State-of-the-art stellarator optimization code

The STELLOPT code is design to optimize 3D MHD equilibria to a set of target physics parameters encompassing stellarator design and 3D equilibrium reconstruction. It is currently interfaced to the VMEC 3D equilibrium solver.

STELLOPT is currently hosted on the Princeton University’s GitHub repository.

Table of Contents

The STELLOPT code can be thought of as a multi-dimensional non-linear curve fitting algorithm. The curve to which a fit is sought is parameterized by a set of target parameters which themselves may be MHD equilibrium quantities (beta, aspect ratio, etc.) or non-linear functions of the equilibrium itself (stability, particle transport, etc). The function used to fit these targets is the MHD equilibrium itself. Here the input parameters to the equilibrium serve as a set of coefficients parameterizing our non-linear fit function. The code uses either a modified Levenberg-Marquardt algorithm, genetic algorithm, or differential evolution to find a best fit set of equilibrium inputs to the set of desired targets. The quality of fit is determined by the Chi-squared metric $$ \chi_i^2=\frac{\left(f_i^{target}-f_i^{equilbria}\right)^2 }{\sigma_i^2} .$$ The sigma here represents the tolerance of the target value. The goal of the code is to minimize the total chi-squared value.

The user specifies which equilibrium input values to vary independently of the targets. Additionally, profiles for Ne, Te, and Ti (as functions of toroidal flux) have been added to STELLOPT. If utilized these profiles are used to calculate the equilibrium total pressure before a calculation is preformed. The code is capable of targeting a number of different parameters by either directly evaluating the equilibrium or calling subcodes once an equilibrium is found. The following list outlines the currently available targets:

Target | Dimension | Description |
---|---|---|

ASPECT RATIO | Single | Equilibrium Aspect Ratio |

BETA | Single | Equilibrium Total Plasma Beta |

CURTOR | Single | Equilibrium Total Toroidal Current |

PHIEDGE | Single | Equilibrium Total Enclosed Toroidal Flux |

R0 | Single | Equilibrium Radial Magnetic Axis Position |

RBTOR | Single | Equilibrium R-B_toroidal |

STORED ENERGY | Single | Equilibrium Stored Energy |

VOLUME | Single | Equilibrium Volume |

EXTCUR | Vector | Vacuum Field Currents |

LINE_NE | Vector | Array of line integrated electron density measurements |

FARADAY | Vector | Array of Faraday Rotation measurements |

PRESS | Vector | Array of pressure profile measurements (R,PHI,Z or S) |

NE | Vector | Array of electron density measurements (R,PHI, Z or S) |

TE | Vector | Array of electron temperature measurements (R, PHI, Z or S) |

TI | Vector | Array of ion temperature measurements (R, PHI, Z or S) |

LINE_TE | Vector | Array of line integrated electron temperature measurements |

LINE_TI | Vector | Array of line integrated ion temperature measurements |

IOTA | Vector | Array of rotational transform measurements (R, PHI, Z or S) |

BPROBE | Vector | Array of B-Field measurements calculated by DIAGNO |

FLUXLOOP | Vector | Array of Flux Loop measurements calculated by DIAGNO |

ROGOWSKI | Vector | Array of Rogowski Coil measurements calculated by DIAGNO |

VESSEL | Matrix | Limiting points in space |

SEPARATRIX | Matrix | Desired edge points in space |

BALLOON | Vector | Array of radial locations for ballooning stability calculation by COBRAVMEC |

KINK | Single | Equilibrium kink stability as calculated by TERPSICHORE |

BOOTSTRAP | Vector | Array of radial locations for bootstrap calculation by BOOTSJ |

NEO | Vector | Array of radial locations for Neoclassical transport calculation by NEO |

HELICITY | Vector | Array of radial locations for Helicity calculation |

JSTAR | Vector | Array of radial locations for particle confinement calculation |

ORBIT | Vector | Array of radial locations on which to calculate confinement BEAMS3D |

COIL_BNORM | Single | Calculation of residual coil normal field after calculation by COILOPT++ |

Compilation of the STELLOPT suite is discussed on the STELLOPT Compilation Page

The STELLOPT code takes an input file which contains all the input namelists necessary to run the equilibrium code, STELLOPT, and any additional codes called by the optimizer. The STELLOPT code itself requires an OPTIMUM namelist which contains the runtime parameters for the code, specifies which input variables to vary, any additional profiles, and the target parameters to match.

The STELLOPT routine is designed to be run on a multi-processor machine or cluster. In general, it should be executed via a call to mpirun (most likely inside a PBS script). The call should take the form:

```
mpirun -np $NPROCS /path-to/xstelloptv2 $runid > log.$runid
```

Here `$NPROCS`

is the number of processors being requested and $runid is
the suffix assigned to the input file. Upon execution STELLOPT will
create a working directory entitled `stellopt_$runid`

where
`$runid`

is the value passed to it from the command line. All work
and final output is preformed in this directory. Optional arguments
which control the code behavior are:

Argument | Default | Description |
---|---|---|

-restart | NONE | Use wout_reset_file.nc in directory to restart VMEC run on first iteration. |

-noverb | NONE | Suppress screen output. |

-log | NONE | Send screen output to log file (useful on systems where `>&` doesn’t work). |

-autodomain XX | NONE | Automatically calculate bounds as percentage (-autodomain 0.2, produces a +/-20% bound) |

-tri FILE1 FILE2 | NONE | Use FILE1 and FILE2 to set Min and Max bounds respectively (used in hyperplane mapping mode) |

-xvec_file FILE1 | NONE | Use FILE1 to initialize population for Differential Evolution and Particle Swarm runs. |

-help | NONE | Print help message |

The STELLOPT code will produce many files so it is suggested that each run be kept in a separate directory. A stereotypical run looks something like:

Once a run has completed the code will output a `stellopt.ext`

file,
and a `input.ext_min`

file. The `_min`

file contains an
equilibrium input file corresponding to the best fit as found by
STELLOPT. The `stellopt.`

file contains an iteration by iteration
analysis of the fit to the target parameters. The file begins with a
version string (VERSION 2.65). Each iteration is demarked by an
iteration line (ITER XXXX). Each target type is then noted by line
indicating a type, the number of target values, and the number of output
values (TYPE 3 1). Immediately following that line is a line of text
indicating the value in each column.

Additionally, many working files denoted by `_optXX`

are produced by
STELLOPT. These files may be removed once a run has completed. If the
user has elected to keep minimum states as the code iterated, various
files produced by the equilibrium calculation, and other codes will be
kept with an iteration number appended to their name. In this way a step
by step analysis of the optimization may be evaluated. The optimization
routines may themselves create output files.

The final output of the STELLOPT routine is a combination of STELLOPT files, VMEC files, and the output files of the physics modules invoked durring the run.

Boundary Representations Explained

Optimization of an NCSX-like configuration

Optimization of Iota using LMDIF