arXiv subjects

Andrew Giuliani

Publications and source records attributed to Andrew Giuliani.

17 recordsLinked to original sources

arXiv2026

Divertor topology and vacuum vessel design for stellarators

We present stellarator optimization algorithms for designing the edge magnetic structure in vacuum fields, together with the vacuum vessel. First, we introduce a numerical method that robustly computes periodic-orbit fixed points of any type (elliptic, hyperbolic, or parabolic), which could form the basis of a divertor. To couple divertor and vacuum vessel design, we introduce parametric families of vacuum vessels for which point-to-vessel distances, and their derivatives, can be computed efficiently. The resulting algorithms use signed distance functions to enforce coil-vessel clearance while allowing coils to be placed on or off the vessel. Using these methods, we jointly optimize modular coils and the vacuum vessel to realize a wide range of magnetic topologies for diverting exhaust, including standard X-point divertors, and single- and double-null configurations. For the first time, we show that precise snowflake divertors can be achieved in stellarators. Using this framework, we generate a number of quasi-axisymmetric stellarator designs with compatible vacuum vessels and diverse divertor architectures, which we consider to be candidates for a next-generation STAR Lite prototype.

physics.plasm-ph↗

arXiv2026

Bayesian optimization of stellarator alpha-particle confinement using data-informed parameter spaces and dimensionality reduction

Modern stellarators are typically designed by optimizing the shape of the plasma boundary surface, with the parameters taken to be Fourier amplitudes. Many promising optimization algorithms such as Bayesian methods require bound constraints on the parameters and are most efficient when each parameter is scaled similarly to the others. With the typical Fourier parameterization, it is unclear how to set these bounds: wide constraints lead to self-intersecting boundaries and frequent failures of the MHD equilibrium calculation, while tight bound constraints limit expressiveness. To address these issues, here we propose two new parameter spaces for stellarator optimization. Both begin with a dataset of existing stellarator boundaries. In the first approach, a quantile transformation is applied to each Fourier degree of freedom, mapping the data distribution to a uniform distribution on the unit interval. In the second approach, principal component analysis (PCA) is applied to points on the boundaries, followed by a quantile transformation. For both approaches, the transformed variables become the degrees of freedom, naturally bounded to [0, 1]. The PCA method has the additional benefit of dimensionality reduction, with high expressiveness for a small number of parameters. The methods are demonstrated via Bayesian optimization for good alpha-particle confinement with guiding-center tracing inside the optimization loop, using asynchronous parallelization. These optimizations yield stellarator configurations with excellent fast-particle confinement in fields that can be far from quasisymmetric or quasi-isodynamic.

physics.plasm-ph↗

arXiv2026

STAR_Lite: A stellarator designed to experimentally validate non-resonant divertors

The non-resonant divertor (NRD) offers a promising exhaust solution for stellarators, combining topological simplicity with resilience to magnetic field perturbations. To experimentally validate the robustness of non-resonant divertors in a quasi-axisymmetric (QA) configuration, we introduce STAR_Lite, a new stellarator experiment at Hampton University. This paper details the design and analysis of the first STAR_Lite coil configuration, STAR_Lite-A. The two field-period configuration manifests an NRD through X-points with zero rotational transform, at the top and bottom of the device. The divertor legs extruding from the X-points are topologically similar to the poloidal divertors of tokamaks. To expand the experimental range, STAR_Lite-A is optimized for experimental flexibility, producing a wide range of distinct QA configurations by only varying the currents in the modular coils. The NRDs not only persist across these configurations, but numerical strike-line simulations confirm that heat exhaust remains resilient to changes in coil currents, with plasma following the divertor legs and creating a toroidal, discontinuous, strike pattern. We further examine the resilience of the NRD to magnetic perturbations caused by manufacturing errors in the modular coils. We find that quasisymmetry and the existence of X-points is well-preserved under these magnetic field changes, but the rotational transform may vary substantially and displacements of the divertor X-points may lead to one X-point having a dominant effect on edge transport. Overall, our analysis indicates a compact, modular design can likely generate a resilient NRD structure while satisfying the practical constraints of a university-scale experiment.

physics.plasm-ph↗

arXiv2025

Diffusion for Fusion: Designing Stellarators with Generative AI

Stellarators are a prospective class of fusion-based power plants that confine a hot plasma with three-dimensional magnetic fields. Typically framed as a PDE-constrained optimization problem, stellarator design is a time-consuming process that can take hours to solve on a computing cluster. Developing fast methods for designing stellarators is crucial for advancing fusion research. Given the recent development of large datasets of optimized stellarators, machine learning approaches have emerged as a potential candidate. Motivated by this, we present an open inverse problem to the machine learning community: to rapidly generate high-quality stellarator designs which have a set of desirable characteristics. As a case study in the problem space, we train a conditional diffusion model on data from the QUASR database to generate quasisymmetric stellarator designs with desirable characteristics (aspect ratio and mean rotational transform). The diffusion model is applied to design stellarators with characteristics not seen during training. We provide evaluation protocols and show that many of the generated stellarators exhibit solid performance: less than 5% deviation from quasisymmetry and the target characteristics. The modest deviation from quasisymmetry highlights an opportunity to reach the sub 1% target. Beyond the case study, we share multiple promising avenues for generative modeling to advance stellarator design.

cs.LG↗

arXiv2024

A comprehensive exploration of quasisymmetric stellarators and their coil sets

We augment the `QUAsi-symmetric Stellarator Repository' (QUASR) to include vacuum field stellarators with quasihelical symmetry using a globalized optimization workflow. The database now has almost 370,000 quasisaxisymmetry and quasihelically symmetric devices along with coil sets, optimized for a variety of aspect ratios, rotational transforms, and discrete rotational symmetries. This paper outlines a couple of ways to explore and characterize the data set. We plot devices on a near-axis quasisymmetry landscape, revealing close correspondence to this predicted landscape. We also use principal component analysis to reduce the dimensionality of the data so that it can easily be visualized in two or three dimensions. Principal component analysis also gives a mechanism to compare the new devices here to previously published ones in the literature. We are able to characterize the structure of the data, observe clusters, and visualize the progression of devices in these clusters. The topology of the data is governed by the interplay of the design constraints and valleys of the quasisymmetry objective. These techniques reveal that the data has structure, and that typically one, two or three principal components are sufficient to characterize it. The latest version of QUASR is archived at https://zenodo.org/doi/10.5281/zenodo.10050655 and can be explored online at quasr.flatironinstitute.org.

physics.app-ph↗

arXiv2023

Direct stellarator coil design using global optimization: application to a comprehensive exploration of quasi-axisymmetric devices

Many stellarator coil design problems are plagued by multiple minima, where the locally optimal coil sets can sometimes vary substantially in performance. As a result, solving a coil design problem a single time with a local optimization algorithm is usually insufficient and better optima likely do exist. To address this problem, we propose a global optimization algorithm for the design of stellarator coils and outline how to apply box constraints to the physical positions of the coils. The algorithm has a global exploration phase that searches for interesting regions of design space and is followed by three local optimization algorithms that search in these interesting regions (a ``global-to-local" approach). The first local algorithm (phase I), following the globalization phase, is based on near-axis expansions and finds stellarator coils that optimize for quasisymmetry in the neighborhood of a magnetic axis. The second local algorithm (phase II) takes these coil sets and optimizes them for nested flux surfaces and quasisymmetry on a toroidal volume. The final local algorithm (phase III) polishes these configurations for an accurate approximation of quasisymmetry. Using our global algorithm, we study the trade-off between coil length, aspect ratio, rotational transform, and quality of quasi-axisymmetry. The database of stellarators, which comprises approximately 200,000 coil sets, is available online and is called QUASR, for `QUAsi-symmetric Stellarator Repository'.

physics.comp-ph↗

arXiv2023

A new provably stable weighted state redistribution algorithm

We propose a practical finite volume method on cut cells using state redistribution. Our algorithm is provably monotone, total variation diminishing, and GKS stable in many situations, and shuts off continuously as the cut cell size approaches a target value. Our analysis reveals why original state redistribution works so well: it results in a monotone scheme for most configurations, though at times subject to a slightly smaller CFL condition. Our analysis also explains why a pre-merging step is beneficial. We show computational experiments in two and three dimensions.

math.NA↗

arXiv2022

Direct stellarator coil optimization for nested magnetic surfaces with precise quasi-symmetry

We present a robust optimization algorithm for the design of electromagnetic coils that generate vacuum magnetic fields with nested flux surfaces and precise quasi-symmetry. The method is based on a bilevel optimization problem, where the outer coil optimization is constrained by a set of inner least-squares optimization problems whose solutions describe magnetic surfaces. The outer optimization objective targets coils that generate a field with nested magnetic surfaces and good quasi-symmetry. The inner optimization problems identify magnetic surfaces when they exist, and approximate surfaces in the presence of magnetic islands or chaos. We show that this formulation can be used to heal islands and chaos, thus producing coils that result in magnetic fields with precise quasi-symmetry. We show that the method can be initialized with coils from the traditional two stage coil design process, as well as coils from a near axis expansion optimization. We present a numerical example where island chains are healed and quasi-symmetry is optimized up to surfaces with aspect ratio 6. Another numerical example illustrates that the aspect ratio of nested flux surfaces with optimized quasi-symmetry can be decreased from 6 to approximately 4. The last example shows that our approach is robust and a cold-start using coils from a near-axis expansion optimization.

physics.plasm-ph↗

arXiv2022

Stochastic and a posteriori optimization to mitigate coil manufacturing errors in stellarator design

It was recently shown in [Wechsung et. al., Proc. Natl. Acad. Sci. USA, 2022] that there exist electromagnetic coils that generate magnetic fields which are excellent approximations to quasi-symmetric fields and have very good particle confinement properties. Using a Gaussian process based model for coil perturbations, we investigate the impact of manufacturing errors on the performance of these coils. We show that even fairly small errors result in noticeable performance degradation. While stochastic optimization yields minor improvements, it is not able to mitigate these errors significantly. As an alternative to stochastic optimization, we then formulate a new optimization problem for computing optimal adjustments of the coil positions and currents without changing the shapes of the coil. These a-posteriori adjustments are able to reduce the impact of coil errors by an order of magnitude, providing a new perspective for dealing with manufacturing tolerances in stellarator design.

physics.plasm-ph↗

arXiv2022

Direct computation of magnetic surfaces in Boozer coordinates and coil optimization for quasi-symmetry

We propose a new method to compute magnetic surfaces that are parametrized in Boozer coordinates for vacuum magnetic fields. We also propose a measure for quasi-symmetry on the computed surfaces and use it to design coils that generate a magnetic field that is quasi-symmetric on those surfaces. The rotational transform of the field and complexity measures for the coils are also controlled in the design problem. Using an adjoint approach, we are able to obtain analytic derivatives for this optimization problem, yielding an efficient gradient-based algorithm. Starting from an initial coil set that presents nested magnetic surfaces for a large fraction of the volume, our method converges rapidly to coil systems generating fields with excellent quasi-symmetry and low particle losses. In particular for low complexity coils, we are able to significantly improve the performance compared to coils obtained from the standard two-stage approach, e.g.~reduce losses of fusion-produced alpha particles born at half-radius from $17.7\%$ to $6.6\%$. We also demonstrate 16-coil configurations with alpha loss < $1\%$ and neoclassical transport magnitude $\epsilon_{\mathrm{eff}}^{3/2}$ less than approximately $5\times 10^{-9}.$

physics.plasm-ph↗

arXiv2021

A Weighted State Redistribution Algorithm for Embedded Boundary Grids

State redistribution is an algorithm that stabilizes cut cells for embedded boundary grid methods. This work extends the earlier algorithm in several important ways. First, state redistribution is extended to three spatial dimensions. Second, we discuss several algorithmic changes and improvements motivated by the more complicated cut cell geometries that can occur in higher dimensions. In particular, we introduce a weighted version with less dissipation in an easily generalizable framework. Third, we demonstrate that state redistribution can also stabilize a solution update that includes both advective and diffusive contributions. The stabilization algorithm is shown to be effective for incompressible as well as compressible reacting flows. Finally, we discuss the implementation of the algorithm for several exascale-ready simulation codes based on AMReX, demonstrating ease of use in combination with domain decomposition, hybrid parallelism and complex physics.

math.NA↗

arXiv2021

Single-stage gradient-based stellarator coil design: stochastic optimization

We extend the single-stage stellarator coil design approach for quasi-symmetry on axis from [Giuliani et al, 2020] to additionally take into account coil manufacturing errors. By modeling coil errors independently from the coil discretization, we have the flexibility to consider realistic forms of coil errors. The corresponding stochastic optimization problems are formulated using a risk-neutral approach and risk-averse approaches. We present an efficient, gradient-based descent algorithm which relies on analytical derivatives to solve these problems. In a comprehensive numerical study, we compare the coil designs resulting from deterministic and risk-neutral stochastic optimization and find that the risk-neutral formulation results in more robust configurations and reduces the number of local minima of the optimization problem. We also compare deterministic and risk-neutral approaches in terms of quasi-symmetry on and away from the magnetic axis, and in terms of the confinement of particles released close to the axis. Finally, we show that for the optimization problems we consider, a risk-averse objective using the Conditional Value-at-Risk leads to results which are similar to the risk-neutral objective.

math.OC↗

arXiv2021

A two-dimensional stabilized discontinuous Galerkin method on curvilinear embedded boundary grids

We propose a state redistribution method for high order discontinuous Galerkin methods on curvilinear embedded boundary grids. State redistribution relaxes the overly restrictive CFL condition that results from arbitrarily small cut cells and explicit time stepping. Thus, the scheme can take time steps that are proportional to the size of cells in the background grid. The discontinuous Galerkin scheme is stabilized by postprocessing the numerical solution after each stage or step of an explicit time stepping method. This is done by temporarily merging the small cells into larger, possibly overlapping neighborhoods using a special weighted inner product. Then, the numerical solution on the neighborhoods is returned to the base grid in a conservative fashion. The advantage of this approach is that it uses only basic mesh information that is already available in many cut cell codes and does not require complex geometric manipulations. Finally, we present a number of test problems that demonstrate the encouraging potential of this technique for applications on curvilinear embedded geometries. Numerical experiments reveal that our scheme converges with order $p+1$ in $L_1$ and between $p$ and $p+1$ in $L_\infty$ for problems with smooth solutions. We also demonstrate that state redistribution is capable of capturing shocks.

math.NA↗

arXiv2020

Single-stage gradient-based stellarator coil design: Optimization for near-axis quasi-symmetry

We present a new coil design paradigm for magnetic confinement in stellarators. Our approach directly optimizes coil shapes and coil currents to produce a vacuum quasi-symmetric magnetic field with a target rotational transform on the magnetic axis. This approach differs from the traditional two-stage approach in which first a magnetic configuration with desirable physics properties is found, and then coils to approximately realize this magnetic configuration are designed. The proposed single-stage approach allows us to find a compromise between confinement and engineering requirements, i.e., find easy-to-build coils with good confinement properties. Using forward and adjoint sensitivities, we derive derivatives of the physical quantities in the objective, which is constrained by a nonlinear periodic differential equation. In two numerical examples, we compare different gradient-based descent algorithms and find that incorporating approximate second-order derivative information through a quasi-Newton method is crucial for convergence. We also explore the optimization landscape in the neighborhood of a minimizer and find many directions in which the objective is mostly flat, indicating ample freedom to find simple and thus easy-to-build coils.

physics.plasm-ph↗

arXiv2020

A state redistribution algorithm for finite volume schemes on cut cell meshes

In this paper we develop a new technique, called \textit{state redistribution}, that allows the use of explicit time stepping when approximating solutions to hyperbolic conservation laws on embedded boundary grids. State redistribution is a postprocessing technique applied after each time step or stage of the base finite volume scheme, using a time step that is proportional to the volume of the full cells. The idea is to stabilize the cut cells by temporarily merging them into larger, possibly overlapping neighborhoods, then replacing the cut cell values with a stabilized value that maintains conservation and accuracy. We present examples of state redistribution using two base schemes: MUSCL and a second order Method of Lines finite volume scheme. State redistribution is used to compute solutions to several standard test problems in gas dynamics on cut cell meshes, with both smooth and discontinuous solutions. We show that our method does not reduce the accuracy of the base scheme and that it successfully captures shocks in a non-oscillatory manner.

math.NA↗

arXiv2017

Edge coloring in unstructured CFD codes

We propose a way of preventing race conditions in the evaluation of the surface integral contribution in discontinuous Galerkin and finite volume flow solvers by coloring the edges (or faces) of the computational mesh. In this work we use a partitioning algorithm that separates the edges of triangular elements into three groups and the faces of quadrangular and tetrahedral elements into four groups; we then extend this partitioning to adaptively refined, nonconforming meshes. We use the ascribed coloring to reduce code memory requirements and optimize accessing the elemental data in memory. This process reduces memory access latencies and speeds up computations on graphics processing units.

cs.DC↗

arXiv2016

Discontinuous Galerkin methods on graphics processing units for nonlinear hyperbolic conservation laws

We present a novel implementation of the modal discontinuous Galerkin (DG) method for hyperbolic conservation laws in two dimensions on graphics processing units (GPUs) using NVIDIA's Compute Unified Device Architecture (CUDA). Both flexible and highly accurate, DG methods accommodate parallel architectures well as their discontinuous nature produces element-local approximations. High performance scientific computing suits GPUs well, as these powerful, massively parallel, cost-effective devices have recently included support for double-precision floating point numbers. Computed examples for Euler equations over unstructured triangle meshes demonstrate the effectiveness of our implementation on an NVIDIA GTX 580 device. Profiling of our method reveals performance comparable to an existing nodal DG-GPU implementation for linear problems.

cs.DC↗