We introduce a topology optimization method to design permanent magnets for advanced stellarators. Recent researches show that permanent magnets have great potentials to simplify stellarator coils. We adopt state-of-the-art numerical techniques to determine the presence of magnets in the entire designing space. The FAMUS code is developed and it can design engineering-feasible permanent magnets for general stellarators satisfying the constraints of the maximum material magnetization and explicitly forbidden regions. FAMUS has been successfully verified against the previously proposed linear method. Three different permanent magnet designs together with planar TF coils for a half-Tesla NCSX configuration have been obtained for demonstrations. The designs have good accuracy in generating the desired equilibrium and offer considerably large plasma access on the outboard side. The results show that FAMUS is a flexible, advanced numerical tool for future permanent magnet stellarator designs.
It is shown that the magnetic-field coils of a stellarator can, at least in principle, be substantially simplified by the use of permanent magnets. Such magnets cannot create toroidal magnetic flux but they can be used to shape the plasma and thus to create poloidal flux and rotational transform, thereby easing the requirements on the magnetic-field coils. As an example, a quasiaxisymmetric stellarator configuration is constructed with only 8 circular coils (all identical) and permanent magnets.
We have developed a fast method to design perpendicular permanent magnets for simplifying stellarator coils based on existing codes. Coil complexity is one of the main challenges for stellarators. To date, only electromagnetic coils have been used to generate 3D fields for stellarators. Permanent magnets provide an alternative way to produce the desired magnetic field for optimized stellarators. In this paper, we revisit the concept of representing surface current using magnetic dipoles and carry out numerical validations. A surface magnetization is proven to be equivalent to the surface curre…
The development of stellarators that use permanent magnet arrays to shape their confining magnetic fields has been a topic of recent interest, but the requirements for how such magnets must be shaped, manufactured, and assembled remain to be determined. To address these open questions, we have performed a study of geometric concepts for magnet arrays with the aid of the newly developed MAGPIE code. A proposed experiment similar to the National Compact Stellarator Experiment (NCSX) is used as a test case. Two classes of magnet geometry are explored: curved bricks that conform to a regular grid …
A common scientific inverse problem is the placement of magnets that produce a desired magnetic field inside a prescribed volume. This is a key component of stellarator design, and recently permanent magnets have been proposed as a potentially useful tool for magnetic field shaping. Here, we take a closer look at possible objective functions for permanent magnet optimization, reformulate the problem as sparse regression, and propose an algorithm that can efficiently solve many convex and nonconvex variants. The algorithm generates sparse solutions that are independent of the initial guess, exp…
A problem arising in several engineering areas is to design magnets outside a volume that produce a desired magnetic field inside it. One instance of this problem is stellarator design, where it has recently been shown that permanent magnets can provide the required shaping of the magnetic field. Here we demonstrate a robust and efficient algorithm REGCOIL_PM to calculate the spatial distribution of these permanent magnets. The procedure involves a small number of fixed-point iterations, with a linear least-squares problem solved at each step. The method exploits the Biot-Savart Law's exact li…
The usage of permanent magnets to shape the confining magnetic field of a stellarator has the potential to reduce or eliminate the need for non-planar coils. As a proof-of-concept for this idea, we have developed a procedure for designing an array of cubic permanent magnets that works in tandem with a set of toroidal-field coils to confine a stellarator plasma. All of the magnets in the design are constrained to have identical geometry and one of three polarization types in order to simplify fabrication while still producing sufficient field accuracy. We present some of the key steps leading t…
In this self-contained document, we aim to present the basic theoretical building blocks to understand modeling of stellarator magnetic fields, some of the challenges associated with modeling, and optimization for designing stellarators. As often as possible, the ideas will be presented using equations and pictures, and references to other relevant introductory material will be included. This document is accessible to those who may not have a physics background but are interested in applications of mathematical and computational tools to stellarator research.
This paper describes a new and efficient method of defining an annular region of a curl-free magnetic field with specific physics and coil properties that can be used in stellarator design. Three statements define the importance: (1) Codes can follow an optimized curl-free initial state to a final full-pressure equilibrium. The large size of the optimization space of stellarators, approximately fifty externally-produced distributions of magnetic field, makes success in finding a global optimum largely determined by the starting point. (2) The design of a stellarator is actually improved when t…
The design of turbulence optimized stellarators has so far relied on three-dimensional equilibrium codes such as VMEC in order to find the minimum of a given objective function. In this work, we propose a complimentary approach based on the near-axis expansion to compute the geometry parameters of neoclassicaly optimized stellarators used in turbulence studies. As shown here, the near-axis expansion can be a reasonable approximation of the geometric parameters relevant for turbulence and stability simulations of the core of existing optimized stellarator designs. In particular, we examine the …
Non-resonant stellarator divertors have magnetic flux tubes, called magnetic turnstiles, that cross cantori, which are fractal remnants of destroyed invariant tori with holes, that lie outside the outermost confining surface. The exiting and entering flux tubes can be adjacent as is generally expected but can also have the unexpected feature of entering or exiting at separate locations of the cantori. Not only can there be two types of turnstiles, but pseudo turnstiles can also exist. A pseudo turnstile is formed when a cantorus has a sufficiently large, although limited, radial excursion to s…
A method is demonstrated to optimize a stellarator's geometry to eliminate magnetic islands and achieve other desired physics properties at the same time. For many physics quantities that have been used in stellarator optimization, including quasisymmetry, neoclassical transport, and magnetohydrodynamic stability, it is convenient to use a magnetic equilibrium representation that assures the existence of magnetic surfaces. However, this representation hides the possible presence of magnetic islands, which are typically undesirable. To include both surface-based objectives and island widths in …
The design of any large stellarator requires a plan for the removal of the particles and heat that are exhausted across the plasma edge. This is called the divertor problem, for the particle exhaust must be diverted into pumping chambers. Although the physics of diverted plasmas has many subtleties, the magnetic field configuration between the plasma edge and the surrounding chamber walls is the foundation upon which divertor design is based. The properties of this magnetic configuration has both practical constraints and mathematical constraints from magnetic field lines obeying a 1~1/2 degre…
Good magnetic surfaces, as opposed to magnetic islands and chaotic field lines, are generally desirable for stellarators. In previous work, M. Landreman et al. [Phys. of Plasmas 28, 092505 (2021)] showed that equilibria computed by the Stepped-Pressure Equilibrium Code (SPEC) [S. P. Hudson et al., Phys. Plasmas 19, 112502 (2012)] could be optimized for good magnetic surfaces in vacuum. In this paper, we build upon their work to show the first finite-$β$, fixed- and free-boundary optimization of SPEC equilibria for good magnetic surfaces. The objective function is constructed with the Greene's …
A stellarator is said to be omnigeneous if all particles have vanishing average radial drifts. In omnigeneous stellarators, particles are perfectly confined in the absence of turbulence and collisions, whereas in non-omnigeneous configurations, particle can drift large radial distances. One of the consequences of omnigeneity is that the unfavorable inverse scaling with collisionality of the stellarator neoclassical fluxes disappears. In the pioneering and influential article [Cary~J~R and Shasharina~S~G 1997 {\it Phys. Plasmas} {\bf 4} 3323], the conditions that the magnetic field of a stellar…
Quasisymmetry can greatly improve the confinement of energetic particles and thermal plasma in a stellarator. The magnetic field of a quasisymmetric stellarator at high plasma pressure is significantly affected by the bootstrap current, but the computational cost of accurate stellarator bootstrap calculations has precluded use inside optimization. Here, a new efficient method is demonstrated for optimization of quasisymmetric stellarator configurations such that the bootstrap current profile is consistent with the geometry. The approach is based on the fact that all neoclassical phenomena in q…
Quasisymmetric stellarators are appealing intellectually and as fusion reactor candidates since the guiding center particle trajectories and neoclassical transport are isomorphic to those in a tokamak, implying good confinement. Previously, quasisymmetric magnetic fields have been identified by applying black-box optimization algorithms to minimize symmetry-breaking Fourier modes of the field strength $B$. Here instead we directly construct magnetic fields in cylindrical coordinates that are quasisymmetric to leading order in distance from the magnetic axis, without using optimization. The met…
We present a technique that can be used to design stellarators with a high degree of experimental flexibility. For our purposes, flexibility is defined by the range of values the rotational transform can take on the magnetic axis of the vacuum field while maintaining satisfactory quasisymmetry. We show that accounting for configuration flexibility during the modular coil design improves flexibility beyond that attained by previous methods. Careful placement of planar control coils and the incorporation of an integrability objective enhance the quasisymmetry and nested flux surface volume of ea…
The ongoing development of electromagnets based on High Temperature Superconductors has led to the conceptual exploration of high-magnetic-field fusion reactors of the tokamak type, operating at on-axis fields above 10 T. In this work we explore the consequences of the potential future availability of high-field three-dimensional electromagnets on the physics design point of a stellarator reactor. We find that, when an increase in the magnetic field strength $B$ is used to maximally reduce the device linear size $R\sim B^{-4/3}$ (with otherwise fixed magnetic geometry), the physics design poin…
An optimized compact stellarator with four simple coils is obtained from direct optimization via coil shape. The new stellarator consists of two interlocking coils and two vertical field coils similar to those of the Columbia Non-neutral Torus (CNT)[Pedersen et al. Phys. Rev. Lett. 88, 205002 (2002)]. The optimized configuration has global magnetic well and a low helical ripple level comparable to that of Wendelstein 7-X (W7-X)[Wolf et al. Nucl. Fusion 57, 102020 (2017)]. The two interlocking coils have a smooth three-dimensional shape much simpler than those of advanced stellarators such as W…
A method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren and Boozer's expansion about the magnetic axis (Phys Fluids B 3, 2805 (1991)), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landreman, Sengupta and Plunk (J Plasma Phys 85, 9058…
A good understanding of the confinement of energetic ions in non-axisymmetric magnetic fields is key for the design of reactors based on the stellarator concept. In this work, we develop a model that, based on the radially-local bounce-averaged drift-kinetic equation, classifies orbits and succeeds in predicting configuration-dependent aspects of the prompt losses of energetic ions in stellarators. Such a model could in turn be employed in the optimization stage of the design of new devices.
A single-field-period quasi-isodynamic stellarator configuration is presented. This configuration, which resembles a twisted strip, is obtained by the method of direct construction, that is, it is found via an expansion in the distance from the magnetic axis. Its discovery, however, relied on an additional step involving numerical optimization, performed within the space of near-axis configurations defined by a set of adjustable magnetic-field parameters. This optimization, completed in 30 seconds on a single cpu core using the SIMSOPT code, yields a solution with excellent confinement, as mea…
This paper deals with the design of the stellarator field with the simple coil set. In order to realize the stellarator field by the simple coil set, the tilted toroidal field coil uses for creating the rotational transform. Sixteen tilted TF coils create the small radial field and large vertical field. With reducing the vertical field of the tilted toroidal field coil by the axisymmetric poloidal field coil, the stellarator field can be made. The formation of clear and nested flux surfaces is confirmed, and the rotational transform is proportional to the tilting angle of the toroidal field co…
AI-Accelerated Gyrokinetic Predictions of Turbulent Transport for Stellarator Design Optimization and Experimental Planning
Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）；University of Maryland, College Park（马里兰大学帕克分校）；Oak Ridge National Laboratory（橡树岭国家实验室）；Type One Energy；California Institute of Technology（加州理工学院）；Argonne National Laboratory（阿贡国家实验室）；Princeton University（普林斯顿大学）；University of Wisconsin, Madison（威斯康星大学麦迪逊分校）
We present a stellarator configuration optimized for a large threshold (``critical gradient'') for the onset of the ion temperature gradient (ITG) driven mode, which achieves the largest critical gradient we have seen in any stellarator. Above this threshold, gyrokinetic simulations show that the configuration has low turbulence levels over an experimentally relevant range of the drive strength. The applied optimization seeks to maximize the drift curvature, leading to enhanced local-shear stabilization of toroidal ITG modes, and the associated turbulence. These benefits are combined with exce…
The design of a stellarator with acceptable confinement properties requires optimization of the magnetic field in the non-convex, high-dimensional spaces describing their geometry. Another major challenge facing the stellarator program is the sensitive dependence of confinement properties on electro-magnetic coil shapes, necessitating the construction of the coils under tight tolerances. In this Thesis, we address these challenges with the application of adjoint methods and shape sensitivity analysis. Adjoint methods enable the efficient computation of the gradient of a function that depends o…
A number of scientific fields rely on placing permanent magnets in order to produce a desired magnetic field. We have shown in recent work that the placement process can be formulated as sparse regression. However, binary, grid-aligned solutions are desired for realistic engineering designs. We now show that the binary permanent magnet problem can be formulated as a quadratic program with quadratic equality constraints (QPQC), the binary, grid-aligned problem is equivalent to the quadratic knapsack problem with multiple knapsack constraints (MdQKP), and the single-orientation-only problem is e…
Polaris: a flexible stellarator demonstration experiment with simple modular coils
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；BergTop fusion（BergTop聚变）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）
Optical analogy for stellarators: Ridges as caustics and coils as singularities
Princeton University（普林斯顿大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Maryland, College Park（马里兰大学帕克分校）；Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford, OX1 3PU, UK（鲁道夫·派尔斯理论物理中心）；Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）
Efficient Computation of Stellarator Coils with an Augmented Lagrangian Optimization Method
Max-Planck-Institut für Plasmaphysik（马克斯·普朗克等离子体物理研究所）；BASIS Independent Silicon Valley（BASIS独立硅谷）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
Tight tolerances have been a leading driver of cost in recent stellarator experiments, so improved definition and control of tolerances can have significant impact on progress in the field. Here we relate tolerances to the shape gradient representation that has been useful for shape optimization in industry, used for example to determine which regions of a car or aerofoil most affect drag, and we demonstrate how the shape gradient can be computed for physics properties of toroidal plasmas. The shape gradient gives the local differential contribution to some scalar figure of merit (shape functi…
Magnetic confinement devices for nuclear fusion can be large and expensive. Compact stellarators are promising candidates for costreduction, but introduce new difficulties: confinement in smaller volumes requires higher magnetic field, which calls for higher coil-currents and ultimately causes higher Laplace forces on the coils-if everything else remains the same. This motivates the inclusion of force reduction in stellarator coil optimization. In the present paper we consider a coil winding surface, we prove that there is a natural and rigorous way to define the Laplace force (despite the mag…
