# Stellarator coil design using cubic splines for improved access on the
outboard side

Nicola Lonigro1, Caoxiang Zhu2 1 Department of Physics and Astronomy,
University of Padua, Via Francesco Marzolo 8, Padua (PD), Italy  2 Princeton
Plasma Physics Laboratory, Princeton University, P.O. Box 451, New Jersey
08543, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In recent years many efforts have been undertaken to simplify coil designs for
stellarators due to the difficulties in fabricating non-planar coils. The
FOCUS code removes the need for a winding surface and represents the coils as
arbitrary curves in 3D. In the following work, the implementation of a spline
representation for the coils in FOCUS is described, along with the
implementation of a new engineering constraint to design coils with a
straighter outer section. The new capabilities of the code are shown as an
example on HSX, NCSX, and a prototype quasi-axisymmetric reactor-sized
stellarator. The flexibility granted by splines along with the new constraint
will allow for stellarator coil designs with improved accessibility and
simplified maintenance.

Keywords: stellarator, coil design, optimization, cubic spline

##  1 Introduction

Stellarators are 3D plasma configurations where both the poloidal and toroidal
field is generated by the magnetic coils thus avoiding the instabilities
driven by large plasma currents [1]. To generate the required magnetic field,
the coils need to twist around the plasma and this leads to complex shapes
which are difficult to manufacture. The typical approach for designing an
optimized stellarator consists of determining a magnetic configuration with
good confinement properties, which can be achieved using the STELLOPT suit of
codes [2], and then design a coil set able to reproduce the desired magnetic
configuration.

Stellarator Coil design was initially performed in NESCOIL[3] by solving for
the current potential on a winding surface which was then discretized in a
finite number of filaments. Later, this approach was improved by the codes
NESVD [4] and REGCOIL[5]. The need to directly control the shape of the coils
lead to the creation of optimization codes such as ONSET [6], COILPLOT [7],
and COILPLOT++ [8]. In this type of code, the coils are represented as
filaments on a toroidal winding surface and a non-linear optimization is
performed on this surface to obtain the final shape for the set of coils. A
drawback of this kind of approach is that the best winding surface to use for
a specific problem is not known a priori and the use of an unsuitable winding
surface will lead to suboptimal coils. FOCUS [9] is a numerical code able to
design coils without the need for a winding surface but instead describing the
coils directly as arbitrary curves in 3D, making it able to sample a wider
range of possible configurations. There are also other recent advances in
stellarator coil design to address specific problems [10, 11, 12].

Large access on the outboard side is vital for remote maintenance in future
fusion reactors. This is even more critical for stellarators as non-planar
coils usually have limited space. COILOPT++ showed the improvements obtained
by using a spline representation for the coils. It has the ability to
implement local constraints in real space in a straightforward way. For
example, it is possible to fix the outer part of the coils to be circular arcs
to create space for ports and allow easier access to the modules of the
blanket[13]. In previous versions of FOCUS the coils were described using a
Fourier representation, which is a global representation and so less suited to
implement constraints on just a portion of the coils. Also, the Fourier
representation is not efficient in describing straight sections. In this
paper, we add a spline representation for the coils, along with a new
engineering constraint to design coils with straighter outer sections. The
differences between this work and the previous COILOPT++ approach are as
follows. COILOPT++ represents coils as planar curves on a given winding
surface, while we follow the logic of FOCUS and adopt a cubic B-spline
representation. The coils can be freely moved in 3D space and thus we could
explore more possible designs. In addition, we use analytical derivatives
during the coil optimization to improve both the speed and accuracy. With this
improvement, FOCUS is now able to design coils with straighter outboard parts
which is extremely favorable for stellarator rectors.

This paper is organized as follows. In section 2 a brief overview of spline
representations is given and in section 3 the optimization procedure is
described. The improved coils for the HSX and NCSX stellarators along with a
quasi-axisymmetric reactor-like configuration obtained using the new additions
to the code are described in section 4. The conclusions and further
improvements to the code are discussed in section 5.

##  2 Cubic B-spline representation

When describing a curve different parametrizations can be used with some being
more appropriate for a certain problem than others. A parametrization is a set
of functions that describe the coordinates of the curve in real space as a
function of a parameter $x$. In a polynomial representation the coordinates of
the curve are a polynomial function of the parameter. While an arbitrarily
complex curve can be described using an high enough order for the polynomial,
the resulting parametrization can be unstable and small changes in the
polynomial coefficients can lead to large variations in the represented curve.
A spline representation avoids this problem by stitching together many low-
order polynomials in series. This allows representing an arbitrary complex
curve by increasing the number of polynomial sections joined together instead
of the degree of each polynomial. The points where the polynomials meet in the
parameter space are called knot points and the interval between two knot
points is called knot interval. Many possible spline representations exist[14]
and the most appropriate one depends on the specific problem at hand. Using a
spline of order 3, only two of the following properties can be achieved:
$\mathbb{C}^{2}$ continuity, interpolation of all the control points or local
control. For this work, a local representation is preferred as it allows to
modify one section of the curve while leaving the rest of the coil unchanged.
This leads to discarding Natural, Hermite, Catmull-Rom and Cardinal cubic
splines due to their lack of locality while Bézier curves were discarded due
to their lack of $\mathbb{C}^{2}$ continuity at the connection points between
neighboring sections. B-splines were selected due to both their locality and
$\mathbb{C}^{2}$ continuity even though the curves do not pass through the
control points used to define them, slightly complicating their
interpretation. In the case of B-splines, the coefficients used in the
definition take the name of control points $C_{i}$ and by changing their
coordinates it is possible to modify the curve. Such a representation is
perfectly suited to represent linear segments as to a series of control points
aligned in real space it immediately corresponds a straight segment of the
corresponding curve.

A B-spline of order $k$ is defined using a set of basis functions as

$\displaystyle S(x)=\sum_{i}C_{i}\mathcal{B}_{i,k}(x),$ (1)

where $C_{i}$ is the i-th control point of the curve and
$\mathcal{B}_{i,k}$($x$) is the i-th basis function of order k. A basis
function of order $k$ will have non-zero values over $k+1$ knot intervals and
so the lower is the degree of the basis functions, the more “local” the effect
of changing the coordinates of the respective control point. When crossing a
knot point or equivalently when passing from a knot interval to the next, one
of the control points stops influencing the next section of the curve while a
new control point takes its place. The values of the basis functions of order
$k$ can be computed iteratively using the Cox-De Boor algorithm [15]:

$\displaystyle\mathcal{B}_{i,0}(x)$ $\displaystyle=\begin{cases}1,\hskip
20.00003ptt_{i}<x<t_{i+1}\\\ 0,\hskip 20.00003ptotherwise\end{cases}$ (2)
$\displaystyle\mathcal{B}_{i,k}(x)$
$\displaystyle=\frac{x-t_{i}}{t_{i+k}-t_{i}}\mathcal{B}_{i,k-1}(x)+\frac{t_{i+k+1}-x}{t_{i+k+1}-t_{i+1}}\mathcal{B}_{i+1,k-1}(x),$
(3)

where $t_{i}$ is an element of the knot vector $(t_{1},t_{2}...t_{m})$ in
which the knot points have been placed in ascending order.

In general a B-spline defined with $n$ control points requires $n+k+1$ knot
points in the knot vector ($t_{1},t_{2}...t_{n+k+1}$) but for a closed curve
the first $k$ control points must be overlapping with the last $k$ ones. In
FOCUS third order (cubic) splines have been implemented and a plot of the
third order basis functions corresponding to different control points are
shown in figure 1.

Figure 1: Third order basis functions corresponding to different control
points. For every point inside a knot interval (e.g. the dotted line) there
are only four basis functions with non-zero value.

It is worth noting how the values of the basis functions at every point x
always sum up to 1 and that every basis function is non-zero only over 4
contiguous knot intervals. Cubic splines are a standard in the computer
graphics field[16] as it is the lowest order that is still able to have a flex
point and in general lower order curves are preferred as they are more stable
to oscillations due to a change in the control points coordinates. During the
optimization the derivatives of the coordinates with respect to the parameter
$x$ are required, so the derivatives of the cubic basis functions were
analytically computed and implemented into the code. The optimization is
performed on the coordinates of the control points, while leaving the points
used in the discretization of the curve at a constant value of the parameter
$x$ (with $x$ $\in[t_{1},t_{n+k+1}]$ ). Evaluating the points along the curve
always at the same values of $x$ in parameter space allows to compute the
basis functions and their derivatives only once, during the initialization,
and to treat them as constants during the execution.

##  3 New cost function straightening coils and optimization procedures

Starting from an initial guess for the coils in B-spline representation (that
can be obtained by manually manipulating the control points or fitting to
known shapes), the code optimizes the coordinates of the control points and
the current in each coil using optimization algorithms. The target function to
be minimized consists of different cost functions that are multiplied by user-
defined parameters called weights and summed. The first-order derivatives are
computed analytically.

###  3.1 Existing FOCUS cost functions used in this work

FOCUS has several cost functions covering both physics requirements and
engineering constraints. All of the cost functions (except the one associated
with the Fourier representation) have been reformed to use the spline
representation. While they all have specific purposes, we mainly use some cost
functions in this work. In particular, the relevant cost functions are:

  * •

Normal field error The following cost function implements the main physical
requirement for the coils: generating a target magnetic field. In particular,
FOCUS will try to minimize the difference between the normal field generated
by the coils and a target normal field on the boundary. This can be expressed
as:

$f_{B}=\int_{S}\frac{1}{2}(\bi{B}_{coils}\cdot\bi{n}-B_{n}^{target})^{2}\differential{a},$
(4)

where $\textbf{B}_{coils}$ is the field generated by the coils, $\mathbf{n}$
the normal vector of the boundary and $B_{n}^{target}$ the target normal
magnetic field assessed from equilibrium codes. For example, if we choose the
last-closed-flux-surface as the boundary on which the total normal field is
zero, we have $B_{n}^{target}=-\mathbf{B}_{plasma}\cdot\mathbf{n}$.

  * •

Length To avoid trivial solutions where the coils tend to move away from the
plasma to reduce the field at the surface, a constraint on the length is
present in FOCUS. For the optimizations in this work the quadratic form for
the length penalty has been used

$f_{L}=\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}\frac{1}{2}\frac{(L_{i}-L_{i,o})^{2}}{L_{i,o}^{2}},$
(5)

where $N_{c}$ is the number of coils, $L_{i,o}$ is the target length and
$L_{i}$ is the length of the $i$-th coil.

  * •

Curvature To design coils that are more easily manufacturable, a constraint to
reduce the curvature of the coils is present. While different forms have been
implemented in FOCUS [17], the one used for the optimizations in this work is
the constrained curvature penalty given by:

$f_{\kappa}=\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}\int_{0}^{2\pi}H_{\kappa_{o}}(\kappa_{i})(\kappa_{i}-\kappa_{o})^{\alpha}dt,$
(6)

where $H_{\kappa_{o}}(\kappa_{i})$ is the step function, $\kappa_{o}$ and
$\alpha$ are user-defined parameters and $\kappa_{i}$ is the curvature at a
point defined as
$\kappa_{i}=\lVert\mathbf{x}^{\prime}\times\mathbf{x}^{\prime\prime}\rVert/\lVert\mathbf{x}^{\prime\prime\prime}\rVert$.
$\kappa_{o}$ has the meaning of a maximum allowed curvature and the cost
function has no effect on coil segments with a curvature smaller than
$\kappa_{o}$.

###  3.2 Cost function for straight sections

The locality of the spline representation was used to constraint the design to
favor straighter outer coils. In particular, a cost function with the same
type of penalty as the curvature cost function but applied only to a subset of
the points making up the curve was implemented. For example, considering the
constrained curvature objective function, the new cost function is given by

$f_{S}=\\\
\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}\int_{0}^{2\pi}W(t)H_{\kappa_{s,0}}(\kappa_{i})(\kappa_{i}-\kappa_{s,o})^{\alpha_{s}}dt\
,$ (7)

where $W(t)$ is a weight function, $H_{\kappa_{s,0}}$ is the step function and
$\kappa_{s,o}$ and $\alpha_{s}$ are user-defined parameters. In this case
$\kappa_{s,o}$ is the maximum curvature allowed on the outboard side while
$\alpha_{s}$ determines the strength of the scaling. The weight function is
defined as

$W(t)=\begin{cases}1,\hskip 20.00003ptP_{xy}(t)>P_{m}+\beta P_{d}\\\ 0,\hskip
20.00003ptotherwise\end{cases}$ (8)

where $P_{m}$ is the mean squared distance of the coil projection in the x-y
plane from a user-defined point and $P_{d}$ is a measure for half the maximum
projection in the x-y plane of the coil radius. Considering the distance of
the projection of a point along the coil in the x-y plane from the user-
defined point ($x_{0},y_{0}$)

$P_{xy}(t)=(x_{i}(t)-x_{0})^{2}+(y_{i}(t)-y_{0})^{2}\ ,$ (9)

they are defined as

$\displaystyle P_{m}=\overline{P_{xy}}\ ,$ $\displaystyle
P_{d}=\frac{\max(P_{xy})-P_{m}}{2}\ .$ (10)

$\beta$ is a user-defined parameter that can be used to tweak the section of
the coil affected by the optimization. With a value $\beta=0$ the section of
the coil with a projected distance greater than the mean value for that coil
will all be affected while for a value $\beta=2$ the new cost function will
not be applied to any point. While it may be tempting to use the three-
dimensional distance to determine the outer section of the coil instead of the
projection along the x-y plane, such a metric is affected by oscillations in
the z-direction which may lead to disjointed sections of the coil begin
affected and so this metric has been adopted instead.

###  3.3 Quick implementation of analytical derivatives using functional
derivatives

For general optimization problems, it is useful to have the information about
the derivatives, especially when one wants to use gradient-based optimization
algorithms. New methods, like the adjoint method [18] and automatic
differentiation [19], have been brought into the field of stellarator
optimization. Here, we introduce a different approach using the functional
derivative.

The derivation of functional derivatives for FOCUS cost functions was
initially introduced in [20]. Later, it was extended to the variation of
surface in [21]. Here, we will briefly revisit the derivation using the cost
function $f_{B}$ as an example. The magnetic field generated by the coils is
calculated using the Biot-Savart law,

$\mathbf{B}_{coils}=\displaystyle\sum_{i=1}^{N_{c}}\mathbf{B}_{i}=\displaystyle\sum_{i=1}^{N_{c}}\frac{\mu_{0}I_{i}}{4\pi}\int_{C}\frac{\differential{\mathbf{l}}_{i}\times\mathbf{r}}{r^{3}}\
,$ (11)

where $\differential{\mathbf{l}}_{i}=\mathbf{x}_{i}^{\prime}\differential{t}$,
$\mathbf{r}=\bar{\mathbf{x}}-\mathbf{x}_{i}$ ($\mathbf{x}_{i}$ a point on the
$i$-th coil, $\bar{\mathbf{x}}$ the evaluation point on the surface). $f_{B}$
is a functional of the coil shapes. If we vary the geometry of the $i$-th
coil, the first variation is

$\delta f_{B}=\displaystyle\int_{S}(\bi{B}_{coils}\cdot\bi{n}-B_{n}^{target})\
\delta\mathbf{B}_{i}\cdot\mathbf{n}\differential{a}\ .$ (12)

The variation of the magnetic field is calculated as

$\delta\mathbf{B}_{i}=\int_{C}(\delta\mathbf{x}_{i}\times\mathbf{x}^{\prime}_{i})\cdot\mathbf{R}_{i}\differential{t}\
,$ (13)

where $\mathbf{R}=3\mathbf{r}\mathbf{r}/r^{5}-\mathbf{I}/r^{3}$ ($\mathbf{I}$
is a unit dyadic). So far, all the equations are general and we haven’t
touched any specific coil parameterization. To calculate the derivatives of
$f_{B}$ with respect to the degrees of freedom (e.g. Fourier coefficients in
the Fourier representation or control points in the spline representation), we
can simply apply the chain rule,

$\partialderivative{f_{B}}{\mathbf{X}}=\frac{\delta
f_{B}}{\delta\mathbf{x}}\partialderivative{\mathbf{x}}{\mathbf{X}}\ .$ (14)

Here, $\mathbf{X}$ denotes the coil parameters. The first term on the right-
hand side of Eq.(14) is parameterization-free and the second term is
parameterization-specific. When we change the coil representation, we only
need to modify the second term, which is usually simple. For example, in this
work, the derivative of the coil shape with respect to the position of a
control point is

$\partialderivative{\mathbf{x}}{C_{i}}=\mathcal{B}_{i,k}\ .$ (15)

Normally, one can directly differentiate functions and write down the
derivatives with respect to parameters. This process, which is usually called
“symbolic differentiation”, will compute derivatives accurately and fast.
However, for complex functions, like the ones we used in this paper, the
derivatives have many terms and are thus difficult to be numerically
implemented. When changing to a new parameterization, one has to derive the
equations and implement the derivatives one more time, which requires non-
trivial effort. In this paper, we split the shape part and the parameter part
using functional derivatives, like in Eq.(14). By doing so, when changing to a
new parameterization, the only parts that have to be re-implemented are the
derivatives of the coil shape with respect to the coil parameters. The method
will be extremely useful for situations with multiple parameterizations (for
curves, surfaces, and other geometries).

##  4 Numerical applications

Now, we can use the new code to design coils with improved access on the
outboard side. In the following, the new representation and cost function were
applied to design an improved set of coils for the Helically Symmetric
eXperiment (HSX) and the National Compact Stellarator Experiment (NCSX)
stellarators. Furthermore, we are going to design a new set of coils for a
prototype quasi-axisymmetric stellarator (QAS) reactor, which has considerably
large outboard access. Due to the toroidal periodicity and stellarator
symmetry, FOCUS only needs to design a set of coils in a half period and the
remaining coils are then obtained by reflection and repetition of the designed
set.

Tests using a variety of weights were performed, studying the variation of the
surface-normalized normal field error,
$<{B_{n}}>=\int\absolutevalue{B_{n}}/{B}\differential{S}/{A}$, which is taken
as the main figure of merit in the rest of the discussion along with the
reduction of the curvature on the outboard side. The optimized coil sets for
the first two experiments were obtained by trying to minimize the curvature on
the outboard side while still maintaining the same maximum curvature along the
entire coil present in the coils of the real experiment and a $<B_{n}>$ lower
than the value corresponding to the real coils. For the QAS reactor-like
configuration, the quality of the coils was determined by how closely the flux
surfaces resembled the target boundary. The best results found by exploring
the parameter space are presented in the rest of the section.

###  4.1 HSX

HSX [22] is a stellarator optimized for quasi-helical symmetry built at the
University of Wisconsin-Madison. It has a major radius of 1.2 m with 4 field
periods and 12 magnets per period divided in 2 sets of 6 magnets related by
the stellarator symmetry. In this section, the coils obtained using the spline
representation are compared with the coils presented in [17], obtained using
the constrained curvature function and already shown to be an improvement over
the actual coils.

In figure 2 a 3D representation of the plasma surface and the two coil sets is
shown. FOCUS optimizes the coils as one-dimensional filaments but they are
shown as finite-build coils to have a more realistic view of what they would
look like once built. A larger straightening effect can be seen on the second
and third coils from the left in the front view. Some characteristics of this
coil set are compared with the reference one in table 1 and it can be noticed
that this configuration is able to have a straighter section on the outside
while still providing a reasonable $<B_{n}>$. Here, we are not trying to match
$<B_{n}>$ from the reference coil set, instead we stopped when $<B_{n}>$ was
lower than the value of the real coils ($<B_{n}>=4.87\times 10^{-3}$). In
figure 4 a comparison of the flux surfaces obtained from field line tracing is
shown and the new configuration can follow the target boundary as well as the
reference configuration. In appendix A, the quality of the magnetic field for
the new coil-set is compared to the reference one to show that the slight
increase of $<B_{n}>$ does not affect significantly the quasi-helical
symmetry.

Figure 2: Comparison of the coils designed with splines for HSX (blue) and the ones shown in [17] (violet). The golden area of the coil is the part affected by the straightening cost function. The color of the surface represents the $B_{n}$ at that position. Table 1: Comparison of reference and optimized coils characteristics for HSX Coil Set | Ref. | Opt.
---|---|---
Maximum curvature | 12.3 m-1 | 12.3 m-1
Average curvature | 6.61 m-1 | 4.39 m-1
Average Length | 2.38 m | 2.23 m
$<B_{n}>$ | 1.15$\times 10^{-3}$ | 4.52$\times 10^{-3}$
Minimum coil-coil |  |
distance on outboard side | 0.076 m | 0.090 m
Table 2: Comparison of outboard side mean and max curvature $\kappa$ [m-1] for HSX Coil | Ref. $\overline{\kappa}$ | Opt. $\overline{\kappa}$ | Ref. max $\kappa$ | Opt. max $\kappa$
---|---|---|---|---
1 | 3.27 | 2.57 | 4.53 | 3.26
2 | 5.15 | 2.42 | 12.02 | 3.17
3 | 6.78 | 3.24 | 12.27 | 5.9
4 | 7.07 | 4.76 | 12.29 | 6.79
5 | 6.44 | 4.67 | 12.24 | 6.98
6 | 5.59 | 4.97 | 8.68 | 6.00

In figure 3 the curvature along each coil is compared with the reference set.
The improvement on the outer part is evident and the values of the mean and
maximum curvature in the outer regions are compared in table 2. The optimized
configuration has been obtained using the same $\kappa_{0}=12.3$ m-1 of the
real coils but restricting the maximum curvature on the outer side to be
$\kappa_{s}=3$ m-1 and using $\beta=0.5$

Figure 3: Comparison of the curvature $\kappa$ for each coil in HSX along the
length of the coil L. The gray areas indicate the regions of the optimized
coils affected by the straight-out cost function. Coil 1 represents the left-
most coil in Figure 2 and coil 6 the right-most coil. Figure 4: Comparison
between the Poincarè plot obtained from the coils in [17](red) and the
optimized coils (blue) for $\phi$ = 0 in HSX using the FOCUS field line
tracing algorithm.

###  4.2 NCSX

NCSX is a quasi-axisymmetric stellarator that was partly built at the
Princeton Plasma Physics Laboratory but was unfortunately canceled [23]. The
target equilibrium, labeled as C09R00, has a major radius of 1.44 m and a
minor radius of 0.32 m. It has 3 toroidal field periods and the main coil
system consists of 12 modular coils in 3 unique shapes. The new spline
representation has also been applied successfully to improve the design for
NCSX. In figure 5 the improved coils are compared to the ones envisioned for
the experiment and the improvement can be noticed also by looking at the
curvature plots for the coils shown in figure 6. In figure 7 the flux surfaces
obtained using the free-boundary VMEC [24] with the two coil sets are compared
with the fixed-boundary target. As expected from similar $<B_{n}>$ values, the
two configurations can follow the target surface similarly well. In this case,
the VMEC code has been used to include the effect of the non-zero target
normal field $T_{B_{n}}$ necessary to describe this plasma surface. The
general characteristics of the two coil sets are compared in table 3 while
their mean and average curvature of the outer section are compared in table 4.
While the minimum distance between two coils on the outer side is slightly
lower in the optimized case, this minimum distance is reached at the edge of
the region affected by the cost function. This upper “junction point” between
the inner and outer side of the coils lies above the plasma and so even if the
coils were to be closer in that position, this would not affect the placement
of ports around $z\approx 0$.

In this case, the optimization allowed for both a significantly simpler
geometry as well as an improvement of the normal field error while leaving the
inner side basically unmodified.

Figure 5: Comparison of the coils designed with splines (blue) and the ones
designed originally for the NCSX experiment (violet). The golden area of the
coil is the part affected by the straightening cost function. The color of the
surface represents the $B_{n}$ at that position.

Figure 6: Comparison of the curvature $\kappa$ for each coil in NCSX. The gray areas indicate the regions of the optimized coils affected by the straight-out cost function. Coil 1 represents the left-most coil in Figure 5 and coil 3 the right-most coil. Figure 7: Comparison between the VMEC solutions obtained from the coils initially designed for the experiment (red) and the optimized coils (blue) for $\phi$ = 0 (full lines) and $\phi=\frac{\pi}{3}$ (dotted lines) in NCSX. Table 3: Comparison of reference and optimized coils characteristics for NCSX Coil Set | Ref. | Opt.
---|---|---
Maximum curvature | 14.8 m-1 | 14.8 m-1
Average curvature | 2.36 m-1 | 2.28 m-1
Average Length | 6.81 m | 6.77 m
$<B_{n}>$ | 5.55 $\times 10^{-3}$ | 4.14 $\times 10^{-3}$
Minimum coil-coil |  |
distance on outboard side | 0.29 m | 0.26 m
Table 4: Comparison of outboard side mean and max curvature $\kappa$ [m-1] for NCSX. Coil | Ref. $\overline{\kappa}$ | Opt. $\overline{\kappa}$ | Ref. max $\kappa$ | Opt. max $\kappa$
---|---|---|---|---
1 | 1.38 | 1.05 | 2.90 | 1.19
2 | 1.81 | 1.23 | 4.74 | 2.16
3 | 3.28 | 1.73 | 14.77 | 5.36

###  4.3 QAS reactor

Having verified the new capabilities of the code by comparing its results to
known configurations, the coils designed by FOCUS for a prototype QAS reactor
are presented in the rest of the section. The target configuration comes from
a study following [13] and is obtained using a free-boundary STELLOPT
optimization [2]. Some main parameters are $N_{fp}=3$, $R=9.32$ m, $a=1.55$ m,
$V=444.13\mathrm{m}^{3}$, and $<B>=6.41$ T.

In figure 8 a 3D plot of the coils and the plasma surface is shown. The coils
clearly appear to lack any rapid oscillations on the outer side. Most of the
outer section is approximately straight and only curving back towards the
bottom of the coils to join the inner side of the coils. This will be
extremely suitable for remote access and blanket replacement. In figure 10 the
curvature for every coil is plotted and its suppression in the outer region is
evident. In the third coil, which corresponds to the region where the plasma
twists at the end of a period, the larger value of the curvature in the outer
region is due to the necessity of the coil to follow the curvature of the
plasma. Increasing the strength of the constraint further leads to straighter
coils also in this region but at the cost of having flux surfaces
significantly different from the target boundary.

In figure 9 the free-boundary VMEC results are shown and the flux surfaces
follow the target surface closely along the entire field period, here
represented by the section at the start of a period ($\phi=0$) and at its
middle point ($\phi={\pi}/{3}$). This suggests the coil set can produce the
desired magnetic field supporting the target MHD equilibrium.

Figure 8: 3D plot of the coils obtained by FOCUS for the prototype QAS
reactor. The golden area of the coil is the part affected by the straightening
cost function. The color of the surface represents the $B_{n}$ at that
position. Figure 9: VMEC solutions for the coils designed using FOCUS for the
prototype QAS reactor for $\phi=0$ and $\phi={\pi}/{3}$

Figure 10: Curvature $\kappa$ for each coil in the QAS prototype reactor. The
gray areas indicate the regions of the optimized coils affected by the
straight-out cost function. Coil 1 represents the left-most coil in Figure 8
and coil 3 the right-most coil.

##  5 Conclusions and future work

A suitable spline representation for the coils in FOCUS was found and
implemented in the code. This new representation allows for a more intuitive
understanding of the optimization parameters and to constrain the resulting
coils directly in real space. A new cost function was implemented to include
an engineering constraint for straighter outer sections. These straighter
sections have the potential to simplify the designs of future reactors and
increase accessibility on the outboard side, leaving space for ports and
direct access to the modules of the blanket, while still providing good
confinement.

The functional derivative used in this paper provides an easy approach to
computing analytic derivatives with multiple parameterizations. The functional
derivative term only involves the geometry and has a close connection to the
so-called “shape gradient” [25]. Using the high flexibility that can be
obtained by varying the parameters, the optimization procedure can be tweaked
to design coils that balance the physical and engineering constraints for the
experiment under study.

In the future, the use of advanced optimization techniques implemented for the
Fourier representation [26] and the coil sensitivity analysis [20] can be
extended to the spline one. The advantages of a local representation could be
exploited even further to specify the allowed interval for each control point
coordinate and so impose arbitrary constraints on the shape of coils according
to the user’s needs, for example setting specific parts to be exactly straight
and optimizing only the remaining part of the coils.

The authors thank T. Kruger for providing the HSX data, N. Pomphrey & T. Brown
for providing the QAS reactor data. This work was made possible by funding
from the Department of Energy for the Summer Undergraduate Laboratory
Internship (SULI) program. This work is supported by the U.S. DOE Contract No.
DE-AC02-09CH11466.

##  Appendix A Quasi-helical symmetry in HSX with the optimized coils

Figure 11: Normal magnetic field produced by the reference coil set (left) and
the optimized coils (right) in HSX. $\theta$ and $\phi$ are the angles
parameterizing the surface. Figure 12: Non-helically symmetric fourier modes
of the magnetic field in HSX

To verify the quality of the magnetic field produced by the new coil set, the
normal magnetic field to the plasma surface is shown for the new configuration
and the reference one in figure 11. In the new configuration a more visible
coil ripple is present along with a slight increase of the normal field error.
However, as shown in figure 4, the new coil set is still able to reproduce
flux surfaces that follow the target surface. Here, we further check the
quality of quasi-helical symmetry, which is one of the main design goals. We
compare the magnetic field components in the Boozer coordinates from the free-
boundary VMEC equilibria. As shown in figure 12, where the Fourier modes of
the magnetic field different from the helical (1,1) mode are summed up, a good
quasi-helical symmetry is maintained in both equilibria and the difference
between the two configurations is negligible compared to the total value of
the field.

## References

## References

  * [1] Freidberg J 1987 Ideal magnetohydrodynamics (Plenum Press)
  * [2] Lazerson S, Schmitt J, Zhu C, Breslau J and STELLOPT Developers A 2020 Stellopt, version 2.7.5
  * [3] Merkel P 1987 Nuclear Fusion 27 867–871
  * [4] Pomphrey N, Berry L, Boozer A, Brooks A, Hatcher R, Hirshman S, Ku L P, Miner W, Mynick H, Reiersen W, Strickler D and Valanju P 2001 Nuclear Fusion 41 339–347
  * [5] Landreman M 2017 Nuclear Fusion 57 046003
  * [6] Drevlak M 1998 Fusion Technology 33 106–117
  * [7] Strickler D J, Berry L A and Hirshman S P 2002 Fusion Science and Technology 41 107–115
  * [8] Gates D, Boozer A, Brown T, Breslau J, Curreli D, Landreman M, Lazerson S, Lore J, Mynick H, Neilson G, Pomphrey N, Xanthopoulos P and Zolfaghari A 2017 Nuclear Fusion 57 126064
  * [9] Zhu C, Hudson S R, Song Y and Wan Y 2017 Nuclear Fusion 58 016008
  * [10] Lobsien J F, Drevlak M and Sunn Pedersen T 2018 Nuclear Fusion 58 106013
  * [11] Yamaguchi H 2019 Nuclear Fusion 59 104002
  * [12] Giuliani A, Wechsung F, Cerfon A, Stadler G and Landreman M 2020 arXiv 2010.02033
  * [13] Brown T, Breslau J, Gates D, Pomphrey N and Zolfaghari A 2015 Engineering optimization of stellarator coils lead to improvements in device maintenance 2015 IEEE 26th Symposium on Fusion Engineering (SOFE) pp 1–6
  * [14] Bartels R H, Beatty J C and Barsky B A 1987 An Introduction to Splines for Use in Computer Graphics $\&$ Geometric Modeling (Morgan Kaufmann Publishers Inc.)
  * [15] De Boor C 1978 A Practical Guide to Splines (Springer Verlag)
  * [16] Glassner A S (ed) 1990 Graphics Gems (Academic Press Professional, Inc.)
  * [17] Kruger T G, Zhu C, Bader A, Anderson D T and Singh L 2021 Journal of Plasma Physics 87 175870201
  * [18] Paul E, Landreman M, Bader A and Dorland W 2018 Nuclear Fusion 58 076015
  * [19] McGreivy N, Hudson S and Zhu C 2021 Nuclear Fusion 61 026020
  * [20] Zhu C, Hudson S R, Lazerson S A, Song Y and Wan Y 2018 Plasma Physics and Controlled Fusion 60 054016
  * [21] Hudson S R, Zhu C, Pfefferlé D and Gunderson L 2018 Physics Letters, Section A: General, Atomic and Solid State Physics 382 2732–2737
  * [22] Almagri A F, Anderson D T, Anderson F S, Probert P H, Shohet J L and Talmadge J N 1999 IEEE Transactions on Plasma Science 27 114–115
  * [23] Neilson G H, Gruber C O, Harris J H, Rej D J, Simmons R T and Strykowsky R L 2010 IEEE Transactions on Plasma Science 38 320–327
  * [24] Hirshman S P, van RIJ W I and Merkel P 1986 Computer Physics Communications 43 143–155
  * [25] Landreman M and Paul E J 2018 Nuclear Fusion 58 076023
  * [26] Zhu C, Hudson S R, Song Y and Wan Y 2018 Plasma Physics and Controlled Fusion 60 065008