUM CODE, VERSION 8.47
Lambda: Full Radial Mesh. L-Force: hybrid full/half.
COMPUTER: computer.domain.net OS: Linux RELEASE: 2.6.18-194.17.4.el5 DATE = Sep 06,2011 TIME = 12:43:39
NS = 9 NO. FOURIER MODES = 137 FTOLV = 1.000E-06 NITER = 2500
ITER FSQR FSQZ FSQL RAX(v=0) WMHD
1 5.24E+01 6.99E+00 1.63E-01 1.608E+00 3.7781E+00
172 9.30E-07 2.96E-07 2.69E-07 1.610E+00 3.5200E+00
NS = 28 NO. FOURIER MODES = 137 FTOLV = 1.000E-08 NITER = 2500
ITER FSQR FSQZ FSQL RAX(v=0) WMHD
1 4.91E-02 2.35E-02 3.37E-04 1.610E+00 3.5199E+00
200 7.25E-08 2.33E-08 1.02E-08 1.609E+00 3.5196E+00
300 9.94E-09 3.01E-09 1.35E-09 1.609E+00 3.5196E+00
NS = 49 NO. FOURIER MODES = 137 FTOLV = 1.000E-11 NITER = 2500
ITER FSQR FSQZ FSQL RAX(v=0) WMHD
1 1.51E-02 8.86E-03 6.66E-06 1.609E+00 3.5196E+00
200 5.23E-08 2.49E-08 9.58E-10 1.609E+00 3.5195E+00
400 2.34E-09 8.33E-10 2.14E-10 1.608E+00 3.5195E+00
600 3.77E-10 1.11E-10 2.99E-11 1.608E+00 3.5195E+00
800 7.51E-11 1.87E-11 3.21E-12 1.608E+00 3.5195E+00
989 9.93E-12 3.22E-12 5.38E-13 1.608E+00 3.5195E+00
EXECUTION TERMINATED NORMALLY
FILE : ncsx_c09r00_fixed
NUMBER OF JACOBIAN RESETS = 4
TOTAL COMPUTATIONAL TIME 81.34 SECONDS
TIME TO READ IN DATA 0.04 SECONDS
TIME TO WRITE DATA TO WOUT 0.12 SECONDS
TIME IN EQFORCE 2.65 SECONDS
TIME IN FOURIER TRANSFORM 24.27 SECONDS
TIME IN INVERSE FOURIER XFORM 17.28 SECONDS
TIME IN FORCES 17.31 SECONDS
TIME IN BCOVAR 12.45 SECONDS
TIME IN RESIDUE 0.50 SECONDS
TIME (REMAINDER) IN FUNCT3D 6.41 SECONDS
>
For this example four files were created (jxbout_ncsx_c09r00_fixed.nc, mercier.ncsx_c09r00_fixed, threed1.ncsx_c09r00_fixed, and wout_ncsx_c09r00_fixed.nc). As was mentioned before, each file had the suffix of the input file appended to it’s name. This allows multiple runs to be stored in the same directory for comparison. The ‘jxbout’ file contains values for various quantities on a grid throughout the simulation domain. The ‘mercier’ file contains radial profiles (radial index in VMEC is denoted by the variable ‘s’) of various quantities. The ‘threed1’ file can be considered an expanded log file where various quantities are calculated which were not output to the screen. This file is fairly self explanatory. The ‘wout’ file is the data file for the run. It contains the Fourier Coefficients for the magnetic field along with various quantities. A few packages exist to visualize this data and the user is encourage to use these as templates for their own visualization routines.