Date / Time (ET) | Speaker | Title of Talk (PDF Link) |
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Sep 12 (13), 2024 10:00-11:00am (EDT) | J. Larson | Efficiently Using Computational Resources by Exploiting Problem-Specific Knowledge In Derivative-Free OptimizationAbstract: We start with a high-level presentation comparing various methods for optimizing computationally expensive functions that lack reliable gradient information. We highlight algorithms that utilize the structure of common problems and demonstrate their efficiency on relevant problems from particle physics and fusion power systems. We highlight how such algorithms can be incorporated into an asynchronous, multi-start framework. We conclude with a discussion of practical performance and possible points of collaboration.(Video recording of talk) |

Aug 29 (30), 2024 10:00-11:00am (EDT) | R. MacKay | Title: Flux Volume Formula (TBR)Abstract: TBSL(Video recording of talk) |

Aug 15 (16), 2024 10:00-11:00am (EDT) | M. Ruth | Title: TBSLAbstract: TBSL(Video recording of talk) |

Aug 1 (2), 2024 10:00-11:00am (EDT) | TBSL | Title: TBSLAbstract: TBSL(Video recording of talk) |

Jul 18 (19), 2024 10:00-11:00am (EDT) | M. Czekanski | Title: TBSLAbstract: TBSL(Video recording of talk) |

Jun 20 (21), 2024 10:00-11:00am (EDT) | E. Qian | Title: TBSLAbstract: TBSL(Video recording of talk) |

Jun 6 (7), 2024 10:00-11:00am (EDT) | I. Farcas | SCIML Reduced Models for Plasma TurbulenceAbstract: TBSL(Video recording of talk) |

May 23 (24), 2024 10:00-11:00am (EDT) | J. Hicken | Optimization in the Presence of ChaosAbstract: Chaotic dynamics govern many engineering systems that we would like to optimize: fusion reactors, wind turbines, and aircraft, to name a few. Unfortunately, chaotic systems are notoriously difficult to optimize. I will spend the first half of this seminar explaining why chaotic systems are challenging to optimize, and I will review the techniques that have been proposed to overcome this challenge. Existing techniques can be categorized as either sensitivity-analysis methods or regularization methods; the former aim to produce useful derivatives that approximate the trend/slope of time-averaged outputs, while the latter "smooth" out, or regularize, the optimization problem by eliminating problematic oscillations in the time-averaged output. The second half of the presentation will focus on my lab's recent work developing a regularization-type method related to unstable periodic orbits.(Video recording of talk) |

May 09 (10), 2024 10:00-11:00am (EDT) | G. Acton | Optimization of Gyrokinetic Microstability Using Adjoint MethodsAbstract: Microinstabilities drive turbulent fluctuations in inhomogeneous, magnetized plasmas. In the context of magnetic confinement fusion devices, this leads to an enhanced transport of particles, momentum, and energy, thereby degrading confinement. In this work, we describe an application of the adjoint method to efficiently determine variations of gyrokinetic linear growth rates on a general set of external parameters in the local δf -gyrokinetic model. We then offer numerical verification of this approach. When coupled with gradient-based techniques, this methodology can facilitate the optimization process for the microstability of the confined plasmas across a high-dimensional parameter space. We present a numerical demonstration wherein the ion-temperature gradient (ITG) instability growth rate in a tokamak plasma is minimized with respect to flux surface shaping parameters. The adjoint method approach demonstrates a significant computational speed-up compared to a finite-difference gradient calculation.(Video recording of talk) |

Apr 25 (26), 2024 10:00-11:00am (EDT) | D. Biek | Parametric Design and Analysis of High Field Stellarator CoilsAbstract: Comparing to tokamaks, the engineering challenges with stellarator coils are related to their complicated 3D shape with small bending radii and demanding requirements on the coil-shape tolerances. In addition, due to the variety of stellarator configurations, there is a need for an analysis tool applicable to various configurations. The objective of this work is to propose an engineering design tool for the magnet system of stellarators. Different options are considered like Nb3Sn superconductors, high-temperature superconductors (HTS) and non-insulated HTS coils. This talk presents the workflow for the development of this tool for analysis of stellarator coils, which bases solely on the central current path. As case study, HELIAS 5-B is analyzed which is the EUROfusion baseline for the stellarator engineering efforts. The electromagnetic analysis for HELIAS 5B has been conducted leading to around 12 T peak field in the winding pack, and a preliminary design of the winding pack, based on Nb3Sn, is proposed. Plans for the mechanical and thermal-hydraulic analyses of the coil system will be outlined.(Video recording of talk) |

Apr 11 (12), 2024 10:00-11:00am (EDT) | L. Horesh | Should we Derive or Let the Data Drive? A Symbiotic ApproachAbstract: The scientific method has been transformative to humankind. However, there are signs that despite great investment in the area, scientific discovery is approaching a state of stagnation. In the context of scientific discovery, a fundamental problem is to explain natural phenomena in a manner consistent with both (noisy) experimental data, and a body of (possibly inexact and incomplete) background knowledge about the laws of the universe. Historically, models were manually derived in a first-principles deductive fashion. The first-principles approach often offers the derivation of interpretable symbolic models of remarkable levels of universality while being substantiated by little data. Nonetheless, derivation of such models is time-consuming and relies heavily upon domain expertise. Conversely, with the rising pervasiveness of statistical AI and data-driven approaches, automated, rapid construction and deployment of models has become a reality. Many data-driven modeling techniques demonstrate remarkable scalability due to their reliance upon predetermined, exploitable model form (functional form) structures. Such structures, entail non-interpretable models, demand Big Data for training, and provide limited predictive power for out-of-set instances. In this lecture, we will delve into the two distinct discovery frameworks, examine recent efforts to bridge their gap, and consider the possibility of achieving a unified understanding of fundamental natural laws at scale.(Video recording of talk) |

Mar 28 (29), 2024 10:00-11:00am (EDT) | S. Walker | Overview of Shape Optimization with Finite Element MethodsAbstract: I will start with an introduction to shape derivatives and shape optimization. Next, I will give a general discussion on how shape optimization is done numerically and some of the issues that can arise (e.g. the classic debate between the optimize-then-discretize and discretize-then-optimize approaches).Then, I will talk about an unfitted finite element method (FEM) for PDE-constrained shape optimization. The geometry is represented (and optimized) using a level set approach and we consider objective functionals that are defined over bulk domains. For a discrete objective functional (i.e. one defined in the unfitted FEM framework), we show that the *exact* shape derivative can be computed rather easily. In other words, one gains the benefits of both the optimize-then-discretize and discretize-then-optimize approaches. We illustrate the method on a simple model (geometric) problem with known exact solution, as well as shape optimization of structural designs (linear elasticity). (Video recording of talk) |

Feb 29(Mar 1), 2024 10:00-11:00am (EST) | J. Burby | Spatial Dynamics Formulation of MagnetohydrostaticsAbstract: The magnetohydrostatics (MHS) PDE system is usually studied as a 3-dimensional boundary value problem. I will describe several benefits to treating MHS as a 2-dimensional "time" evolution problem, where toroidal angle plays the role of "time" and poloidal coordinates parameterize "space". In this spatial dynamics reformulation of MHS, planar hydrodynamic equations prescribe the evolution of toroidal flux, pressure, and field line velocity. Newcomb's variational principle for the usual boundary-value formulation of MHS implies a dynamical variational principle for the hydrodynamic system. I will summarize the implications of this variational principle in light of Noether's theorem connecting symmetries with conservation laws. As with all variational dynamical systems, the MHS spatial dynamics equations admit a dual Hamiltonian formulation. Remarkably, this Hamiltonian structure is Lie-Poisson on the dual to a certain infinite-dimensional Lie algebra. This implies that a technique developed by Scovel-Weinstein originally for beam physics produces exact particle-swarm solutions of the spatial dynamics equations suitably regularized at small poloidal length scales, thereby suggesting a method for sidestepping Grad's conjecture. The particle-like solutions obey an ODE system that I call smoothed particle magnetohydrostatics (SPMHS), due to its resemblance with smoothed particle hydrodynamics. In large aspect ratio domains, the spatial dynamics equations exhibit a pair of disparate timescales. Reducing to the corresponding normally-hyperbolic slow manifold corresponds to solving the elliptic part of the equations, leaving a hyperbolic evolution equation that every physical solution must obey. Eliminating the elliptic component of MHS in this manner eliminates the confounding mixed type of the original PDE system, thereby opening the door to applying (Hamiltonian) hyperbolic PDE theory to stellarator equilibria. (Video recording of talk) |

Feb 15(16), 2024 10:00-11:00am (EST) | B. Lee | Stellarator Coil Optimization Supporting Multiple Magnetic ConfigurationsAbstract: 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 each configuration. We show that it is possible to achieve flexibility, quasisymmetry, and nested flux surface volume to reasonable degrees with a relatively simple coil set through an NCSX-like example. This example coil design is optimized to achieve three rotational transform targets and nested flux surface volumes in each magnetic configuration larger than the NCSX design plasma volume. Our work suggests that there is a tradeoff between flexibility, quasisymmetry, and volume of nested flux surfaces.(Video recording of talk) |

Feb 1(2), 2024 10:00-11:00am (EST) | K. Garcia | Exploration of Non-Resonant Divertor Features on the Compact Toroidal HybridAbstract: Non-resonant divertors (NRDs) separate the confined plasma from the surrounding plasma facing components (PFCs). The resulting striking field line intersection pattern on these PFCs is insensitive to plasma equilibrium effects. However, a complex scrape-off layer (SOL), created by chaotic magnetic topology in the plasma edge, connects the core plasma to the PFCs through varying magnetic flux tubes. The Compact Toroidal Hybrid (CTH) serves as a test-bed to study this by scanning across its inductive current. Simulations observe a significant change of the chaotic edge structure and an effective distance between the confined plasma and the instrumented wall targets. The intersection pattern is observed to be a narrow helical band, which we claim is a resilient strike line pattern. However, signatures of finger-like structures, defined as heteroclinic tangles in chaotic domains, within the plasma edge connect the island chains to this resilient pattern. The dominant connection length field lines intersecting the targets are observed via heat flux modeling with EMC3-EIRENE. At low inductive current levels, the excursion of the field lines resembles a limited plasma wall scenario. At high currents, a private flux region is created in the area where the helical strike line pattern splits into two bands. These bands are divertor legs with distinct SOL parallel particle flow channels. The results demonstrate the NRD strike line pattern resiliency within CTH, but also show the underlying chaotic edge structure determining if the configuration is diverted or limited. Additionally, to extrapolate these fundamental findings to optimized stellarator configurations, the impact of strong shaping on NRD performance and resiliency is investigated. The Helically Symmetric eXperiment (HSX) can be used to explore NRD features in a quasihelically symmetric stellarator. We show preliminary results and ongoing questions for the work. This work supports future design efforts for a mechanical structure for the NRD. (Video recording of talk) |

Jan 18(19), 2024 10:00-11:00am (EST) | E. Kolemen | Some Midsize Research Stellarator Options for the USAbstract: While tokamak designs are mature enough to try to build high Q fusion reactors, the large space of stellarators has not been explored enough to find the ideal stellarator reactor options. While private companies are studying different technologies, and many countries (e.g. China, Spain, Germany) studying next level stellarators for physics studies, U.S. DOE does not have a national lab scale research stellarator project. Given the the huge unknown in stellarator transport/divertor, I will try to explain why building a \$100-\$200M midsize stellarator would be a great research option for DOE. A stellarator that can study many different plasma configurations can fulfill that mission within the funding restrictions (no T, no \$1B project so no HTS). I will explain some of the ideas we are exploring for such a reactor. One idea is many small sources (power supplies) connected on a torus, a second is to vary the conductivity of the torus surface with a 1000s of pins of different conductivity (copper, carbon) inside a shell. These setups at least theoretically can make a wide variety of stellarators, more analysis is underway to study the engineering feasibility.(Video recording of talk) |

Jan 4(5), 2024 10:00-11:00am (EST) | K. Hammond | Improved Stellarator Permanent Magnet Designs Through Combined Discrete and Continuous OptimizationsAbstract: A common optimization problem in the areas of magnetized plasmas and fusion energy is the design of magnets to produce a given three-dimensional magnetic field distribution to high precision. When designing arrays of permanent magnets for stellarator plasma confinement, such problems have tens of thousands of degrees of freedom whose solutions, for practical reasons, should be constrained to discrete spaces. We perform a direct comparison between two algorithms that have been developed previously for this purpose, and demonstrate that composite procedures that apply both algorithms in sequence can produce substantially improved results. One approach uses a continuous, quasi-Newton procedure to optimize the dipole moments of a set of magnets and then projects the solution onto a discrete space. The second uses an inherently discrete greedy optimization procedure. The approaches are both applied to design arrays of cubic rare-Earth permanent magnets to confine a quasi-axisymmetric plasma with a magnetic field on axis of 0.5 T. The first approach tends to find solutions with higher field accuracy, whereas the second can find solutions with substantially fewer magnets. When the approaches are combined, they can obtain solutions with magnet quantities comparable to the second approach while exceeding the field accuracy of what either approach can achieve on its own.(Video recording of talk) |

Dec 21(22), 2023 10:00-11:00am (EST) | K. Camacho Mata | Near-Axis QIAbstract: We develop the formalism of the first order near-axis expansion of the MHD equilibrium equations described in Garren & Boozer (1991), Plunk et al. (2019) and Plunk et al. (2021), for the case of a quasi-isodynamic, N-field period, stellarator symmetric, single-well magnetic field equilibrium. The importance of the magnetic axis shape is investigated, and we conclude that control of the curvature and torsion is crucial to obtain omnigenous configurations with finite aspect ratio and low effective ripple, especially for a higher number of field periods. For this reason a method is derived to construct classes of axis shapes with favourable curvature and torsion. Solutions are presented, including a three-field-period configuration constructed at an aspect ratio of A=20, with a maximum elongation of e=3.2 and an effective ripple under 1%, which demonstrates that high elongation is not a necessary feature of QI stellarators. (Video recording of talk) |

Dec 7(8), 2023 | --- | ANU Conference - no Simon's Hour meeting |

Nov 23, 2023 | --- | Thanksgiving - USA Holiday - no Simon's Hour meeting |

Nov 9(10), 2023 10:00-11:00am (EST) | D. Dudt | Magnetic Fields with General OmnigenityAbstract: Omnigenity is a desirable property of toroidal magnetic fields that ensures confinement of trapped particles. All of the ideal magnetohydrodynamic equilibria previously found to approximate omnigenity have been either axisymmetric, quasi-symmetric, or have poloidally closed contours of magnetic field strength B. However, general omnigenous equilibria are a much larger design space than these subsets with hidden symmetries. A new model is presented and employed in the DESC stellarator optimization suite to represent and discover the full parameter space of omnigenous equilibria. Examples far from quasi-symmetry with poloidally, helically, and toroidally closed B contours are shown to have low neoclassical collisional transport and fast particle losses.(Video recording of talk) |

Oct 26(27), 2023 10:00-11:00am (EDT) | J. Velasco | Robust Stellarator Optimization via Flat Mirror Magnetic FieldsAbstract: Stellarator magnetic configurations need to be optimized in order to meet all the required properties of a fusion reactor. We have recently introduced [1] the notion of robust optimization via a flat mirror term: we have shown that a quasi-isodynamic configuration with sufficiently small radial variation of the mirror term can achieve the maximum-J property at low plasma β. This results in small radial transport of energy and good confinement of bulk and fast ions even if the configuration is not very close to perfect omnigeneity, and for a wide range of plasma scenarios, including low β and small radial electric field. This opens the door to constructing better stellarator reactors. On the one hand, they would be easier to design, as they would be more robust against error fields. On the other hand, they would be easier to operate since, both during startup and steady-state operation, they would require less auxiliary power, and the damage to plasma-facing components caused by fast ion losses would be reduced to acceptable levels. The most prominent result of this optimization strategy has been CIEMAT-QI [2], a flat-mirror quasi-isodynamic magnetic configuration that, in terms of physics performance, qualifies as a potential candidate for a stellarator reactor design. It is the first member of a growing family of optimized configurations [3] resulting from the exploration of this newly-identified region of the stellarator configuration space. [1] J.L. Velasco et al. 2023 Nucl. Fusion in press https://doi.org/10.1088/1741-4326/acfe8a[2] E. Sánchez et al. 2023 Nucl. Fusion 63 066037.[3] G. Godino-Sedano et al. 2023 European Fusion Theory Conference.(Video recording of talk) |

Oct 12(13), 2023 10:00-11:00am (EDT) | D. Bindel A. Bhattacharjee M. Churchill S. Henneberg | SciDAC / Eurofusion Collaboration / Hidden Symmetries - Simons Hour discussion Simons: Hidden Symmetries - Slides (Bindel) SciDAC: HiFiStell - Slides (Bhattacharjee) SciDAC: StellFoundary - Slides (Churchill) EuroFUsion: TSVV - Slides (Henneberg)(Video recording of talk) |

Sept 28(29), 2023 10:00-11:00am (EDT) | N. Riva | Development of the First Non-Planar REBCO Stellarator Coil Using VIPER cableAbstract: The benefits of operating fusion devices, such as tokamaks and stellarators, at high fields make high-temperature superconducting magnets necessary to realize a compact fusion power system. Superconducting stellarators, such as W7-X, have used standard low-temperature superconductor technology niobium-titanium. ARPA-E has recently funded a two-year project led by the startup Type One Energy and involving the Fusion Technology Institute at the University of Wisconsin-Madison, the Plasma Science to design and fabricate the first non-planar high-temperature superconductor (HTS) rare-earth barium copper oxide (REBCO) coil for a high-field stellarator based on the SPARC tokamak's VIPER cable concept.(Video recording of talk) |

Sept 14(15), 2023 10:00-11:00am (EDT) | S. McIntosh | ITER’s Assembly Tolerances, Translating Physics Limits to MillimetresAbstract: The manipulation of massive components under tight tolerances presents a formidable challenge. Whilst reference to millimetres can seem commonplace, the realization that these targets must be satisfied over long distances and with very large components, such ITER’s 17 meter high, 360 tonne D-shaped Toroidal Field magnets, places this alignment challenge firmly in the world of precision engineering.The need for such an alignment precision for ITER’s magnets comes from physics. The magnets produce the magnetic field to confine the plasma and this needs to be as close to perfectly symmetric as possible. ITER is equipped with a set of error field correction coils to correct for such inaccuracies, but their capabilities are limited. Limitations on the number and current capability of these coils restrict corrections to the first three toroidal harmonics and with a limited magnitude. It follows thus that asymmetries introduced during the manufacturing and assembly process must be kept as low as possible and mandatorily within the correction coils’ capabilities. The impact of component and manufacturing inaccuracies are assessed via a series of large scale Monte Carlo simulations. These simulations propagate uncertainties from assembly tolerances to physics limits. Taking an iterative approach, we show how multiple Monte Carlo simulations are used to tune assembly tolerances applied to each component. The result is a coherent assembly strategy for the Toroidal Field coils, Poloidal Field coils, the Central Solenoid and the first wall. We demonstrate how trade-offs may be made between tolerances placed on various components. One such trade-off links the TF coils to the first wall. Here, an increased alignment tolerance is applied to the TF coils to relax the first wall alignment tolerance necessary to limit start-up heat loads. (Video recording of talk) |

Aug 31(Sep 1), 2023 7:00- 8:00pm (EDT) 7:00- 8:00am (AWST) 9:00-10:00am (AEST) | L. Roberts | Large-scale Derivative-Free Optimization Using Random Subspace MethodsAbstract: Many standard optimization algorithms require being able to cheaply and accurately compute derivatives for the objective and/or constraint functions. However, in the presence of noise, or computationally expensive or black-box procedures, derivative information may be inaccurate or impractical to compute. Derivative-Free Optimization (DFO) encompasses a variety of techniques for nonlinear optimization in the absence of derivatives. However, such techniques can struggle on large-scale problems for reasons including high linear algebra costs and strong dimension-dependency of worst-case complexity bounds. In this talk, I will discuss model-based and direct search DFO algorithms based on iterative searches in randomly drawn subspaces and show how these methods can be used to improve the scalability of DFO. This is joint work with Coralia Cartis (Oxford) and Clément Royer (Paris Dauphine-PSL).(Video recording of talk) |

Aug 17(18), 2023 10:00-11:00am (EDT) | C. Lowe, A. Punjabi & S. Naik | STAR_Lite Stellarator projectAbstract: STAR_Lite is a project to build a small stellarator experiment at Hampton University (HU). The Stellarator for Training And Research builds on the theoretical work of Profs. Alkesh Punjabi and Halima Ali in the HU Mathematics Department and their collaborator, Prof. Allen Boozer, at Columbia’s Department of Applied Math and Applied Physics. STAR_Lite will begin its physics design phase in the Fall. It is supported by internal, federal, and private funding. In the first part of the talk, Prof. Lowe will give a status update on the project. In the second part of the talk, Prof. Punjabi will discuss some of the science topics around effects of magnetic perturbations on plasma stability and confinement and non-resonant divertors that will be studied in STAR_Lite. Lastly, Dr. Shibabrat Naik, who will Join the faculty of the HU Mathematics Department in the Fall, will discuss some additional plasma confinement concepts that may be explored in the STAR_Lite project and his motivation for joining our efforts.We are pleased to acknowledge the interest and support of the U.S. Dept. of Energy, the Simons Foundation, and Hampton University for plasma fusion science at HU. (Video recording of talk) |

Aug 3(4), 2023 10:00-11:00am (EDT) | C. Albert | Optimizing phase-space barriers for alpha particles via fast classificationAbstract: Optimization of stellarators towards low fusion alpha particle losses is a critical task beyond usual transport optimization. Combining symplectic integration methods with classifiers acting on footprints in a Poincaré section in phase space have been shown to accelerate this task significantly. The underlying principle is the distinction of regular orbits that are either promptly or never lost, and chaotic orbits for which final predictions can only be made by tracing them directly. Collisionless orbits can move only along contours of constant magnetic moment in phase-space. This leads to the requirement of regular orbits on a discrete set of magnetic flux surfaces. These act as staggered barriers that cannot be crossed by collisionless alpha particle orbits. We demonstrate that the approach remains valid also to minimize collisional energy losses.(Video recording of talk) |

Jul 20(21), 2023 10:00-11:00am (EDT) | M. Ruth | Finding Invariant Circles from a Single TrajectoryAbstract: A problem in stellarator optimization is how to automatically analyze MHD equilibria when they do not consist of nested flux surfaces. Typically researchers look at Poincaré plots to classify trajectories as invariant circles (flux surfaces), islands, or chaos. Moreover, after trajectory classification, additional work is needed to determine the Fourier coefficients of the invariant structures. The weighted Birkhoff average [1] has been shown to classify trajectories, but finding invariant circles and rotation numbers is still missing. In this talk, we will show how a technique from sequence extrapolation, the reduced rank extrapolation method (RRE), can also be used to classify trajectories with a single linear least-squares solve. For the non-chaotic trajectories, a subsequent eigenvalue problem returns the number of islands, the rotation number, and the Fourier coefficients of the trajectory. This method will be demonstrated with a variety of examples.[1] E. Sander and J. Meiss, Physica D: Nonlinear Phenomena, 411 (2020), p. 132569 (Video recording of talk) |

Jul 6(7), 2023 10:00-11:00am (EDT) | J. Meiss | Weighted Birkhoff AveragesAbstract: Birkhoff’s ergodic theorem implies that when an orbit is ergodic on an invariant set, spatial averages of a phase-space function can be computed as time averages. However the convergence of a time average can be very slow. In 2016, Das et al introduced a C^∞ weighting technique that they later showed can give super-polynomial convergence of averages for orbits that lie on (Diophantine) invariant tori. Evelyn Sander and I showed that this Weighted Birkhoff Average (WBA) can give a sharp distinction between chaotic and regular dynamics and that it allows accurate computation of rotation vectors for regular orbits.Nathan Duignan and I applied* this to several flows: the two-wave Hamiltonian system, the Paul-Hudson-Helander model magnetic field line flow and a quasiperiodically forced, dissipative system with a “strange nonchaotic attractor”. In practice the WBA is shown to achieve machine precision for quasiperiodic orbits after an integration time of O(10^3) periods. The contrasting, relatively slow convergence for chaotic trajectories allows an efficient discrimination criterion. We propose that the WBA could be more efficient than visualizing Poincare sections or computing Lyapunov exponents. *Duignan, N. and J. D. Meiss (2023). “Distinguishing between Regular and Chaotic orbits of Flows by the Weighted Birkhoff Average.” Physica D 449(July): 133749. (Video recording of talk) |

Jun 22(23), 2023 10:00-11:00am (EDT) | F. Parra-Diaz | Linear Equations for Stellarator Local MHD EquilibriaAbstract: Building on previous work [1, 2, 3], we develop a new set of linear equations to determine the magnetic geometry coefficients needed for local gyrokinetic simulations on a flux surface of interest. The inputs required for the model are the shape of the flux surface, the radial derivative of that shape and four constants. One possible choice for these four constants is the pressure gradient, the gradient of the toroidal flux, and the rotational transform and its radial derivative at the flux surface of interest. When we apply our equations to rational flux surfaces, we find that, for flux surfaces to exist, two conditions must be satisfied. One of the conditions is the well-known Hamada condition [4], but the other has not been discussed in the literature to our knowledge.References [1] C.C. Hegna, Phys. Plasmas 7, 3921 (2000), [2] A.H. Boozer, Phys. Plasmas 9, 3726 (2002), [3] J. Candy and E.A. Belli, J. Plasma Phys. 81, 905810323 (2015), [4] S. Hamada, Nucl. Fusion 2, 23 (1962). (Video recording of talk) |

Jun 8(9), 2023 10:00-11:00am (EDT) | N. Mandell | Trinity+GX for Stellarator Profile PredictionAbstract: Trinity3D+GX is a framework that leverages multi-scale gyrokinetic theory to model macro-scale profile evolution in fusion plasmas (tokamaks and stellarators) due to micro-scale turbulent processes. In this talk I will first provide a brief background on the multi-scale gyrokinetic theory underpinning the model. I will then discuss the GX gyrokinetic code, which has been developed as a GPU native code that uses an efficient pseudo-spectral discretization scheme to target fast turbulence calculations for fusion reactor design and optimization. This enables GX to be embedded as the micro-turbulence model in the Trinity3D transport solver for tractable fusion profile prediction (and evolution) calculations. I will highlight some preliminary results of modeling W7X plasmas with the Trinity3D+GX system and discuss future plans for using the framework in experimental studies as well as stellarator FPP design and optimization.(Video recording of talk) |

May 25(26), 2023 10:00-11:00am (EDT) | C. Nührenberg | Linearized Ideal Magnetohydrodynamic Stability in StellaratorsIn the past, aspects of linearized ideal MHD stability, e.g. the vacuum field magnetic well, reduced parallel current density, or Mercier's criterion, were targeted for in the optimization of stellarator experiments such as W7-X and HSX. In this tutorial, the stability of plasma equilibria is discussed based on the energy principle of ideal MHD in the form of global mode analyses and local stability criteria. The field-line ballooning equation and Mercier's criterion are derived. Example applications typical for various classes of equilibria (tokamak, stellarator, weak or medium magnetic shear) are used to point out relationships between the violation of the local stability criteria and the spatial structure of global modes aiming at a categorization of ideal MHD stability limits. In the light of experimental results of the W7-AS stellarator and of the LHD torsatron, experiments are planned in W7-X to clarify the question of how important ideal MHD is in stellarator optimization. (Video recording of talk) |

May 11(12), 2023 7:00-8:00pm (EDT) | M. Landreman | Efficient Calculation of Internal Magnetic Field & Self-Force for Electromagnetic CoilsFor designing magnetic fusion facilities, it is important to compute the magnetic field inside the electromagnetic coils due to superconducting quench limits. It is also necessary to compute the Lorentz force on coils to ensure a support structure is feasible. For both the internal field and force calculations, the coils cannot be approximated in the usual way as infinitesimally thin filaments due to divergences when the source and evaluation points coincide, so more computationally demanding calculations are usually required, resolving the finite cross-section of the conductors. Here, we present a new alternative method that enables both the internal magnetic field vector and self-force to be computed rapidly and accurately within a 1D filament model. The method is obtained by rigorous analysis of the singularity, such that the filament model matches the true high-dimensional integrals for the field and force at high coil aspect ratio. The new filament model exactly recovers analytic results for a circular coil, and is shown to accurately reproduce direct finite-cross-section calculations for a non-planar coil of the HSX stellarator. Due to the efficiency of the model here, it is well suited for use inside optimization, such as in the optimization of stellarator coil shapes. (Video recording of talk) |

Apr 13(14), 2023 7:00- 8:00pm (EDT) 7:00- 8:00am (AWST) 9:00-10:00am (AEST) | D. Perrella | Existence of Global Symmetries of Divergence-Free Fields with First IntegralsThe relationship between symmetry ﬁelds and ﬁrst integrals of divergence-free vector ﬁelds is explored in three dimensions in light of its relevance to plasma physics and magnetic conﬁnement fusion. A Noether-type Theorem is known: for each such symmetry, there corresponds a conserved quantity. The extent to which the converse is true is investigated. In doing so, a reformulation of this Noether-type Theorem is found for which the converse holds on what is called the toroidal region. Some consequences of the methods presented are quick proofs of the existence of ﬂux coordinates for magnetic ﬁelds in high generality; without needing to assume a symmetry such as in the cases of magneto-hydrostatics (MHS) or quasi-symmetry. https://arxiv.org/pdf/2303.03191 (Video recording of talk) |

Mar 30(31), 2023 10:00-11:00am (EDT) | M. Padidar | Direct Optimization of Fast-Ion Confinement in StellaratorsConfining energetic ions such as alpha particles is a prime concern in the design of stellarators. However, directly measuring alpha confinement through numerical simulation of guiding-center trajectories has been considered to be too computationally expensive and noisy to include in the design loop, and instead has been most often used only as a tool to assess stellarator designs post hoc. In its place, proxy metrics, simplified measures of confinement, have often been used to design configurations because they are computationally more tractable and have been shown to be effective. Despite the success of proxies, it is unclear what is being sacrificed by using them to design the device rather than relying on direct trajectory calculations. In this study, we optimize stellarator designs for improved alpha particle confinement without the use of proxy metrics. In particular, we numerically optimize an objective function that measures alpha particle losses by simulating alpha particle trajectories. While this method is computationally expensive, we find that it can be used successfully to generate configurations with low losses. (Video Recording of talk) |

Mar 23 - 24, 2023 | Various | Hidden Symmetries Annual Meeting @ Simons Foundation (NYC)Rogerio Manuel Cabete de Jesus Jorge, IST Lisbon, Portugal Nathan Duignan, University of Sydney Benjamin Faber, University of Wisconsin-Madison Daniel Ginsberg, Princeton University Egemen Kolemen, Princeton University Gabriel Plunk, Max Planck Institute for Plasma Physics Josefine Proll, Technical University of Eindhoven Adelle Wright, PPPL |

Mar 20 - 22, 2023 | Various | Hidden Symmetries Team Meeting @ Princeton University (Carnegie Center) Agenda: https://hiddensymmetries.princeton.edu/meetings/2023-team-meeting-mar |

Mar 16(17), 2023 7:00-8:00pm (EST) | D. Anderson | History of the HSX, Optimization and Design(Video recording of talk) |

Mar 2(3), 2023 10:00-10:05am (EST) 10:05-11:00am (EST) | D. Bindel S. Hudson | Hidden Symmetries Study UpdatesA Tutorial/Review of Almost-Invariant Surfaces (Quadratic-Flux-Minimizing and Ghost Surfaces) and Heat Transport in Non-Integrable FieldsAn introduction/tutorial on ghost surfaces (GS) and (weighted) quadratic-flux-minimizing (QFM) surfaces, perhaps better described as action-gradient-squared surfaces, and why QFM surfaces (without the "weight") are like snowflake fractals on a hot summer's day, and under what conditions ghost-surfaces and QFM surfaces coincide, and how these surfaces are isotherms of the anisotropic diffusion equation in the limit that \kappa_\perp approaches zero, and why don't we use perturbation theory to solve for the anisotropic diffusion equation by expanding about \kappa_\perp = 0. (Video recording of talk) |

Feb 16(17), 2023 7:00-7:05pm (EST) 7:05-8:00pm (EST) | D. Bindel M. Nemec | Hidden Symmetries Study UpdatesGradient-Based Shape Optimization using High-Fidelity Simulations with Goal-Oriented Error ControlAerodynamic shape optimization is emerging as an indispensable tool in the design of aerospace vehicles. This presentation focuses on several innovations that have made numerical optimization practical in the context of computational fluid dynamics. We begin with the adjoint method. This method provides optimization gradients and is also used to estimate the level of discretization error in the outputs of interest. The benefits are threefold. First, the computational cost of gradient evaluations is essentially independent of the number of design variables. Second, it offers direct control over discretization error through use of adaptive mesh refinement to improve confidence in the optimized designs and to eliminate the requirement of hand-crafting a sufficiently general grid that is appropriate for all candidate designs. Third, we obtain additional cost savings by using progressive optimization, where the depth of the adaptive mesh refinement is systematically increased as the design improves. In addition, we highlight a component-based geometry approach for flexibility in choosing both the shape parameterization and geometric modelers, along with symbolic definition of objectives and constraints for general problem specification. We present design examples involving real-world aircraft configurations in shock-dominated flows, including sonic boom shaping for NASA’s new X-59 aircraft. (Video recording of talk) |

Feb 2(3), 2023 10:00-10:05am (EST) 10:05-11:00am (EST) | D. Bindel I. Farcas / B. Peherstorfer | Hidden Symmetries Study UpdatesLeveraging Machine Learning for Large-Scale Multi-Fidelity Uncertainty QuantificationRecent advances in computational science and high-performance computing enable the simulation of large-scale real-world problems such as turbulent transport in magnetic confinement devices with ever-increasing realism and accuracy. However, these simulations remain computationally expensive even on large supercomputers, which prevents straightforward approaches to important many-query applications such as uncertainty quantification. In contrast, data-driven machine-learning methods such as those based on deep neural networks provide computationally cheaper low-fidelity models, but typically require large training sets of high-fidelity model evaluations to be predictive, which hampers a straightforward application in large-scale, computationally expensive problems. In this presentation, we demonstrate that data-driven low-fidelity models learned from few data samples can nevertheless be effectively used for large-scale uncertainty quantification: The key is to combine these low-fidelity models with the high-fidelity model in a multi-fidelity fashion. The first part of this presentation focuses on a multi-fidelity Monte Carlo sampling approach in which a hierarchy of data-driven low-fidelity models is constructed using both the full set of uncertain inputs and subsets comprising only selected, important parameters. We illustrate the proposed method in a plasma micro-turbulence simulation scenario concerning turbulence suppression via energetic particles with $14$ stochastic parameters, demonstrating that it is about two orders of magnitude more efficient than standard Monte Carlo methods measured in single-core performance. This translates into a runtime reduction from around eight days to one hour on 240 cores on parallel machines. The second part of this presentation introduces a context-aware multi-fidelity Monte Carlo method that optimally balances the costs of training low-fidelity models with the costs of Monte Carlo sampling. Our theory shows that low-fidelity models can be overtrained, which is in stark contrast to traditional surrogate modeling and model reduction techniques that construct low-fidelity models with the primary goal of approximating well the high-fidelity model outputs. Numerical experiments in a plasma micro-turbulence simulation scenario with 12 uncertain inputs show speedups of up to two orders of magnitude compared to standard methods, which corresponds to a runtime reduction from 72 days to about four hours on 32 cores on parallel machines. (Video recording of talk) |

Jan 19(20), 2023 7:00-7:05pm (EST) 7:05-7:30pm (EST) 7:35-8:00pm (EST) | D. Bindel F. Law / M. Nakata | Hidden Symmetries Study UpdatesTalk 1 - Meta Variance Reduction Schemes for Estimation of Alpha Particle ConfinementNumerical estimation of energetic particle confinement using Monte Carlo methods remains a computationally demanding task in stellarator optimization loops, due to the large number of simulated trajectories required at each iteration. We introduce meta estimators to accelerate the estimation of confinement statistics, by simultaneously leveraging multiple constituent variance reduction techniques including multifidelity Monte Carlo, importance sampling, and information reuse. Each constituent technique takes advantage of a different facet of the estimation problem, leading to quasi-multiplicative speedup when combined. We test our meta estimators, using data-driven surrogates and biasing densities, on a coil optimization seeking to reproduce a quasi-axisymmetric configuration (Landreman and Paul, 2022). We observe that each meta estimator outperforms its constituent variance reduction techniques, with the highest performing meta estimator yielding two orders of magnitude speedup compared to standard Monte Carlo estimation at an equivalent computational cost. (Video recording of Law talk)Talk 2 - Geometry Induced Activation of Zonal Flows in Stellarator PlasmasOne of the prominent physical processes in confined plasmas is spontaneous formation of mesoscopic coherent structures, e.g., zonal flows, geodesic acoustic oscillations, and radially elongated streamers, which are nonlinearly excited in microscopic turbulence. Then, how can we “activate” or “enhance" such nonlinear structures? The diversity of 3D magnetic geometries in stellarators motivates us to explore the capabilities of enhancing the zonal-flow formation. This talk presents a recent activity on the geometry-induced activation of the zonal flows in a context of stellarator optimizations with a nonlinear proxy model. (Video recording of NAKATA talk) |

Jan 5(6), 2023 10:00-10:05am (EST) 10:05-11:00am (EST) | D. Bindel D. Ginsberg | Hidden Symmetries Study UpdatesOn the Distribution of Heat in Integrable and Non-Integrable Magnetic FieldsWe study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimension two and higher. Provided the magnetic field is sufficiently structured (integrable in the sense that it is fibered by co-dimension one invariant tori, on most of which the field lines ergodically wander) and the effective thermal diffusivity transverse to the tori is small, it is proved that the temperature distribution is well approximated by a function that only varies across the invariant surfaces. The same result holds for "nearly integrable" magnetic fields up to a "critical" size. In this case, a volume of non-integrability is defined in terms of the temperature defect distribution and related to the non-integrable structure of the magnetic field, confirming a physical conjecture of Paul-Hudson-Helander. Our proof crucially uses a certain quantitative ergodicity condition for the magnetic field lines on full measure set of invariant tori, which is automatic in two dimensions for magnetic fields without null points and, in higher dimensions, is guaranteed by a Diophantine condition on the rotational transform of the magnetic field. (Video recording of Ginsberg talk) |

Dec 22(23), 2022 7:00-7:05pm (EST) 7:05-8:00pm (EST) | D. Bindel H. Yamaguchi | Hidden Symmetries Study UpdatesDevelopment of Coil Shaping Based Optimization Code at NIFS(Video recording of talk) |

Dec 8(9), 2022 10:00-10:05am (EST) 10:05-11:00am (EST) | D. Bindel Y. Suzuki | Hidden Symmetries Study UpdatesStrong Nonlinearity of MHD Stability in StellaratorsIn stellarators, one of characteristics on MHD is strong nonlinearity. Because, sophisticated mode couplings of the 3D equilibrium and instability must be considered, and those are saturated nonlinearly. For example, in the LHD experiment, although the Mercier criterion predicts the linearly unstable plasma, the non-disruptive discharge could be realized. The dissipation and damping mechanisms are still mysteries. In this talk, some of results from 3D nonlinear simulations will be discussed to understand the strong nonlinearity of MHD in stellarators. (Video recording of Suzuki talk) |

Nov 10(11), 2022 10:00-10:05am (EST) 10:05-11:00am (EST) | M. Landreman D. Spong | Hidden Symmetries Study UpdatesHistory of the HSX and QPS Stellarator DesignsQPS (Quasi Poloidal Stellarator) design history from 1990-2008. (Video recording of Spong talk) |

Oct 27(28), 2022 7:00-7:05pm (EDT) 7:05-8:00pm (EDT) | D. Bindel
| Hidden Symmetries Study UpdatesIntroduction to Turnstiles & TransportsTurnstiles were introduced by MacKay, Meiss and Percival in 1984 as a mechanism to quantify the flux through broken invariant surfaces, such as the homoclinic tangle formed by an unstable periodic orbit or a cantorus that is the remnant of an invariant torus. A turnstile is constructed from the stable and unstable manifolds of a pair of orbits, and allows one to visualize the transport as the flux through a partial barrier, and show that it can be localized to a single “gap” in the orbit pair. MacKay’s renormalization theory gives a scaling law for the growth of the flux as a function of perturbation strength as a torus is broken. A related power law appears to hold for the exit-time distribution from a chaotic region containing regular islands, though a complete theoretical justification is still lacking. (Video recording of Meiss talk) |

Oct 13(14), 2022 10:00-10:10am (EDT) 10:10-11:00am (EDT) | D. Bindel R. MacKay / N. Shibabrat / E. Paul | Hidden Symmetries Study UpdatesUpdate on Isodrasticity (MacKay Slides) (Shibabrat Slides) (Paul Slides)Group Presentation on Isodrasticity (Video recording of MacKay/Shibabrat/Paul talk) |

Sep 29(30), 2022 7:00-7:10pm (EDT) 7:10-8:00pm (EDT) | D. Bindel / A. Bhattacharjee D. Bindel | Hidden Symmetries Study UpdatesOptimization Under Stability ConstraintsIn this talk, we survey some prior work on the treatment of optimization problems with stability constraints in fields other than plasma physics, and we describe the types of technical issues that frequently occur when incorporating eigenvalue-based functions in the objectives or constraints of an optimization problem. We then give a very preliminary discussion of how we are thinking of incorporating stability constraints in stellarator optimization problems. (Video recording of Bindel talk) |

Sep 15, 2022 10:00-10:10am (EDT) 10:10-11:00am (EDT) | D. Bindel F. Pasqualotto | Hidden Symmetries Study UpdatesOn The Construction Of 3D MHD Equilibria In General Bounded DomainsThis talk will discuss a recent paper (https://arxiv.org/abs/2208.11109) concerning the construction of magnetohydrostatic (MHS) equilibria in a general 3D domain via a long-time limit of a suitably regularized MHD system. It is a rigorous justification of “magnetic relaxation” in the context of a regularized MHD system. (Video recording of Pasqualotto talk) |

Sep 1, 2022 8:00-8:10 (EDT) 8:10-9:00 (EDT) | D. Bindel
| Hidden Symmetries Study UpdatesCoarse-Grained Gyrokinetics For The Critical Ion Temperature Gradient In StellaratorsThis talk will present a modified gyrokinetic theory to predict the critical gradient that determines the linear onset of the ion temperature gradient (ITG) mode in stellarator plasmas. A coarse-graining technique is applied to the drift curvature, entering the standard gyrokinetic equations, around local minima. Thanks to its simplicity, this novel formalism yields an estimate for the critical gradient with a computational cost low enough for application to stellarator optimization. When comparing against a gyrokinetic solver, our results show good agreement for an assortment of stellarator designs. Insight gained here into the physics of the onset of the ITG-driven instability enables us to devise a compact configuration, similar to the Wendelstein 7-X device, but with almost twice the ITG linear critical gradient, an improved nonlinear critical gradient, and reduced ITG mode transport above the nonlinear critical gradient. Preliminary optimization results applying the coarse-graining technique to quasisymmetric stellarator design using SIMSOPT will also be discussed. (Video recording of Roberg-Clark talk) |

Aug 18, 2022 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee
| Hidden Symmetries Study UpdatesOmnigeneous Toroidal Plasma Equilibria (Slides)Omnigeneous Toroidal Plasma Equilibria (Paper)A simple condition is derived for omnigeneous toroidal plasma equilibria, which means that in a collisionless plasma the turning points of a trapped particle remain on the same magnetic surface. Omnigeneity is important for it assures that collisionless particle trajectories are consistent with achieving ignition in toroidal fusion systems. When the magnetic field strength depends on only one angular coordinate in Boozer coordinates, the magnetic field is quasi-symmetric, and drift trajectories are confined by a conserved canonical momentum. It is shown that a magnetic field is omnigeneous when it obeys the single-angle constraint at extrema of the field strength. Elsewhere it can be far from quasi-symmetric, but must obey a symmetry in a function R about field strength minima. When the field strength depends only on the poloidal angle near extrema, it is called quasi-poloidally symmetric. For this case, it is shown that bootstrap current need not be zero and the sign of the electric potential is more obscure than generally assumed. https://arxiv.org/pdf/2208.02391(Video recording of Boozer talk) |

Aug 4, 2022 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee
| Hidden Symmetries Study UpdatesMinimizing Island Width Sensitivity to Maximize Stellarator Coil TolerancesStellarators have advantageous physical properties but require complicated engineering. Stellarator equilibrium magnetic fields are non-integrable in general, while the integrability of stellarator magnetic fields is targeted in optimization and typically determines coil tolerances. This talk discusses the use of FOCUS to optimize coils using gradient based algorithms in order to minimize magnetic field errors. (Video recording of Kruger talk) |

July 21, 2022 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee A. Zocco | Hidden Symmetries Study UpdatesKinetic Infernal Modes and Magnetic Reconnection in Magnetically Confined Fusion Plasmas (Abstract)Magnetically confined fusion plasmas must meet some basic magnetohydrodynamic (MHD) stability requirements in order to be of any use in a reactor. In particular, they must not experience long wavelength instabilities. These can involve the whole plasma column, and have to be either operationally avoided or altogether inhibited by design. The free energy needed for such instabilities to grow is provided by current density and plasma pressure gradients. In this talk I will review the basic concepts of the theory of infernal modes, paying attention to their relation to the reconnecting internal kink mode (especially in W7-X) and discuss their extension to fusion relevant regimes in which both electron and ion kinetics are important. (Video recording of Zocco talk) |

June 9, 2022 8:00-8:15 (EDT) 8:15-9:00 (EDT) | P. Helander/ A. Bhattacharjee E. Paul | Greifswald Retreat, June 27-July 8 (15 minutes)Energetic Particle Loss Mechanisms In Reactor-Scale Equilibria Close To QuasisymmetryThe transport of fusion-born alpha particles in 3D equilibria is largely determined by collisionless physics. Several transport mechanisms have been implicated in stellarator configurations, including stochastic diffusion due to transitions, ripple trapping, and superbanana orbits. On longer time scales, many lost trajectories undergo transitions between trapping classes, either with periodic or irregular behavior. Possible optimization strategies for each of the relevant transport mechanisms and perform a comparison between classified guiding center losses and metrics for superbanana transport will be discussed. (Video recording of Helander/Bhattacharjee/Paul talk) |

May 12, 2022 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee W. Sengupta | Hidden Symmetries Study UpdatesPreferred Magnetic Axes For Optimal Quasi-AxisymmetryThis talk will discuss the preferred choice of the magnetic axes for optimal quasisymmetry, which is evident from the numerical optimization of asymptotic expansions near the magnetic axis. The talk will show that the magnetic axis is well described for small rotational transforms by the same equations that govern Euler-Kirchhoff elastic rod centerlines, and present analytical and numerical evidence applicable for a broad range of quasi-axisymmetric stellarators. (Video recording of Sengupta talk) |

Apr 28, 2022 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee F. Volpe & C. Smiet | Hidden Symmetries Study UpdatesRenaissance Fusion - Technology and PlansRenaissance Fusion strives to make stellarators smaller via High-Temperature Superconducting (HTS) coils. It makes them less radioactive and more resilient to alpha particle losses by flowing mesoscale liquid metal walls and simpler to build - by using simpler coil winding surfaces and HTS manufacturing. Initial results will be presented in the areas of coil force minimization, simplification of the coil winding surface, neutronic optimization of the liquid wall materials, design point of a compact, profitable stellarator reactor and retrofitting of a fission power-plant. Paradigm-shifting ways of manufacturing HTS stellarator coils and extracting Tritium will be presented. (Video recording of Volpe & Smiet talk) |

Apr 14, 2022 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee R. Mackenbach | Hidden Symmetries Study UpdatesAvailable Energy and its Relation to Turbulent TransportAny collisionless plasma possesses some "available energy" (AE), which is that part of the thermal energy that may be converted into instabilities and turbulence. This talk investigates the AE carried by electrons, which are trapped in a magnetic mirror, and the ability to quickly and cheaply assess the turbulence levels, driven by these trapped electrons. (Video recording of Mackenbach talk) |

Mar 17, 2022 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee R. Dewar | Hidden Symmetries Study UpdatesQuasi-Relaxed Magnetohydrodynamics (QRxMHD) incorporating Ideal Ohm's Law Constraint (IOL) (part 2) The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium. (Video recording of Dewar talk part 2) |

Mar 3, 2022 8:00-8:10 (EST) 8:10-9:00 (EST) | A. Bhattacharjee R. Dewar | Hidden Symmetries Study UpdatesQuasi-Relaxed Magnetohydrodynamics (QRxMHD) incorporating Ideal Ohm's Law Constraint (IOL) (part 1) The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium. (Video recording of Dewar talk part 1) |

Feb 17, 2022 8:00-8:10 (EST) 8:10-9:00 (EST) | A. Bhattacharjee A. Punjabi | Hidden Symmetries Study UpdatesMagnetic Turnstiles in Nonresonant Stellarator Divertor Nonresonant stellarator divertors have a pair of magnetic flux tubes. One has field lines that go from just outside the outermost confining surface to the surrounding chamber wall, and the other has lines that come inward from the wall. This outward-inward action led to the name magnetic turnstile. Plasma is diverted along both tubes of the pair. The pair of flux tubes cross the annulus between the outermost confining surface and the walls through holes in magnetic cantori, which are the fractal remnants of magnetic surfaces. The exiting and entering flux tubes can be adjacent as in the literature on turnstiles. But, tubes were also found that have the unexpected feature of the entering and the exiting the region near the outermost confining surface at separated locations. Not only can there be two types of turnstiles but pseudo-turnstiles can also exist. A pseudo-turnstile is formed when an outer surface has a sufficiently large, although limited, radial excursion to strike the wall. The existence of non-adjacent and adjacent turnstiles and pseudo-turnstiles resolves issues that arose in earlier simulations of nonresonant stellarator divertors. (Video recording of Punjabi talk) |

Feb 03, 2022 8:00-8:10 (EST) 8:10-9:00 (EST) | A. Bhattacharjee N. Nikulsin | Hidden Symmetries Study UpdatesModels and Methods for Nonlinear Magnetohydrodynamic Simulations of StellaratorsThe JOREK code has recently been extended to allow nonlinear fully 3D stellarator simulations. This is made possible by generalizing the JOREK reduced MHD model to support stellarator geometries, and by allowing the grid to be non-axisymmetric, so that it can be aligned to the flux surfaces in a stellarator. The stellarator reduced model differs mainly in that the magnetic field can be represented as any curl-free field plus a perturbation in the stellarator model, whereas in the tokamak model it is a toroidal field plus a perturbation. It is shown that this model conserves energy, but introduces an error into momentum conservation. An alternate model, which does not guarantee energy conservation, but has a smaller momentum conservation error is also derived. The energy and momentum conservation properties of the main and alternate models are then studied numerically in the tokamak limit. The main model was then tested on a set of l=2 stellarator equilibria based on Wendelstein 7-A. The simulations demonstrate that stable full MHD equilibria are preserved in the reduced model: the flux surfaces do not move throughout the simulation, and closely match the full MHD flux surfaces. Further, both tearing and ballooning modes were simulated, and their growth rates benchmarked against the linear full MHD code CASTOR3D, showing good agreement. (Video recording of Nikulsin talk) |

Jan 20, 2022 8:00-8:10 (EST) 8:10-9:00 (EST) | A. Bhattacharjee E. Rodriguez | Hidden Symmetries Study UpdatesUnderstanding the Space of Quasisymmetric Configurations: Phases and Phase TransitionsOptimisation plays a central role in the pursuit of viable stellarator designs. Customarily, such designs result from a search in a large parameter space, guided by desirable physics requirements. This approach has proven useful but the complexity of the space makes the approach, to a large extent, a 'black box'. The talk presents our attempt to shed light on the space of quasisymmetric configurations. We identify designs with a reduced set of functions and parameters that describe the configurations approximately by expansions about their magnet axes. This allows us to structure the space in an effective and powerful way. The presentation will focus on the first step in this approach, which identifies configurations at the most basic level through fundamental geometric properties of their magnetic axes. We show that this reduced level of description is sufficient to naturally divide the space of configurations into quasisymmetric phases and phase transitions. The basic structure of the space can be leveraged to contextualise typical quasisymmetric designs and describe several important properties. We will also present some results on extensions of the basic reduced model. (Video recording of Rodriguez talk) |

Dec 9, 2021 8:00-8:10 (EST) 8:10-9:00 (EST) | A. Bhattacharjee Z. Qu | Hidden Symmetries Study UpdatesOn The Non-Existence Of Stepped-Pressure Equilibria Far From SymmetryThe Stepped Pressure Equilibrium Code (SPEC) has been successful in the construction of equilibria in 3D configurations that contain a mixture of flux surfaces, islands and chaotic magnetic field lines. In this model, the plasma is sliced into sub-volumes separated by ideal interfaces, and in each volume the magnetic field is a Beltrami field. In the cases where the system is far from possessing a continuous symmetry, such as in stellarators, the existence of solutions to a stepped-pressure equilibrium with given constraints, such as a multi-region relaxed MHD minimum energy state, is not guaranteed but is often taken for granted. Using SPEC, we have studied two different scenarios in which a solution fails to exist in a slab with analytic boundary perturbations. We found that with a large boundary perturbation, a certain interface becomes fractal, corresponding to the break-up of a Kolmogorov–Arnold–Moser (KAM) surface. Moreover, an interface can only support a maximum pressure jump while a solution of the magnetic field consistent with the force balance condition can be found. An interface closer to break-up can support a smaller pressure jump. We discovered that the pressure jump can push the interface closer to being non-smooth through force balance, thus significantly decreasing the maximum pressure it can support. Our work shows that a convergence study must be performed on a SPEC equilibrium with interfaces close to break-up. These results may also provide insights into the choice of interfaces and have applications in finding out the maximum pressure a machine can support. (Video recording of Qu talk) |

Nov 24, 2021 8:00-8:10 (EST) 8:10-9:00 (EST) | A. Bhattacharjee R. Nies | Hidden Symmetries Study UpdatesAdjoint Methods for Quasisymmetry of Vacuum Fields on a SurfaceStellarator optimisation can be significantly sped up by using adjoint methods instead of finite-differences to obtain derivative information. In this work, we apply adjoint methods to stellarator vacuum fields, considering objective functions targeting quasisymmetry and rotational transform on the boundary. To define quasisymmetry on the surface, a novel way of constructing approximate flux coordinate on an isolated flux surface is proposed. The obtained adjoint equations are of a simple form such that they can be solved with existing numerical tools, and yield highly accurate derivatives. Using adjoint methods, we obtain configurations with highly accurate quasisymmetry on the boundary, and we are able to systematically investigate the interplay between quasisymmetry and other optimisation targets, such as the aspect ratio and rotational transform. (Video recording of Nies talk) |

Oct 28, 2021 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee D. Spong | Hidden Symmetries Study UpdatesSuppression of Energetic Particle-Driven Instabilities in StellaratorsGood energetic particle (EP) confinement is essential for stellarators to achieve efficient heating and protection of plasma facing components. EP confinement is influenced by several mechanisms: classical orbit confinement, slowing-down timescales, and EP-driven instabilities. Stellarator optimization has recently made significant progress towards improved quasi-symmetry (which improves classical EP confinement). However, experimental evidence has shown that EP instabilities can enhance EP transport to levels of the same order as classical orbit losses. While EP instability driven transport can to some extent be influenced by profile control, three-dimensional shaping is also expected to affect these instabilities. Shaping can be used to control either the MHD wave structures that EP populations resonate with, or the particle-wave resonance dynamics. The physics of EP instabilities will be reviewed and some of the new possibilities for EP instability optimization targets described. (Video recording of Spong talk) |

Oct 14, 2021 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee P. Helander | Hidden Symmetries Study UpdatesUpper Bounds on Gyrokinetic InstabilitiesEnergy confinement in Stellarators is mostly limited by turbulence, in particular when the plasma temperature is low or the neoclassical transport has been reduced by optimization of the magnetic field. The turbulence and the underlying micro-instabilities are thought to be well described by the gyrokinetic set of equations, which have been the subject of thousands of papers and millions of lines of computer code. Yet, little is known about the general properties of the solutions to these equations. Here, a family of rigorous upper bounds on the growth rate of local gyrokinetic instabilities is derived. These bounds hold for both electrostatic and electromagnetic instabilities, regardless of the number of particle species, their collision frequency, and the geometry of the magnetic field. A large number of results that have earlier been derived in special cases and observed in numerical simulations are thus brought into a unifying framework. These bounds apply not only to linear instabilities but also imply an upper limit to the nonlinear growth of the free energy. (Video recording of Helander talk) |

Sep 30, 2021 8:00-8:10 (EDT) 8:10-9:00 (EDT) | A. Bhattacharjee Y-M. Huang | Hidden Symmetries Study UpdatesStructure of Pressure-Driven Current SingularitiesThe objective of this study is to work out a prototype problem with p' <> 0 on the resonant surface as a concrete example. For that purpose, we use The Hahm-Kulsrud-Taylor (HKT) problem, which 1 - is amenable to analytic solutions; 2 - has been studied with various codes including a Grad-Shafranov solver, a fully Lagrangian code, and SPEC for the case with p = 0. (Video recording of Huang talk) |

Sep 16, 2021 8:00-8:30 (EDT) | A. Giuliani | Optimization For Quasi-Symmetry on Surfaces in Single-Stage Coil DesignIn this talk, we'll give an update on our single-stage approach to optimizing for quasi-symmetry on surfaces. Previously, we've presented our new technique for computing surfaces in Boozer coordinates, currently available in SIMSOPT. In this talk, we'll give addition details and describe how these surfaces can be used in a gradient-based approach to optimize for quasi-symmetry. (Video recording of Giuliani talk) |

Sep 16, 2021 8:30-9:00 (EDT) | F. Law | Multifidelity Monte Carlo Estimation of Energetic Particle Confinement in StellaratorsIn the design of Stellarators, energetic particle confinement is a critical point of concern which remains challenging to analyze from a numerical point of view. Due to the absence of fully reliable proxy functions in quantifying the energetic particle confinement properties of magnetic configurations, studies are typically based on a standard Monte Carlo analysis. (Video recording of Law talk) |

Sep 2, 2021 8:00-9:00 (EDT) | N. Sato 佐藤直木 | Quasisymmetric Magnetic Fields in Asymmetric Toroidal Domains(Video recording of Sato talk) |

Aug 5, 2021 8:00-9:00 (EDT) | M. Landreman | Magnetic Fields With Excellent Quasisymmetry Throughout a Volume(Video recording of Landremantalk) |

July 22, 2021 8:00-9:00 (EDT) | S. Henneberg | Representing The Plasma Boundary in Stellarator Optimization(Video recording of Henneberg talk) |

July 8, 2021 8:00-9:00 (EDT) | R. White | Particle Resonances in Toroidal Plasmas(Video recording of White talk) |

Jun 24, 2021 8:00-9:00 (EDT) | D. Peralta-Salas | MHD Equilibria with Non-Constant Pressure in Nondegenerate Toroidal Domains(Video recording of Peralta-Salas talk) |

May 27, 2021 8:00-9:00 (EDT) | A. Carlton-Jones E. Stenson | Computing The Shape Gradient of Stellarator Coil Complexity With Respect to the Plasma Boundary(Video recording of Carlton-Jonestalk)Computational Studies of Mu-Breaking in a Magnetic Dipole and Other Simple Coil Configurations(Video recording of Stenson talk) |

Apr 29, 2021 8:00-9:00 (EDT) | M. Landreman | Update on SIMSOPT(Video recording of Landreman talk) |

Apr 15, 2021 8:00-9:00 (EDT) | Y-M. Huang | Numerical Approach to ∂-function Current Sheets Arising From Resonant Magnetic Perturbations(Video recording of Huang talk) |

Apr 1, 2021 2:00-3:00 (EDT) | A. Geraldini | Adjoint Calculation of Magnetic Island Width Sensitivity(Video recording of Geraldini talk) |

Mar 18, 2021 2:00-3:00 (EDT) | R. MacKay / J. Burby | Isodrastic Magnetic Fields(Video recording of MacKay talk) |

Mar 4, 2021 2:00-3:00 (EST) | J. Lion | Generalization of the Systems Code PROCESS to Stellarators |

Feb 18, 2021 2:00-3:00 (EST) | F. Wechsung | Single-Stage Gradient-Based Stellarator Coil Design |

Feb 4, 2021 2:00-3:00 (EST) | S. Henneberg | Algorithms for Combined Plasma and Coil optimization |

Jan 7, 2021 2:00-3:00 (EST) | N. Duignan | A Presymplectic View of Magnetic Fields |

Dec 11, 2020 9:10-10:00 (EST) | W. Sengupta | Obtaining exact quasisymmetry on a single flux surface: a near-surface expansion approach |

Nov 20, 2020 9:10-10:00 (EST) | A. Guiliani | Single-stage gradient-based stellarator coil design:Optimization for near-axis quasi-symmetry |

Oct 30, 2020 9:20-10:00 (EDT) | E. Rodriguez | Avoiding the problem of overdetermination in quasisymmetric near-axis |

Oct 2, 2020 9:20-10:00Am (EDT) | J. -F. Lobsien | Stochastic Stellarator Coil Optimization |

July 24, 2020 9:30-10:00Am (EDT) | N. McGreivy | Finite-Build Stellarator Coil Design & Automatic Differentiation |

June 26, 2020 9:20-10:00Am (EDT) | T. Pedersen | Introduction to (Stellarator) Divertors |

June 12, 2020 9:20-10:00Am (EDT) | J. Burby | Grad-Shafranov Equation For Non-Axisymmetric MHD Equilibria (slides)https://arxiv.org/pdf/2005.13664.pdf (paper) |

May 29, 2020 9:00-9:20Am (EDT) | A. Bhattacharjee | Summer School 2020 Schedule |

May 29, 2020 9:20-10:00Am (EDT) | R. Granetz | Applying High Temperature Superconductor Technology to Stellarators |

May 15, 2020 9:00-10:00Am (EDT) | B. Khesin | Madelung Transform and Binormal Flowsin the Euler Hydrodynamics |

May 1, 2020 9:05-9:30Am (EDT) | E. Rodriguez | Constructing Quasisymmetry |

May 1, 2020 9:35-10:00Am (EDT) | N. Kallinikos | Approximate Quasisymmetry |

Apr. 17, 2020 9:00-10:00Am (EDT) | S. Glas | Optimization Under Uncertainty |

Apr. 6, 2020 3:00‑4:00Pm (EDT) | D. Ginsberg | Quasisymmetric Equilibria |

Mar. 23, 2020 3:00‑4:00Pm (EDT) | P. Helander | Stellarators with Permanent Magnets |

Mar. 9, 2020 3:00‑4:00Pm (EDT) | A. Bader | Stellarator Optimization in PracticeAnd Where We Can Improve |

Mar. 2, 2020 3:00‑4:00Pm (EST) | D. Malhotra | Integral Equation Methods for CalculatingStepped-Pressure Equilibrium in Stellarators |

Feb. 24, 2020 3:00‑4:00Pm (EST) | A. Giuliani | Adjoint-Based Vacuum-FieldStellarator Optimization |

Feb. 10, 2020 3:00‑4:00Pm (EST) | E. Paul | Efficient Stellarator Shape Optimizationand Sensitivity Analysis |

Jan. 27, 2020 3:00‑4:00Pm (EST) | R. Dewar and Z. Qu | MRxMHD With Flow |

Jan. 10, 2020 9:00‑10:00Am (EST) | M. Landreman | SIMSOPT phase 1: MANGO |

Dec. 13, 2019 9:00‑10:00Am (EST) | F. Hindenlang | GVEC: A newly developed 3D ideal MHDGalerkin Variational Equilibrium Code |

Nov. 29, 2019 9:00‑10:00Am (EST) | J. Loizu | Direct Prediction of Nonlinearly SaturatedTearing Modes with SPEC |

Nov. 15, 2019 9:00‑10:00Am (EST) | B. Faber | Stellarator Optimization at theUniversity of Wisconsin-Madison |

Nov. 8, 2019 8:30‑9:30Am (EST) | Special Discussion on SIMSOPT | |

Nov. 1, 2019 9:00‑10:00Am (EDT) | R. Jorge | Near-Axis Expansion Framework at ArbitraryOrder in the Distance to the Magnetic Axis |

Sep. 20, 2019 9:00‑10:00Am (EDT) | Dhariya Malhotra | Higher-Order Integration for SingularIntegrals in Magnetostatics |

Sep. 6, 2019 9:00‑10:00Am (EDT) | Allen Boozer | Curl-free Magnetic Fields for Stellarator Optimization |

Aug. 5, 2019 3:00‑4:00Pm (EDT) | David Bindel | Surrogate Optimization |

May 13, 2019 3:00‑4:00Pm (EDT) | Georg Stadler | Optimization |

Apr. 29, 2019 3:00‑4:00Pm (EDT) | Amitava Bhattacharjee | Summer School Schedule |

Mar. 18, 2019 3:00‑4:00Pm (EDT) | Amitava Bhattacharjee | Team meeting in PCTS, Annual Meeting at Simons Foundation |