Direct stellarator coil optimization for nested magnetic surfaces with precise quasi-symmetry Andrew Giuliani1aFlorian Wechsung1Antoine Cerfon1Matt Landreman2and Georg Stadler1

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Direct stellarator coil optimization for nested magnetic surfaces with precise

quasi-symmetry

Andrew Giuliani,1, a) Florian Wechsung,1Antoine Cerfon,1Matt Landreman,2and Georg Stadler1

1)Courant Institute of Mathematical Sciences New York University New York,

N.Y. 10012-1185

2)Institute for Research in Electronics and Applied Physics University of Maryland College Park, Maryland 20742,

USA

(Dated: 15 March 2023)

We present a robust optimization algorithm for the design of electromagnetic coils that generate vacuum

magnetic ﬁelds with nested ﬂux surfaces and precise quasi-symmetry. The method is based on a bilevel

optimization problem, where the outer coil optimization is constrained by a set of inner least-squares op-

timization problems whose solutions describe magnetic surfaces. The outer optimization objective targets

coils that generate a ﬁeld with nested magnetic surfaces and good quasi-symmetry. The inner optimization

problems identify magnetic surfaces when they exist, and approximate surfaces in the presence of magnetic

islands or chaos. We show that this formulation can be used to heal islands and chaos, thus producing coils

that result in magnetic ﬁelds with precise quasi-symmetry. We show that the method can be initialized with

coils from the traditional two stage coil design process, as well as coils from a near axis expansion optimiza-

tion. We present a numerical example where island chains are healed and quasi-symmetry is optimized up

to surfaces with aspect ratio 6. Another numerical example illustrates that the aspect ratio of nested ﬂux

surfaces with optimized quasi-symmetry can be decreased from 6 to approximately 4. The last example shows

that our approach is robust and a cold-start using coils from a near-axis expansion optimization.

I. INTRODUCTION

Nested magnetic surfaces are not guaranteed to ex-

ist in stellarators, unlike in tokamaks, and conﬁnement

properties of the stellarator can be negatively aﬀected

by the presence of chaos and island chains in the mag-

netic ﬁeld. Furthermore, the calculation of many physics

performance metrics that might be used to optimize stel-

larators rely on the assumption of nested magnetic sur-

faces. MHD stability, neoclassical conﬁnement calcula-

tions, and optimization for quasi-symmetry on surfaces1

are simpler when the ﬁeld presents nested magnetic sur-

faces. For all these reasons, it is crucial to develop stel-

larator optimization algorithms that are robust in the

presence of chaotic ﬁeld lines and island chains, and that

will optimize the stellarator to a state with nested mag-

netic surfaces and precise quasi-symmetry, i.e., that can

target island and chaos healing. By “precise” quasi-

symmetry, we mean that the optimized coil designs gen-

erate magnetic ﬁelds that present an accurate approxima-

tion of quasi-symmetry where this property is targeted.

Quasi-symmetry is a favorable property of a magnetic

ﬁeld that ensures the collisionless trajectories of charged

particles up to a certain energy will be conﬁned2. How-

ever, it is not completely known how closely this property

can be approximated or realized in non-axisymmetric

magnetic ﬁelds. Recently, stellarators with very accurate

(though imperfect) quasi-symmetry on a volume were

discovered3using a novel stellarator optimization soft-

ware framework4.

a)Electronic mail: giuliani@cims.nyu.edu

Island healing was a target during the design of the

NCSX coils, where the aim was to ﬁnd a plasma equilib-

rium and coil set such that speciﬁc resonant components

of the ﬁeld from the former and latter cancelled out5,6. In

Ref. 7, a spectral analysis was conducted to manipulate

island width size. An optimization-based approach was

recently proposed, which relies on a combined VMEC-

SPEC optimization of a toroidal boundary surface8. An

initial conﬁguration with islands and chaos was computed

in a stage I optimization using VMEC. Then, a combined

VMEC-SPEC optimization was completed with the goal

of optimizing away the signiﬁcant island chains present

in the initial state. The VMEC solution was used to op-

timize for quasi-symmetry, and the SPEC solution was

used to optimize away the magnetic islands by penaliz-

ing the island width via a Greene’s residue computation.

The VMEC ﬁeld was needed because the quasi-symmetry

penalty term relies on the assumption of nested magnetic

surfaces, which are not guaranteed in the SPEC ﬁeld.

The latter ﬁeld was needed because island chains cannot

be resolved by the VMEC code. It is also possible to

only target island width using SPEC for ﬁnite βequilib-

ria, without optimizing for quasi-symmetry, which bore

promising results9. One disadvantage of the Greene’s

residue approach is that one must know a priori which

resonance one would like to heal. The Greene’s residue

approach can be diﬃcult to use if resonances enter or

leave the magnetic conﬁguration at a given iteration of

the optimization, which happens particularly at the start

of the optimization procedure.

Chaos healing also has been the focus of several studies

in the past10–15. Instead of eliminating localized island

chains corresponding to speciﬁc resonances, the idea here

is to increase the plasma volume that is free from gener-

arXiv:2210.03248v3 [physics.plasm-ph] 13 Mar 2023

alized stochasticity. This was ﬁrst done for analytic vac-

uum ﬁelds, using perturbative analysis10–12. Later, the

Greene residue method was applied in an optimization

framework to ﬁnd coils generating vacuum ﬁelds with re-

duced chaos13,14. These early works did not consider the

simultaneous goal of having magnetic ﬁelds with a high

level of quasi-symmetry. More recently, chaos healing was

attempted in Ref. 15, where a quadratic ﬂux minimizing

(QFM) penalty was added to a coil optimization objec-

tive to favor nested magnetic surfaces further away from

the magnetic axis and optimize away chaotic regions of

the ﬁeld. However, doing so does not control the quality

of quasi-symmetry on low aspect ratio surfaces.

In this manuscript, we address some of the shortcom-

ings of the previous approaches. Speciﬁcally, the main

contributions of this work are: (1) we outline a robust

approach for computing approximations of ﬂux surfaces

even in the presence of island chains and chaos, (2)

based on that surface computation, we present a novel

optimization-based approach to island and chaos heal-

ing, (3) our approach optimizes directly the geometry of

electromagnetic coils, rather than a toroidal boundary

surface, and (4) the algorithm promotes precise quasi-

symmetry on nested magnetic surfaces.

In our previous work (Ref. 1), we outlined a nu-

merical method to optimize stellarator coils for quasi-

axisymmetry under the assumption that over the course

of the optimization, the rotational transform remained

strongly irrational. Despite this strong requirement, we

found precisely quasisymmetric magnetic ﬁelds generated

by coils. The procedure in Ref. 1 relies on the solution to

a partial diﬀerential equation (PDE) that can be diﬃcult

to solve numerically. In this work, we propose a least-

squares formulation to solve the PDE in a more robust

fashion, resulting in a numerical optimization procedure

that can be used even when nested ﬂux surfaces do not

exist. Our approach is formulated as a bilevel optimiza-

tion problem, where the outer coil optimization problem

is constrained by the solution to a PDE, solved in a least

squares sense in the inner optimization problem. This

is similar in spirit to the work in Ref. 16, where qua-

sisymmetric stellarators were found by solving an outer

optimization problem constrained by the least squares so-

lution to the force balance equations computed using the

DESC code. We also focus on curl-free magnetic ﬁelds

as they are an important ﬁrst step in the development

of stellarator optimization algorithms. Furthermore, vac-

uum ﬁelds can serve as suitable initial states in stellarator

optimization approaches in which the plasma pressure is

gradually ramped up17.

II. COMPUTING SURFACES

The goal of this section is to outline a robust numeri-

cal method for computing magnetic surfaces in curl-free

magnetic ﬁelds B∈R3. Even though the external mag-

netic ﬁelds that we use here are always generated by elec-

tromagnetic coils, our method is not restricted to ﬁelds

represented in this manner. We use a ﬁnite-dimensional

representation of a toroidal surface Σs: [0,1)2→(x, y, z)

that satisﬁes nfp-rotational symmetry and stellarator

symmetry. nfp stands for the “number of field periods”

and indicates the the number of times the ﬁeld repeats

itself after a full toroidal rotation. The unknowns, or sur-

face parameters, in this representation are combined into

a vector s∈Rnswhere nsis the number of parameters

that describe the surface; for details of this representation

see Ref. 1. Given a vacuum magnetic ﬁeld B, we seek to

compute a magnetic surface in Boozer coordinates Σs(s),

its rotational transform ι, and the constant G, that solve

r(s) = [rx(s), ry(s), rz(s)] = 0, where1

r(s) := GB

kBk− kBk∂Σs

∂ϕ +ι∂Σs

∂θ .(1)

This equation can be derived by equating two diﬀer-

ent representations of the magnetic ﬁeld, which assume

that it is curl- and divergence-free1,17. Then, with the

dual relations, it can be shown that magnetic surfaces

parametrized in Boozer coordinates satisfy (1). Solutions

to this partial diﬀerential equation can only be expected

when ιis strongly irrational. Based on this assumption,

we have presented a pseudospectral approach to solve (1)

in Ref. 1, and used that numerical method in an optimiza-

tion loop to ﬁnd coils with accurate quasi-symmetry. The

pseudospectral method aimed to ﬁnd surfaces for which

the residual was exactly zero at a ﬁxed number of col-

location points. We called these surfaces “BoozerExact”

surfaces. However, this numerical method can be brittle

when nested ﬂux surfaces do not exist, e.g. in regions

with chaotic ﬁeld lines and island chains, which occur

at places where the rotational transform is not strongly

irrational. The approach that we describe now is more

robust and can be used to determine surfaces even in

regions where the pseudospectral method (BoozerExact)

would have diﬃculty.

Discretizing this partial diﬀerential equation, surfaces

are computed by solving the following constrained least

squares minimization problem

min

2Z1

0Z1/nfp

kr(s)k2dϕ dθ

subject to V(Σs)−V0= 0,

(2)

where r(s) : [0,1) ×[0,1/nfp)→R3,

kr(s)k2=rx(s)2+ry(s)2+rz(s)2,(3)

V0is a given target volume, and

V(Σs) = Z1

0Z1/nfp

3Σs·ndϕ dθ, (4)

where n=∂Σs/∂ϕ ×∂Σs/∂θ. Note that while nis in

the direction of the surface normal, here is not in general

the unit normal. The aim is to solve (1) in a least-squares

sense by minimizing the quadratic residual. The formula

for the volume enclosed by the surface can be derived by

recognizing

V=ZD

dx dy dz =ZD

3∇ · ~r dx dy dz,

where ~r =xˆx+yˆy+zˆz∈Dand Dis the region en-

closed by the toroidal surface Σs. Applying the diver-

gence theorem to the right-hand-side, we obtain (4) after

substituting ~r =Σs.

We use collocation on a tensor grid to approximate

the integrals in (2), which results in the nonlinear least

squares problem

min

6nϕnθ

kR(s)k2+1

2wv(V(Σs)−V0)2(5)

where

R(s)=[rx(s)1, ry(s)1, rz(s)1, . . . , rx(s)Nc, ry(s)Nc, rz(s)Nc],

R(s)∈R3nϕnθand the indices correspond to the col-

location points. We use a tensor product grid of nϕ

and nθcollocation points in the ϕand θdirections,

respectively. We have experimented with two grids of

quadrature points of the form (ϕi, θj) = (i∆ϕ, j∆θ), for

i= 0,1, . . . , nϕ−1, j= 0,1, . . . , nθ−1. Rule 1 is on a

full-period (ϕ, θ)∈[0,1/nfp)×[0,1), ∆ϕ= (1/nfp)/nϕ

and ∆θ= 1/nθ,nϕ= 2ntor + 1, nθ= 2npol + 1. This

rule is spectrally accurate. Rule 2 is on a half-period

(ϕ, θ)∈[0,1/2nfp)×[0,1) with ∆ϕ= (1/2nfp)/nϕ,

∆θ= 1/nθ,nϕ=ntor + 1, nθ= 2npol + 1. This rule

exploits stellarator symmetry, using half the number of

points as rule 1 and is not spectrally accurate. A com-

parison of the two rules follows in Section II A.

A related concept is that of quadratic-ﬂux-minimizing

(QFM) surfaces18, which are surfaces that minimize

0Z1/nfp

(B·n)2dθdϕ,

without constraints on the angles that parametrize the

surface. QFM and BoozerLS surfaces are related to one

another in regimes where nested ﬂux surfaces exist. In

inﬁnite dimensions, BoozerLS surfaces are also QFM sur-

faces, with the added requirement that the BoozerLS sur-

face is parameterized in Boozer coordinates. In ﬁnite

dimensions, numerical experiments show that BoozerLS

surfaces do indeed approximate magnetic surfaces and we

expect both BoozerLS and QFM to be close to each other.

The existence of QFM surfaces in more general contexts

is delicate: when the quadratic ﬂux is unweighted, it is

shown in Ref. 18 that only true ﬂux surfaces extremize

the QFM functional. An analogous conclusion may be

true for BoozerLS surfaces, but we have not attempted

to show this here.

The ﬁrst-order optimality condition of (5) is

g(s) := J(s)TR(s) + wv(V(s)−V0)∂V

∂s=0(6)

where g∈Rns,J=∂R

∂s∈R(3nϕnθ)×nsis the Jacobian of

R(·) and

V(s) := 1

3nϕnθ

i=1

(Σs)i·ni.(7)

is the discretized surface volume.

In (Ref. 1), we showed numerically that computing

BoozerLS surfaces is robust, even in the presence of is-

lands and chaos. Framing the surface computation as a

least-squares optimization problem allows us to use line

search algorithms that track progress of the solution al-

gorithm, and to prevent step sizes that are too large.

Thanks to the optimization framework which deﬁnes the

BoozerLS surfaces, we are free to introduce regulariza-

tions that prevent the numerically computed surfaces

from self-intersecting.

The Newton step of (6) for the increment δsis

J(s)TJ(s) + wv

∂V

∂s

T∂V

∂s

3nϕnθ

i=1

Ri(s)∂2Ri

∂s2(s) + wv(V(s)−V0)∂2V

∂s2δs=−g(s)

(8)

where the Hessian matrix on the left-hand side multiply-

ing δs∈Rnsis denoted by H∈Rns×ns. We initially

use the Broyden–Fletcher–Goldfarb–Shanno (BFGS) al-

gorithm to solve (5). Then, using this solution as an

initial guess, we use Newton’s method on (8) to reduce

the gradient of the nonlinear residual further. We observe

that while the Hessian is not absolutely necessary for the

computation of the least square surfaces, its availability

is crucial for the computation of gradients in the outer

coil optimization problem we consider in Section III B,

when the least-squares optimality condition is used as

constraint within the coil optimization problem. We will

highlight this point again in that section. We also note

that in contrast to Ref. 1, the residual in (3) is divided

by kBk. This is to prevent rfrom scaling with the coil

currents.

The approach described in this section shows how to

compute a magnetic surface that encloses a user-deﬁned

volume V0. The coil optimization studies in Section III B

use this numerical method to compute multiple magnetic

surfaces, each with a diﬀerent target volume.

A. Convergence study

In this section, we present a convergence test of the

BoozerLS formulations using two diﬀerent quadrature

rules to approximate the integral in (2), called rules 1 and

2. We compare this new least squares formulation to the

one in Ref. 1 i.e., when the number of unknowns coincides

with the number of collocation points, which we refer to

as BoozerExact. We compute the innermost surface used

in the coil optimization problem (section IV) using the

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Directstellaratorcoiloptimizationfornestedmagneticsurfaceswithprecisequasi-symmetryAndrewGiuliani,1,a)FlorianWechsung,1AntoineCerfon,1MattLandreman,2andGeorgStadler11)CourantInstituteofMathematicalSciencesNewYorkUniversityNewYork,N.Y.10012-11852)InstituteforResearchinElectronicsandAppliedPhysicsUniv...

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Direct stellarator coil optimization for nested magnetic surfaces with precise quasi-symmetry Andrew Giuliani1aFlorian Wechsung1Antoine Cerfon1Matt Landreman2and Georg Stadler1

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