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Caoxiang Zhu

Publications and source records attributed to Caoxiang Zhu.

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Quasi-single-stage optimization for advanced stellarators

Advanced stellarator design requires a balance between plasma performance and the manufacturability of three-dimensional modular coils. In conventional two-stage optimization, the coils required to realize an optimized equilibrium can be limited by engineering feasibility. Here, we develop a quasi-single-stage (QSS) framework that incorporates coil feasibility directly into plasma-boundary optimization. QSS uses the maximum normalized normal-field error, evaluated rapidly from surface currents on a uniformly offset winding surface, as a coil-feasibility surrogate. We apply this method to optimize configurations targeting quasi-axisymmetry, quasi-helical symmetry, quasi-isodynamicity, and a combination of omnigenity with piecewise omnigenity. The QSS-optimized configurations exhibit smoother plasma boundaries and winding surfaces, lower normal-field reconstruction errors, and reduced coil complexity, while preserving favourable magnetic-symmetry and transport properties. QSS provides a practical proof-of-principle strategy for co-optimizing plasma physics and coil engineering in stellarator design.

physics.plasm-ph

Optimized finite-$β$ tokamak-stellarator hybrid configurations achieved by planar dipole-field coils

Tokamak--stellarator hybrids seek to combine tokamak-like compactness and confinement with stellarator-like externally generated rotational transform and steady-state operation. In this work, we build on the recent tokamak--stellarator hybrid study using planar dipole-field coils (PDCs) [Yu et al., arXiv:2605.03599], in which the fixed-position, programmable coils on an axisymmetric winding surface generate flexible three-dimensional shaping fields. Using single-stage free-boundary optimization of coil currents and plasma-equilibrium parameters, we construct vacuum and finite-$β$ configurations. The vacuum cases show controllable external transform and magnetic well. The finite-$β$ cases accommodate various density, temperature, and pressure profiles, producing quasi-axisymmetric (QA) equilibria with self-consistent bootstrap current, favorable Mercier stability, and reduced demand for external current drive. Re-optimization enables $β$ ramp-up and access to different field-period QA branches with moderate coil-current changes. At large rotational transform, a toroidally omnigenous (TO)-like configuration exhibits more favorable infinite-$n$ ideal-ballooning behavior than a QA reference with matched profiles, even though ballooning stability is not directly optimized for. These results demonstrate that PDCs provide a flexible platform for achieving optimized finite-$β$ hybrid configurations.

physics.plasm-ph

A programmable stellarator-tokamak hybrid for million-scale magnetic-configuration discovery

Tokamaks and stellarators are the leading magnetic-confinement concepts for fusion, but they rely on complementary design principles. Tokamaks use simple axisymmetric coils and plasma current, whereas stellarators use externally generated three-dimensional fields for steady-state operation. Here, we propose a programmable stellarator--tokamak hybrid that uses a fixed set of simple planar coils to access a broad magnetic-configuration space. The device adds 288 dipole-field coils to a tokamak-like coil set, with only six independent coil geometries required by symmetry. By programming coil currents, the same hardware generates more than 1.66 million optimized stellarator configurations spanning quasi-axisymmetry, quasi-helical symmetry, and quasi-isodynamicity, as well as tokamak-relevant three-dimensional perturbations. Representative configurations exhibit nested magnetic surfaces, low neoclassical transport, and favorable energetic-particle confinement. This approach enables rapid magnetic-configuration discovery without hardware redesign.

physics.plasm-ph

Optimizing stellarators with hidden symmetry

Stellarators confine fusion plasmas using three-dimensional magnetic fields composed of nested toroidal magnetic surfaces. In generic stellarators, trapped particles can drift across these surfaces and degrade plasma confinement. Certain topological properties of the magnetic field strength can suppress these drifts. However, conventional stellarator design approaches typically enforce restrictive constraints to realize such properties, thereby segmenting and limiting the accessible configuration space. In this work, we reformulate the conditions for efficient confinement as constraints on a homeomorphic straightening transformation of the field contours. Within this framework, the various families of stellarator magnetic fields optimized for plasma confinement arise naturally as specific realizations of a unified mapping. This new perspective provides a significantly more comprehensive description of viable stellarator configurations, enabling systematic exploration of trade-offs among confinement quality, geometric complexity, and engineering requirements. We illustrate this approach by presenting a highly compact stellarator design that nevertheless achieves plasma performance comparable to that of leading reactor-scale designs with much larger aspect ratios.

physics.plasm-ph

Optimization of stellarator configurations combining omnigenity and piecewise omnigenity

We present a method for optimizing stellarator configurations that combine omnigenity and piecewise omnigenity (pwO). Within the \texttt{OOPS} optimization framework [Liu \textit{et al.}, arXiv:2502.09350 (2025)], we introduce a mapping technique that can ``squeeze'' general omnigenous fields to approximate pwO in the high-field side. Using this approach, we obtain a range of optimized configurations that combine poloidal omnigenity (PO) and pwO, spanning different field periods and aspect ratios. We further show that these configurations are compatible with a magnetic well. The resulting configurations exhibit favorable neoclassical transport and bootstrap current properties while partially relaxing the strict constraints of omnigenity. These results suggest that such configurations are promising candidates for future stellarator reactors.

physics.plasm-ph

Quasi-single-stage optimization for permanent magnet stellarators

Advanced stellarators are typically optimized in two stages. The plasma equilibrium is optimized first, followed by the design of coils/permanent magnets. However, the coils/permanent magnets in the second stage may become too complex to achieve the desired equilibrium. To address this problem, a quasi-single-stage optimization method has been proposed. In this paper, we introduce this method for designing permanent magnet (PM) stellarators. The new approach combines straightforward PM metrics to penalize the maximum required PM thickness and the mismatch between the fixed-boundary equilibrium and the free-boundary one, along with typical physical targets. Since the degrees of freedom of the PMs are not included and directly used to minimize the objective function in this method, so we call it "quasi-single-stage" optimization. We apply this quasi-single-stage optimization method to find a new quasi-axisymmetric PM design. The new design starts from MUSE, which was initially designed using a two-stage optimization approach. The resulting design, MUSE++, exhibits an order of magnitude lower quasi-symmetric error and a one-order reduction in normal field error. We show that MUSE++ has approximately 30% fewer magnets compared to a proxy model "MUSE-0" that uses the same FAMUS optimization without the benefit of a single-stage equilibrium optimization. These results demonstrate that the new single-stage optimization method can concurrently improve plasma properties and simplify permanent magnet complexity.

physics.plasm-ph

Effects of Magnetic Helicity on 3D Equilibria and Self-Organized States in KTX Reversed Field Pinch

The RFP is a toroidal magnetic configuration in which plasmas can spontaneously transform into different self-organized states. Among various states, the QSH state has a dominant component for the magnetic field and significantly improves confinement. Many theoretical and experimental efforts have investigated the transitions among different states. This paper employs the MRxMHD model to study the properties of QSH and other states. The SPEC is used to compute MHD equilibria for the KTX. The toroidal volume of KTX is partitioned into two subvolumes by an internal transport barrier. The geometry of this barrier is adjusted to achieve force balance across the interface, ensuring that the plasma in each subvolume is force-free and that magnetic helicity is conserved. By varying the parameters, we generate distinct self-organized states in KTX. Our findings highlight the crucial role of magnetic helicity in shaping these states. In states with low magnetic helicity in both subvolumes, the plasma exhibits axisymmetric behavior. With increasing core helicity, the plasma gradually transforms from an axisymmetric state to a double-axis helical state and finally to a single-helical-axis state. Elevated core magnetic helicity leads to a more pronounced dominant mode of the boundary magnetic field and a reduced core magnetic shear. This is consistent with previous experimental and numerical results in other RFP devices. We find a linear relationship between the plasma current and helicity in different self-organized states. Our findings suggest that KTX may enter the QSH state when the toroidal current reaches 0.72 MA. This study demonstrates that the stellarator equilibrium code SPEC unveils crucial RFP equilibrium properties, rendering it applicable to a broad range of RFP devices and other toroidal configurations.

physics.plasm-ph

Computation of the Biot-Savart line integral with higher-order convergence using straight segments

One common approach to computing the magnetic field produced by a filamentary current-carrying coil is to approximate the coil as a series of straight segments. The Biot-Savart field from each straight segment is known analytically. However, if the endpoints of the straight segments are chosen to lie on the coil, then the accuracy of the Biot-Savart computation is generally only second-order in the number of endpoints. We propose a simple modification: shift each endpoint off the coil in the outwards normal direction by an amount proportional to the local curvature. With this modification, the Biot-Savart accuracy increases to fourth order and the numerical error is dramatically reduced for a given number of discretization points.

physics.comp-ph

Stellarator optimization for good magnetic surfaces at the same time as quasisymmetry

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 a single optimization, two fixed-boundary equilibrium calculations are run at each iteration of the optimization: one that enforces the existence of magnetic surfaces (VMEC [S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983)]), and one that does not (SPEC [S. R. Hudson, et al, Phys. Plasmas 19, 112502 (2012)]). By penalizing the island residues in the objective function, the two magnetic field representations are brought into agreement during the optimization. An example is presented in which, particularly on the surface where quasisymmetry was targeted, quasisymmetry is achieved more accurately than in previously published examples.

physics.plasm-ph

Optimized finite-build stellarator coils using automatic differentiation

A new stellarator coil design code is introduced that optimizes the position and winding pack orientation of finite-build coils. The new code, called FOCUSADD, performs gradient-based optimization in a high-dimensional, non-convex space. The derivatives with respect to parameters of finite-build coils are easily and efficiently computed using automatic differentiation. FOCUSADD parametrizes coil positions in free space using a Fourier series and uses a multi-filament approximation to the coil winding pack. The orientation of the winding pack is parametrized with a Fourier series and can be optimized as well. Optimized finite-build coils for a W7-X-like stellarator are found, and compared with filamentary coil results. The final positions of optimized finite-build W7-X-like coils are shifted, on average, by approximately 2.5mm relative to optimized filamentary coils. These results suggest that finite-build effects should be accounted for in the optimization of stellarators with low coil tolerances.

physics.plasm-ph

Calculation of permanent magnet arrangements for stellarators: A linear least-squares method

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 linearity in magnetization density and approximate linearity in magnet size, for magnets far from the target region. No constraint is placed on the direction of magnetization, so Halbach solutions are found naturally, and the magnitude of the magnetization can be made uniformly equal to a target value.

physics.plasm-ph

Topology optimization of permanent magnets for stellarators

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.

physics.plasm-ph

Designing stellarators using perpendicular 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 current that can be linearly solved by existing coil design codes. An incremental multi-layer method has been developed to obtain a practical solution that is attainable with present permanent magnets. With this method, we can reproduce a half-Tesla NCSX configuration using specially designed neodymium magnets together with simple planar coils. It shows that stellarator coils could be substantially simplified by adopting permanent magnets.

physics.plasm-ph

Identification of important error fields in stellarators using Hessian matrix method

Error fields are predominantly attributed to inevitable coil imperfections. Controlling error fields during coil fabrication and assembly is crucial for stellarators. Excessively tight coil tolerance increases time and cost, and, in part, led to the cancellation of NCSX and delay of W7-X. In this paper, we improve the recently proposed Hessian matrix method to rapidly identify important coil deviations. Two of the most common figures of merit, magnetic island size and quasi-symmetry, are analytically differentiated over coil parameters. By extracting the eigenvectors of the Hessian matrix, we can directly identify sensitive coil deviations in the order of the eigenvalues. The new method is applied to the upcoming CFQS configuration. Important perturbations that enlarge n/m=4/11 islands and deteriorate quasi-axisymmetry of the magnetic field are successfully determined. The results suggest each modular coil should have separate tolerance and some certain perturbation combinations will produce significant error fields. By relaxing unnecessary coil tolerance, this method will hopefully lead to a substantial reduction in time and cost.

physics.plasm-ph

New method to design stellarator coils without the winding surface

Finding an easy-to-build coils set has been a critical issue for stellarator design for decades. Conventional approaches assume a toroidal "winding" surface. We'll investigate if the existence of winding surface unnecessarily constrains the optimization, and a new method to design coils for stellarators is presented. Each discrete coil is represented as an arbitrary, closed, one-dimensional curve embedded in three-dimensional space. A target function to be minimized that covers both physical requirements and engineering constraints is constructed. The derivatives of the target function are calculated analytically. A numerical code, named FOCUS, has been developed. Applications to a simple configuration, the W7-X, and LHD plasmas are presented.

physics.plasm-ph