Quadratic Flux Minimizing Surfaces
Quadratic-Flux-Minimizing (QFM) surfaces are a useful tool in vacuum stellarator optimization, where, given a coil magnetic field, one may extract a surface in space which minimizes the quadratic magnetic flux passing through it, that is,
\[\begin{equation}
S = \arg\min_S \int_S |B_n|^2 dS
\end{equation}\]
Where \(B_n = \mathbf{B} \cdot \mathbf{n}\) is the normal magnetic field on the surface \(S\), and the optimization is understood to be done under the constraint of the surface enclosing some amount of net toroidal magnetic flux. The surface \(S\) itself is described typically with a double Fourier series in the poloidal and toroidal angles (See the docs on basis functions for more details) describing the cylindrical \(R\) and \(Z\) coordinates of the surface, so the optimization is done by varying the Fourier coefficients describing the surface (or potentially, the magnetic field’s degrees of freedom as well). This short notebook will give an example of using QFM surfaces to extrace a surface from a magnetic field, and then solving the interior equilibrum with DESC and comparing it to the original magnetic field.
[1]:
import sys
import os
sys.path.insert(0, os.path.abspath("."))
sys.path.append(os.path.abspath("../../../"))
If you have access to a GPU, uncomment the following two lines.
[2]:
# from desc import set_device
# set_device("gpu")
[3]:
from desc.objectives import (
SurfaceQuadraticFlux,
ToroidalFlux,
ObjectiveFunction,
)
from desc.magnetic_fields import SplineMagneticField
from desc.geometry import FourierRZToroidalSurface
from desc.optimize import Optimizer
from desc.grid import LinearGrid
from desc.plotting import plot_2d, poincare_plot, plot_boundary, plot_surfaces
import numpy as np
DESC version 0.12.3+864.g55c9d3a11.dirty,using JAX backend, jax version=0.4.35, jaxlib version=0.4.35, dtype=float64
Using device: CPU, with 7.33 GB available memory
First, we will load a magnetic field from an mgrid file (the same as is used in the stellarator free boundary notebook)
[4]:
extcur = [4700.0, 1000.0]
field = SplineMagneticField.from_mgrid(
"../../../tests/inputs/mgrid_test.nc", extcur=extcur
)
Our initial surface is a circular surface, which we see is not well-approximating any flux surfaces in this vacuum field.
[5]:
surface0 = FourierRZToroidalSurface(
R_lmn=[0.7, 0.1],
modes_R=[[0, 0], [1, 0]],
Z_lmn=[-0.1],
modes_Z=[[-1, 0]],
sym=True,
NFP=field.NFP,
M=6,
N=6,
)
fig, ax = plot_surfaces(surface0, rho=1.0, theta=0)
poincare_plot(
field,
R0=np.linspace(0.7, 0.8, 10),
Z0=np.zeros(10),
ax=ax,
NFP=field.NFP,
ntransit=50,
);
Let’s setup the optimization problem to find the QFM surface.
[6]:
# SurfaceQuadraticFlux objective is the one to use when trying to minimize quadratic flux for a surface
qflux = SurfaceQuadraticFlux(
surface0,
field,
eval_grid=LinearGrid(M=2 * surface0.M, N=2 * surface0.M, NFP=surface0.NFP),
field_fixed=True, # field is not being optimized
)
# Must include a ToroidalFlux or a Volume target to ensure we don't get a trivial solution of the surface collapsing to a point
target_psi = -0.035
tflux = ToroidalFlux(
surface0,
field,
target=target_psi, # the same toroidal flux as used in the free bdry eq
eq_fixed=False, # we will allow the thing passed in (QFM surface in this case) to vary
field_fixed=True, # field is not being optimized, so we fix it
)
obj = ObjectiveFunction(qflux)
opt = Optimizer("lsq-auglag")
(qfm_surf,), _ = opt.optimize(
surface0,
objective=obj,
constraints=(tflux,),
ftol=1e-5,
maxiter=500,
verbose=3,
copy=True,
)
Building objective: Surface Quadratic Flux
Precomputing transforms
Timer: Precomputing transforms = 1.02 sec
Timer: Objective build = 1.03 sec
Building objective: toroidal-flux
Precomputing transforms
Timer: Precomputing transforms = 442 ms
Timer: Objective build = 1.32 sec
Number of parameters: 169
Number of objectives: 625
Number of equality constraints: 1
Number of inequality constraints: 0
Timer: Initializing the optimization = 2.44 sec
Starting optimization
Using method: lsq-auglag
Iteration Total nfev Cost Cost reduction Step norm Optimality Constr viol. Penalty param max(|mltplr|)
0 1 6.632e+01 2.104e+03 1.383e-02 1.000e+01 0.000e+00
1 2 6.099e+01 5.329e+00 2.309e-03 2.006e+03 1.441e-02 1.000e+01 0.000e+00
2 3 4.257e+01 1.843e+01 9.279e-03 1.634e+03 1.665e-02 1.000e+01 0.000e+00
3 4 5.031e+00 3.754e+01 3.739e-02 4.605e+02 2.437e-02 1.000e+01 0.000e+00
4 5 5.032e+00 -5.593e-04 1.384e-01 6.829e+02 2.530e-03 1.000e+01 0.000e+00
5 6 1.301e-01 4.902e+00 2.761e-02 1.545e+02 1.653e-04 1.000e+01 0.000e+00
6 8 3.489e-03 1.266e-01 8.836e-03 1.881e+01 8.797e-04 1.000e+01 0.000e+00
7 10 6.393e-04 2.850e-03 3.736e-03 6.773e+00 1.053e-03 1.000e+01 0.000e+00
8 12 3.011e-04 3.382e-04 3.605e-03 5.149e+00 7.633e-04 1.000e+01 0.000e+00
9 13 1.259e-04 1.752e-04 3.131e-03 4.394e+00 3.205e-04 1.000e+01 0.000e+00
10 15 2.756e-05 9.835e-05 6.664e-04 4.466e-01 2.356e-04 1.000e+01 0.000e+00
11 17 2.582e-05 1.739e-06 5.895e-04 5.394e-01 1.918e-04 1.000e+01 0.000e+00
12 18 2.444e-05 1.385e-06 5.742e-04 3.878e-01 1.702e-04 1.000e+01 0.000e+00
13 20 2.341e-05 1.022e-06 6.493e-04 2.401e-01 1.584e-04 1.000e+01 0.000e+00
14 22 2.259e-05 8.222e-07 6.590e-04 1.766e-01 1.499e-04 1.000e+01 0.000e+00
15 24 2.184e-05 7.512e-07 6.695e-04 2.146e-01 1.425e-04 1.000e+01 0.000e+00
16 26 2.126e-05 5.819e-07 7.306e-04 2.684e-01 1.356e-04 1.000e+01 0.000e+00
17 27 2.076e-05 4.956e-07 7.722e-04 2.277e-01 1.292e-04 1.000e+01 0.000e+00
18 28 2.027e-05 4.947e-07 7.612e-04 2.088e-01 1.236e-04 1.000e+01 0.000e+00
19 29 1.978e-05 4.910e-07 7.302e-04 2.367e-01 1.188e-04 1.000e+01 0.000e+00
20 30 1.929e-05 4.909e-07 6.950e-04 2.560e-01 1.147e-04 1.000e+01 0.000e+00
21 31 1.880e-05 4.833e-07 6.653e-04 2.674e-01 1.110e-04 1.000e+01 0.000e+00
22 32 1.836e-05 4.374e-07 6.569e-04 2.798e-01 1.074e-04 1.000e+01 0.000e+00
23 33 1.799e-05 3.705e-07 6.684e-04 2.892e-01 1.038e-04 1.000e+01 0.000e+00
24 34 1.767e-05 3.214e-07 6.786e-04 2.792e-01 1.002e-04 1.000e+01 0.000e+00
25 35 1.739e-05 2.831e-07 6.812e-04 2.525e-01 9.675e-05 1.000e+01 0.000e+00
26 36 1.714e-05 2.505e-07 6.813e-04 2.202e-01 9.349e-05 1.000e+01 0.000e+00
27 37 1.691e-05 2.251e-07 6.825e-04 1.880e-01 9.047e-05 1.000e+01 0.000e+00
28 38 1.671e-05 2.084e-07 6.856e-04 1.814e-01 8.763e-05 1.000e+01 0.000e+00
29 39 1.643e-05 2.784e-07 1.706e-04 1.131e-02 8.660e-05 1.000e+01 8.660e-04
30 40 1.649e-05 -6.093e-08 7.438e-04 1.716e-01 1.049e-06 1.000e+01 8.660e-04
31 41 1.625e-05 2.413e-07 1.719e-04 1.122e-02 2.143e-06 1.000e+01 8.660e-04
32 43 1.620e-05 4.927e-08 1.718e-04 1.087e-02 3.019e-06 1.000e+01 8.660e-04
33 45 1.615e-05 4.865e-08 1.722e-04 1.070e-02 3.635e-06 1.000e+01 8.660e-04
34 47 1.610e-05 4.798e-08 1.725e-04 1.057e-02 4.216e-06 1.000e+01 8.660e-04
35 49 1.605e-05 4.733e-08 1.728e-04 1.045e-02 4.787e-06 1.000e+01 8.660e-04
36 51 1.601e-05 4.670e-08 1.731e-04 1.033e-02 5.348e-06 1.000e+01 8.660e-04
37 53 1.596e-05 4.607e-08 1.735e-04 1.022e-02 5.903e-06 1.000e+01 8.660e-04
38 55 1.592e-05 4.544e-08 1.738e-04 1.010e-02 6.450e-06 1.000e+01 8.660e-04
39 57 1.587e-05 4.481e-08 1.742e-04 9.986e-03 6.991e-06 1.000e+01 7.961e-04
40 59 1.582e-05 4.900e-08 1.735e-04 9.898e-03 1.351e-06 1.000e+01 7.961e-04
41 61 1.578e-05 4.406e-08 1.748e-04 9.775e-03 1.204e-06 1.000e+01 7.961e-04
42 63 1.573e-05 4.294e-08 1.753e-04 9.667e-03 1.652e-06 1.000e+01 7.961e-04
43 65 1.569e-05 4.225e-08 1.757e-04 9.542e-03 2.145e-06 1.000e+01 7.961e-04
44 67 1.565e-05 4.160e-08 1.760e-04 9.420e-03 2.638e-06 1.000e+01 7.961e-04
45 69 1.561e-05 4.095e-08 1.764e-04 9.291e-03 3.123e-06 1.000e+01 7.961e-04
46 71 1.557e-05 4.030e-08 1.768e-04 9.164e-03 3.600e-06 1.000e+01 7.961e-04
47 73 1.553e-05 3.966e-08 1.773e-04 9.045e-03 4.072e-06 1.000e+01 7.961e-04
48 75 1.549e-05 3.903e-08 1.777e-04 8.936e-03 4.539e-06 1.000e+01 7.961e-04
49 77 1.545e-05 3.839e-08 1.782e-04 8.832e-03 5.001e-06 1.000e+01 7.961e-04
50 79 1.541e-05 3.776e-08 1.786e-04 8.723e-03 5.455e-06 1.000e+01 7.961e-04
51 81 1.538e-05 3.713e-08 1.791e-04 8.612e-03 5.904e-06 1.000e+01 7.961e-04
52 83 1.534e-05 3.651e-08 1.796e-04 8.502e-03 6.354e-06 1.000e+01 7.961e-04
53 85 1.531e-05 3.592e-08 1.802e-04 8.396e-03 6.806e-06 1.000e+01 7.961e-04
54 87 1.527e-05 3.536e-08 1.808e-04 8.298e-03 7.254e-06 1.000e+01 7.961e-04
55 89 1.524e-05 3.482e-08 1.814e-04 8.186e-03 7.690e-06 1.000e+01 7.961e-04
56 91 1.520e-05 3.430e-08 1.821e-04 8.072e-03 8.123e-06 1.000e+01 7.961e-04
57 93 1.517e-05 3.381e-08 1.829e-04 7.932e-03 8.530e-06 1.000e+01 7.961e-04
58 95 1.513e-05 3.331e-08 1.836e-04 7.797e-03 8.928e-06 1.000e+01 7.961e-04
59 97 1.510e-05 3.284e-08 1.842e-04 7.668e-03 9.317e-06 1.000e+01 7.961e-04
60 99 1.507e-05 3.241e-08 1.850e-04 7.525e-03 9.696e-06 1.000e+01 7.961e-04
61 101 1.504e-05 3.199e-08 1.858e-04 8.042e-03 1.007e-05 1.000e+01 7.961e-04
62 103 1.500e-05 3.159e-08 1.865e-04 8.669e-03 1.045e-05 1.000e+01 7.961e-04
63 105 1.497e-05 3.122e-08 1.873e-04 9.381e-03 1.084e-05 1.000e+01 7.961e-04
64 107 1.494e-05 3.086e-08 1.882e-04 1.017e-02 1.124e-05 1.000e+01 7.961e-04
65 109 1.491e-05 3.048e-08 1.887e-04 1.105e-02 1.164e-05 1.000e+01 7.961e-04
66 111 1.488e-05 3.009e-08 1.890e-04 1.197e-02 1.207e-05 1.000e+01 7.961e-04
67 113 1.485e-05 2.966e-08 1.891e-04 1.289e-02 1.251e-05 1.000e+01 7.961e-04
68 115 1.482e-05 2.921e-08 1.887e-04 1.379e-02 1.295e-05 1.000e+01 7.961e-04
69 117 1.479e-05 2.871e-08 1.881e-04 1.464e-02 1.340e-05 1.000e+01 7.961e-04
70 119 1.477e-05 2.817e-08 1.872e-04 1.540e-02 1.385e-05 1.000e+01 7.961e-04
71 121 1.474e-05 2.758e-08 1.860e-04 1.606e-02 1.431e-05 1.000e+01 7.961e-04
72 123 1.471e-05 2.696e-08 1.845e-04 1.657e-02 1.478e-05 1.000e+01 7.961e-04
73 125 1.469e-05 2.630e-08 1.828e-04 1.690e-02 1.526e-05 1.000e+01 7.961e-04
74 127 1.466e-05 2.563e-08 1.809e-04 1.701e-02 1.574e-05 1.000e+01 7.961e-04
75 129 1.464e-05 2.495e-08 1.790e-04 1.694e-02 1.620e-05 1.000e+01 7.961e-04
76 131 1.461e-05 2.427e-08 1.771e-04 1.662e-02 1.667e-05 1.000e+01 7.961e-04
77 133 1.459e-05 2.361e-08 1.753e-04 1.606e-02 1.712e-05 1.000e+01 7.961e-04
78 135 1.456e-05 2.299e-08 1.735e-04 1.523e-02 1.756e-05 1.000e+01 7.961e-04
79 137 1.454e-05 2.241e-08 1.719e-04 1.468e-02 1.799e-05 1.000e+01 7.961e-04
80 139 1.452e-05 2.190e-08 1.708e-04 1.558e-02 1.841e-05 1.000e+01 7.961e-04
81 141 1.450e-05 2.144e-08 1.699e-04 1.649e-02 1.881e-05 1.000e+01 7.961e-04
82 143 1.448e-05 2.104e-08 1.691e-04 1.739e-02 1.918e-05 1.000e+01 7.961e-04
83 145 1.446e-05 2.070e-08 1.685e-04 1.826e-02 1.952e-05 1.000e+01 7.961e-04
84 147 1.444e-05 2.041e-08 1.683e-04 1.908e-02 1.985e-05 1.000e+01 7.961e-04
85 149 1.442e-05 2.016e-08 1.683e-04 1.980e-02 2.017e-05 1.000e+01 7.961e-04
86 151 1.440e-05 1.994e-08 1.685e-04 2.039e-02 2.046e-05 1.000e+01 7.961e-04
87 153 1.438e-05 1.974e-08 1.686e-04 2.085e-02 2.073e-05 1.000e+01 7.961e-04
88 155 1.436e-05 1.954e-08 1.689e-04 2.111e-02 2.099e-05 1.000e+01 7.961e-04
89 157 1.434e-05 1.935e-08 1.694e-04 2.114e-02 2.123e-05 1.000e+01 7.961e-04
90 159 1.432e-05 1.918e-08 1.699e-04 2.099e-02 2.147e-05 1.000e+01 7.961e-04
91 161 1.430e-05 1.902e-08 1.705e-04 2.064e-02 2.170e-05 1.000e+01 7.961e-04
92 163 1.428e-05 1.888e-08 1.710e-04 2.014e-02 2.193e-05 1.000e+01 7.961e-04
93 165 1.426e-05 1.876e-08 1.713e-04 1.953e-02 2.214e-05 1.000e+01 7.961e-04
94 167 1.424e-05 1.866e-08 1.712e-04 1.884e-02 2.235e-05 1.000e+01 7.961e-04
95 169 1.422e-05 1.861e-08 1.708e-04 1.810e-02 2.254e-05 1.000e+01 7.961e-04
96 171 1.421e-05 1.859e-08 1.700e-04 1.735e-02 2.273e-05 1.000e+01 7.961e-04
97 173 1.419e-05 1.862e-08 1.688e-04 1.663e-02 2.290e-05 1.000e+01 7.961e-04
98 175 1.417e-05 1.868e-08 1.671e-04 1.597e-02 2.306e-05 1.000e+01 7.961e-04
99 177 1.415e-05 1.879e-08 1.653e-04 1.537e-02 2.323e-05 1.000e+01 7.961e-04
100 179 1.413e-05 1.892e-08 1.634e-04 1.486e-02 2.339e-05 1.000e+01 7.961e-04
101 181 1.411e-05 1.908e-08 1.614e-04 1.442e-02 2.354e-05 1.000e+01 7.961e-04
102 183 1.409e-05 1.924e-08 1.594e-04 1.403e-02 2.367e-05 1.000e+01 7.961e-04
103 185 1.407e-05 1.939e-08 1.575e-04 1.367e-02 2.379e-05 1.000e+01 7.961e-04
104 187 1.405e-05 1.950e-08 1.558e-04 1.333e-02 2.390e-05 1.000e+01 7.961e-04
105 189 1.403e-05 1.956e-08 1.544e-04 1.299e-02 2.401e-05 1.000e+01 7.961e-04
106 191 1.401e-05 1.955e-08 1.532e-04 1.263e-02 2.412e-05 1.000e+01 7.961e-04
107 193 1.400e-05 1.949e-08 1.522e-04 1.223e-02 2.422e-05 1.000e+01 7.961e-04
108 195 1.398e-05 1.934e-08 1.517e-04 1.178e-02 2.433e-05 1.000e+01 7.961e-04
109 197 1.396e-05 1.910e-08 1.514e-04 1.127e-02 2.445e-05 1.000e+01 7.961e-04
110 199 1.394e-05 1.878e-08 1.511e-04 1.070e-02 2.456e-05 1.000e+01 7.961e-04
111 201 1.392e-05 1.839e-08 1.509e-04 1.008e-02 2.468e-05 1.000e+01 7.961e-04
112 203 1.390e-05 1.793e-08 1.510e-04 9.380e-03 2.480e-05 1.000e+01 7.961e-04
113 205 1.388e-05 1.742e-08 1.511e-04 9.108e-03 2.491e-05 1.000e+01 7.961e-04
114 207 1.387e-05 1.687e-08 1.513e-04 9.679e-03 2.504e-05 1.000e+01 7.961e-04
115 209 1.385e-05 1.630e-08 1.516e-04 1.025e-02 2.516e-05 1.000e+01 7.961e-04
116 211 1.384e-05 1.571e-08 1.521e-04 1.080e-02 2.528e-05 1.000e+01 7.961e-04
117 213 1.382e-05 1.511e-08 1.529e-04 1.135e-02 2.541e-05 1.000e+01 7.961e-04
118 215 1.381e-05 1.452e-08 1.543e-04 1.182e-02 2.553e-05 1.000e+01 7.961e-04
119 217 1.379e-05 1.392e-08 1.563e-04 1.220e-02 2.565e-05 1.000e+01 7.961e-04
120 219 1.378e-05 1.334e-08 1.588e-04 1.250e-02 2.577e-05 1.000e+01 7.961e-04
121 221 1.377e-05 1.278e-08 1.619e-04 1.270e-02 2.588e-05 1.000e+01 7.961e-04
122 223 1.375e-05 1.225e-08 1.654e-04 1.277e-02 2.598e-05 1.000e+01 7.961e-04
123 225 1.374e-05 1.175e-08 1.690e-04 1.308e-02 2.608e-05 1.000e+01 7.961e-04
124 227 1.373e-05 1.127e-08 1.729e-04 1.470e-02 2.618e-05 1.000e+01 7.961e-04
125 229 1.372e-05 1.083e-08 1.767e-04 1.648e-02 2.626e-05 1.000e+01 7.961e-04
126 231 1.371e-05 1.043e-08 1.806e-04 1.835e-02 2.634e-05 1.000e+01 7.961e-04
127 233 1.370e-05 1.008e-08 1.844e-04 2.014e-02 2.642e-05 1.000e+01 7.961e-04
128 235 1.369e-05 9.764e-09 1.880e-04 2.186e-02 2.649e-05 1.000e+01 7.961e-04
129 237 1.368e-05 9.483e-09 1.913e-04 2.338e-02 2.656e-05 1.000e+01 7.961e-04
130 239 1.367e-05 9.248e-09 1.943e-04 2.486e-02 2.662e-05 1.000e+01 7.961e-04
131 241 1.366e-05 9.054e-09 1.969e-04 2.620e-02 2.669e-05 1.000e+01 7.961e-04
132 243 1.365e-05 8.888e-09 1.991e-04 2.742e-02 2.675e-05 1.000e+01 7.961e-04
133 245 1.364e-05 8.745e-09 2.010e-04 2.849e-02 2.681e-05 1.000e+01 7.961e-04
134 247 1.364e-05 8.615e-09 2.026e-04 2.941e-02 2.687e-05 1.000e+01 7.961e-04
135 249 1.363e-05 8.479e-09 2.039e-04 3.004e-02 2.692e-05 1.000e+01 7.961e-04
136 251 1.362e-05 8.372e-09 2.047e-04 3.050e-02 2.697e-05 1.000e+01 7.961e-04
137 253 1.361e-05 8.260e-09 2.055e-04 3.085e-02 2.703e-05 1.000e+01 7.961e-04
138 255 1.360e-05 8.184e-09 2.061e-04 3.108e-02 2.709e-05 1.000e+01 7.961e-04
139 257 1.359e-05 8.121e-09 2.063e-04 3.117e-02 2.716e-05 1.000e+01 7.961e-04
140 259 1.359e-05 8.036e-09 2.064e-04 3.122e-02 2.723e-05 1.000e+01 7.961e-04
141 261 1.358e-05 7.953e-09 2.065e-04 3.120e-02 2.731e-05 1.000e+01 7.961e-04
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312 538 1.306e-05 2.689e-10 3.246e-04 3.436e-02 3.164e-05 1.000e+01 7.961e-04
313 539 1.306e-05 4.808e-09 8.302e-05 2.163e-03 3.152e-05 1.000e+01 7.961e-04
314 540 1.306e-05 1.828e-10 3.213e-04 3.043e-02 3.165e-05 1.000e+01 7.961e-04
315 541 1.305e-05 4.619e-09 8.256e-05 1.859e-03 3.151e-05 1.000e+01 7.961e-04
316 543 1.305e-05 9.176e-10 8.181e-05 1.801e-03 3.150e-05 1.000e+01 7.961e-04
317 545 1.305e-05 8.967e-10 8.128e-05 1.735e-03 3.150e-05 1.000e+01 7.961e-04
318 547 1.305e-05 8.781e-10 8.081e-05 1.777e-03 3.149e-05 1.000e+01 7.961e-04
319 549 1.305e-05 8.598e-10 8.043e-05 1.809e-03 3.148e-05 1.000e+01 7.961e-04
320 551 1.305e-05 8.416e-10 8.018e-05 1.831e-03 3.148e-05 1.000e+01 7.961e-04
321 553 1.305e-05 8.252e-10 7.995e-05 1.848e-03 3.148e-05 1.000e+01 7.961e-04
322 555 1.305e-05 8.101e-10 7.976e-05 1.862e-03 3.147e-05 1.000e+01 7.961e-04
323 557 1.305e-05 7.942e-10 7.959e-05 1.871e-03 3.147e-05 1.000e+01 7.961e-04
324 559 1.305e-05 7.779e-10 7.940e-05 1.876e-03 3.146e-05 1.000e+01 7.961e-04
325 561 1.305e-05 7.632e-10 7.923e-05 1.882e-03 3.146e-05 1.000e+01 7.961e-04
326 563 1.305e-05 7.430e-10 7.921e-05 1.891e-03 3.147e-05 1.000e+01 7.961e-04
327 565 1.304e-05 7.295e-10 7.917e-05 1.896e-03 3.148e-05 1.000e+01 7.961e-04
328 567 1.304e-05 7.217e-10 7.918e-05 1.877e-03 3.149e-05 1.000e+01 7.961e-04
329 569 1.304e-05 7.046e-10 7.905e-05 1.854e-03 3.150e-05 1.000e+01 7.961e-04
330 571 1.304e-05 6.812e-10 7.873e-05 1.837e-03 3.152e-05 1.000e+01 7.961e-04
331 573 1.304e-05 6.624e-10 7.830e-05 1.816e-03 3.154e-05 1.000e+01 7.961e-04
332 575 1.304e-05 6.447e-10 7.794e-05 1.796e-03 3.156e-05 1.000e+01 7.961e-04
333 577 1.304e-05 6.294e-10 7.758e-05 1.771e-03 3.158e-05 1.000e+01 7.961e-04
334 579 1.304e-05 6.110e-10 7.725e-05 1.738e-03 3.162e-05 1.000e+01 7.961e-04
335 581 1.304e-05 5.988e-10 7.689e-05 1.709e-03 3.165e-05 1.000e+01 7.961e-04
336 583 1.304e-05 5.899e-10 7.658e-05 1.682e-03 3.168e-05 1.000e+01 7.961e-04
337 585 1.304e-05 5.748e-10 7.625e-05 1.656e-03 3.171e-05 1.000e+01 7.961e-04
338 587 1.304e-05 5.586e-10 7.578e-05 1.622e-03 3.174e-05 1.000e+01 7.961e-04
339 589 1.304e-05 5.406e-10 7.519e-05 1.582e-03 3.177e-05 1.000e+01 7.961e-04
340 591 1.304e-05 5.235e-10 7.450e-05 1.534e-03 3.181e-05 1.000e+01 7.961e-04
341 593 1.304e-05 5.052e-10 7.378e-05 1.486e-03 3.184e-05 1.000e+01 7.961e-04
342 595 1.304e-05 4.819e-10 7.286e-05 1.437e-03 3.188e-05 1.000e+01 7.961e-04
343 597 1.303e-05 4.624e-10 7.193e-05 1.383e-03 3.192e-05 1.000e+01 7.961e-04
344 599 1.303e-05 4.430e-10 7.099e-05 1.325e-03 3.197e-05 1.000e+01 7.961e-04
345 601 1.303e-05 4.216e-10 6.984e-05 1.263e-03 3.201e-05 1.000e+01 7.961e-04
346 603 1.303e-05 4.000e-10 6.859e-05 1.202e-03 3.205e-05 1.000e+01 7.961e-04
347 605 1.303e-05 3.805e-10 6.736e-05 1.142e-03 3.208e-05 1.000e+01 7.961e-04
348 607 1.303e-05 3.612e-10 6.622e-05 1.084e-03 3.212e-05 1.000e+01 7.961e-04
349 609 1.303e-05 3.418e-10 6.505e-05 1.025e-03 3.216e-05 1.000e+01 7.961e-04
350 611 1.303e-05 3.236e-10 6.387e-05 9.643e-04 3.219e-05 1.000e+01 4.742e-04
351 612 1.302e-05 1.412e-08 2.486e-04 1.336e-02 3.029e-07 1.000e+01 4.742e-04
352 613 1.302e-05 1.538e-09 2.197e-04 1.188e-02 2.563e-07 1.000e+01 4.742e-04
353 614 1.302e-05 8.966e-10 2.075e-04 1.268e-02 3.157e-07 1.000e+01 4.742e-04
354 615 1.301e-05 1.627e-09 5.203e-05 8.506e-04 2.878e-07 1.000e+01 4.742e-04
355 617 1.301e-05 1.648e-10 5.118e-05 9.522e-04 3.032e-07 1.000e+01 4.742e-04
356 619 1.301e-05 1.581e-10 5.106e-05 1.023e-03 3.205e-07 1.000e+01 4.742e-04
357 621 1.301e-05 1.532e-10 5.126e-05 1.074e-03 3.387e-07 1.000e+01 4.742e-04
358 623 1.301e-05 1.497e-10 5.138e-05 1.130e-03 3.560e-07 1.000e+01 4.742e-04
359 625 1.301e-05 1.464e-10 5.146e-05 1.188e-03 3.726e-07 1.000e+01 4.742e-04
360 627 1.301e-05 1.440e-10 5.165e-05 1.239e-03 3.892e-07 1.000e+01 4.742e-04
361 629 1.301e-05 1.416e-10 5.208e-05 1.273e-03 4.070e-07 1.000e+01 4.742e-04
362 631 1.301e-05 1.400e-10 5.242e-05 1.309e-03 4.248e-07 1.000e+01 4.742e-04
363 633 1.301e-05 1.393e-10 5.272e-05 1.347e-03 4.424e-07 1.000e+01 4.742e-04
364 635 1.301e-05 1.388e-10 5.318e-05 1.388e-03 4.612e-07 1.000e+01 4.742e-04
365 637 1.301e-05 1.387e-10 5.364e-05 1.423e-03 4.789e-07 1.000e+01 4.742e-04
366 639 1.301e-05 1.389e-10 5.326e-05 1.504e-03 4.916e-07 1.000e+01 4.742e-04
367 641 1.301e-05 1.370e-10 5.290e-05 1.579e-03 5.028e-07 1.000e+01 4.742e-04
368 643 1.301e-05 1.360e-10 5.278e-05 1.634e-03 5.112e-07 1.000e+01 4.742e-04
369 645 1.301e-05 1.342e-10 5.272e-05 1.684e-03 5.191e-07 1.000e+01 4.742e-04
370 647 1.301e-05 1.324e-10 5.274e-05 1.725e-03 5.268e-07 1.000e+01 4.742e-04
371 649 1.301e-05 1.308e-10 5.279e-05 1.762e-03 5.346e-07 1.000e+01 4.742e-04
372 651 1.301e-05 1.297e-10 5.291e-05 1.792e-03 5.419e-07 1.000e+01 4.742e-04
Optimization terminated successfully.
`ftol` condition satisfied.
Current function value: 1.301e-05
Constraint violation: 5.419e-07
Total delta_x: 9.170e-02
Iterations: 372
Function evaluations: 651
Jacobian evaluations: 375
Timer: Solution time = 5.35 min
Timer: Avg time per step = 861 ms
==============================================================================================================
Start --> End
Total (sum of squares): 6.632e+01 --> 1.301e-05,
Maximum absolute QFM surface normal field error: 3.970e-02 --> 5.303e-05 (T m^2)
Minimum absolute QFM surface normal field error: 9.255e-18 --> 1.114e-17 (T m^2)
Average absolute QFM surface normal field error: 1.676e-02 --> 4.209e-06 (T m^2)
Maximum absolute QFM surface normal field error: 5.672e-01 --> 7.576e-04 (normalized)
Minimum absolute QFM surface normal field error: 1.322e-16 --> 1.591e-16 (normalized)
Average absolute QFM surface normal field error: 2.394e-01 --> 6.013e-05 (normalized)
Maximum Toroidal Flux: -2.117e-02 --> -3.500e-02 (Wb)
Minimum Toroidal Flux: -2.117e-02 --> -3.500e-02 (Wb)
Average Toroidal Flux: -2.117e-02 --> -3.500e-02 (Wb)
Maximum Toroidal Flux: -2.117e-02 --> -3.500e-02 (normalized)
Minimum Toroidal Flux: -2.117e-02 --> -3.500e-02 (normalized)
Average Toroidal Flux: -2.117e-02 --> -3.500e-02 (normalized)
==============================================================================================================
We can see that after the optimization, the normal field error is quite small
[7]:
plot_2d(qfm_surf, "B*n", field=field, cmap="viridis", log=True);
We also see that the optimized surface is now a good approximation of a flux surface in this vacuum field
[6]:
data = qfm_surf.compute(["R", "Z"], grid=LinearGrid(rho=1.0, theta=0, zeta=0))
fig, ax = plot_surfaces(qfm_surf, rho=1.0, theta=0)
poincare_plot(
field, R0=np.append(data["R"], np.linspace(0.7, 0.8, 9)), Z0=np.zeros(10), ax=ax
);
We can use this QFM surface as the surface for a fixed-boundary DESC equilibrium solve, to find the equilibrium field which matches the vacuum field inside this surface. (Alternatively, one could also perform a free-boundary solve, though we know the surface should not change much from this QFM surface)
[7]:
from desc.equilibrium import Equilibrium
eq = Equilibrium(surface=qfm_surf, Psi=-0.035, L=8, M=6, N=6)
eq.solve(verbose=3, ftol=1e-8);
Building objective: force
Precomputing transforms
Timer: Precomputing transforms = 432 ms
Timer: Objective build = 696 ms
Building objective: lcfs R
Building objective: lcfs Z
Building objective: fixed Psi
Building objective: fixed pressure
Building objective: fixed current
Building objective: fixed sheet current
Building objective: self_consistency R
Building objective: self_consistency Z
Building objective: lambda gauge
Building objective: axis R self consistency
Building objective: axis Z self consistency
Timer: Objective build = 517 ms
Timer: Linear constraint projection build = 2.88 sec
Number of parameters: 628
Number of objectives: 3250
Timer: Initializing the optimization = 4.12 sec
Starting optimization
Using method: lsq-exact
Iteration Total nfev Cost Cost reduction Step norm Optimality
0 1 3.077e-01 1.318e+02
1 2 1.532e-02 2.924e-01 3.456e-01 1.998e+01
2 3 2.502e-04 1.507e-02 8.742e-02 2.972e+00
3 4 1.785e-05 2.324e-04 4.781e-02 1.069e+00
4 6 1.077e-06 1.677e-05 2.240e-02 1.615e-01
5 8 1.693e-07 9.082e-07 1.696e-02 1.260e-01
6 10 8.032e-08 8.894e-08 8.395e-03 4.990e-02
7 12 7.171e-08 8.603e-09 4.109e-03 1.261e-02
8 14 7.046e-08 1.251e-09 2.091e-03 2.995e-03
9 15 6.913e-08 1.335e-09 4.134e-03 9.426e-03
10 16 6.902e-08 1.079e-10 7.997e-03 2.686e-02
11 17 6.534e-08 3.681e-09 1.926e-03 1.536e-03
12 18 6.421e-08 1.132e-09 3.788e-03 4.386e-03
13 19 6.340e-08 8.082e-10 7.313e-03 1.146e-02
14 20 6.113e-08 2.267e-09 7.081e-03 7.488e-03
15 21 5.921e-08 1.917e-09 6.946e-03 5.354e-03
16 22 5.753e-08 1.683e-09 6.875e-03 4.841e-03
17 23 5.604e-08 1.492e-09 6.854e-03 5.263e-03
18 24 5.472e-08 1.323e-09 6.878e-03 6.351e-03
19 25 5.355e-08 1.167e-09 6.924e-03 7.539e-03
20 26 5.253e-08 1.022e-09 6.953e-03 8.220e-03
21 27 5.163e-08 8.997e-10 6.947e-03 7.957e-03
22 28 5.082e-08 8.046e-10 6.948e-03 6.748e-03
23 29 5.012e-08 7.085e-10 7.009e-03 4.981e-03
24 30 4.952e-08 5.909e-10 7.125e-03 3.173e-03
25 31 4.906e-08 4.650e-10 7.249e-03 1.706e-03
26 33 4.878e-08 2.789e-10 3.656e-03 2.598e-04
27 34 4.856e-08 2.176e-10 7.385e-03 5.874e-04
28 36 4.839e-08 1.736e-10 3.715e-03 1.522e-04
29 37 4.822e-08 1.659e-10 7.458e-03 8.609e-04
30 39 4.809e-08 1.381e-10 3.740e-03 2.143e-04
31 40 4.798e-08 1.014e-10 7.495e-03 1.038e-03
32 41 4.786e-08 1.218e-10 7.474e-03 1.086e-03
33 42 4.779e-08 7.661e-11 7.516e-03 1.135e-03
34 43 4.774e-08 4.621e-11 7.517e-03 1.182e-03
35 44 4.769e-08 4.505e-11 6.471e-03 8.956e-04
36 45 4.764e-08 5.289e-11 6.638e-04 5.596e-05
37 46 4.764e-08 6.661e-13 8.618e-04 1.944e-05
38 47 4.764e-08 2.355e-13 1.423e-04 7.326e-06
39 48 4.764e-08 6.031e-14 2.527e-04 2.656e-06
40 49 4.764e-08 1.945e-14 3.577e-05 1.134e-06
41 50 4.764e-08 6.202e-15 8.630e-05 4.266e-07
42 51 4.764e-08 2.045e-15 1.261e-05 2.039e-07
43 52 4.764e-08 6.851e-16 3.179e-05 8.505e-08
44 53 4.764e-08 2.363e-16 6.586e-06 4.203e-08
Optimization terminated successfully.
`ftol` condition satisfied.
Current function value: 4.764e-08
Total delta_x: 4.081e-01
Iterations: 44
Function evaluations: 53
Jacobian evaluations: 45
Timer: Solution time = 34.1 sec
Timer: Avg time per step = 759 ms
==============================================================================================================
Start --> End
Total (sum of squares): 3.077e-01 --> 4.764e-08,
Maximum absolute Force error: 7.821e+04 --> 5.930e+01 (N)
Minimum absolute Force error: 1.817e+00 --> 8.899e-05 (N)
Average absolute Force error: 4.273e+03 --> 1.593e+00 (N)
Maximum absolute Force error: 1.823e-01 --> 1.383e-04 (normalized)
Minimum absolute Force error: 4.236e-06 --> 2.075e-10 (normalized)
Average absolute Force error: 9.963e-03 --> 3.714e-06 (normalized)
R boundary error: 0.000e+00 --> 2.190e-19 (m)
Z boundary error: 0.000e+00 --> 4.375e-19 (m)
Fixed Psi error: 0.000e+00 --> 0.000e+00 (Wb)
Fixed pressure profile error: 0.000e+00 --> 0.000e+00 (Pa)
Fixed current profile error: 0.000e+00 --> 0.000e+00 (A)
Fixed sheet current error: 0.000e+00 --> 0.000e+00 (~)
==============================================================================================================
Finally, we compare the solve equilibrium’s flux surfaces with the vacuum field Poincare plot and confirm that indeed, the equilibrium matches the vacuum field as expected.
[8]:
grid = LinearGrid(L=5)
data = eq.compute(["R", "Z"], grid=grid)
fig, ax = plot_surfaces(eq, rho=grid.nodes[:, 0], rho_lw=2)
poincare_plot(field, R0=data["R"], Z0=data["Z"], ax=ax, NFP=eq.NFP, size=10);
