arXiv subjects

Antoine Cerfon

Publications and source records attributed to Antoine Cerfon.

17 recordsLinked to original sources

arXiv2025

Efficient calculation of magnetic fields from ferromagnetic materials near strong electromagnets, and application to stellarator coil optimization

In fusion reactor design, steels under consideration for the blanket are ferromagnetic, so the steel's effect on the plasma physics must be examined. For efficient calculation of these fields, we can exploit the fact that the magnetic material gives a small perturbation relative to the fields from the electromagnetic coils and plasma. Moreover the magnetization is saturated due to the strong fields in typical fusion systems. These approximations significantly reduce the nonlinearity of the problem, so the magnetic materials can be described by an array of point dipoles of known magnitude, oriented in the direction of the coil and plasma field. The approach is verified by comparison to finite-element calculations with commercial software and shown to be accurate. As no linear or nonlinear solve is required, only evaluation of Biot-Savart-type integrals, the method here is significantly simpler to implement than other methods, and extremely fast. The method is compatible with arbitrary CAD geometry, and also allows rapid computation of the magnetic forces. We demonstrate adding the ferromagnetic effects to free-boundary MHD equilibrium calculations, assessing the effect on plasma properties such as confinement and stability. Moreover, it is straightforward to differentiate through the model to get the derivative of the field with respect to the electromagnet parameters. We thereby demonstrate gradient-based coil optimization for a quasi-isodynamic stellarator in which the field contribution from a ferromagnetic blanket is included. Even a significant steel volume is found to have little impact on the plasma physics properties, with the main effects being a slight destabilization of ballooning modes and a radial shift of the edge islands due to decrease in rotational transform. Both issues are corrected by minor reoptimization of the coil shapes to account for the field from the steel.

physics.plasm-ph↗

arXiv2024

A particle-in-Fourier method with semi-discrete energy conservation for non-periodic boundary conditions

We introduce a novel particle-in-Fourier (PIF) scheme that extends its applicability to non-periodic boundary conditions. Our method handles free space boundary conditions by replacing the Fourier Laplacian operator in PIF with a mollified Green's function as first introduced by Vico-Greengard-Ferrando. This modification yields highly accurate free space solutions to the Vlasov-Poisson system, while still maintaining energy conservation up to an error bounded by the time step size. We also explain how to extend our scheme to arbitrary Dirichlet boundary conditions via standard potential theory, which we illustrate in detail for Dirichlet boundary conditions on a circular boundary. We support our approach with proof-of-concept numerical results from two-dimensional plasma test cases to demonstrate the accuracy, efficiency, and conservation properties of the scheme. By avoiding grid heating and finite grid instability we are able to show an order of magnitude speedup compared to the standard PIC scheme for a long time integration cyclotron simulation.

math.NA↗

arXiv2024

A Massively Parallel Performance Portable Free-space Spectral Poisson Solver

Vico et al. (2016) suggest a fast algorithm for computing volume potentials, beneficial to fields with problems requiring the solution of the free-space Poisson's equation, such as beam and plasma physics. Currently, the standard is the algorithm of Hockney and Eastwood (1988), with second order in convergence at best. The algorithm proposed by Vico et al. converges spectrally for sufficiently smooth functions i.e. faster than any fixed order in the number of grid points. We implement a performance portable version of the traditional Hockney-Eastwood and the novel Vico-Greengard Poisson solver as part of the IPPL (Independent Parallel Particle Layer) library. For sufficiently smooth source functions, the Vico-Greengard algorithm achieves higher accuracy than the Hockney-Eastwood method with the same grid size, reducing the computational demands of high resolution simulations since one could use coarser grids to achieve them. Additionally, we propose an improvement to the Vico-Greengard method which further reduces its memory footprint. This is important for GPUs, which have limited memory, and should be taken into account when selecting numerical algorithms for performance portable codes. Finally, we showcase performance through GPU and CPU scaling studies on the Perlmutter (NERSC) supercomputer, with efficiencies staying above 50% in the strong scaling case. To showcase portability, we also run the scaling studies on the Alps supercomputer at CSCS, Switzerland and the GPU partition of the Lumi supercomputer at CSC, Finland.

physics.comp-ph↗

arXiv2024

Direct, simple, and efficient computation of all components of the virtual-casing magnetic field in axisymmetric geometries with Kapur-Rokhlin quadrature

In a recent publication (Toler et al. 2023), we demonstrated that for axisymmetric geometries, the Kapur-Rokhlin quadrature rule provided an efficient and high-order accurate method for computing the normal component, on the plasma surface, of the magnetic field due to the toroidal current flowing in the plasma, via the virtual-casing principle. The calculation was indirect, as it required the prior computation of the magnetic vector potential from the virtual-casing principle, followed by the computation of its tangential derivative by Fourier differentiation, in order to obtain the normal component of the magnetic field. Our approach did not provide the other components of the virtual-casing magnetic field. In this letter, we show that a more direct and more general approach is available for the computation of the virtual-casing magnetic field. The Kapur-Rokhlin quadrature rule accurately calculates the principal value integrals in the expression for all the components of the magnetic field on the plasma boundary, and the numerical error converges at a rate nearly as high as the indirect method we presented previously.

physics.plasm-ph↗

arXiv2023

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↗

arXiv2023

Meta variance reduction for Monte Carlo estimation of energetic particle confinement during stellarator optimization

This work introduces meta estimators that combine multiple multifidelity techniques based on control variates, importance sampling, and information reuse to yield a quasi-multiplicative amount of variance reduction. The proposed meta estimators are particularly efficient within outer-loop applications when the input distribution of the uncertainties changes during the outer loop, which is often the case in reliability-based design and shape optimization. We derive asymptotic bounds of the variance reduction of the meta estimators in the limit of convergence of the outer-loop results. We demonstrate the meta estimators, using data-driven surrogate models and biasing densities, on a design problem under uncertainty motivated by magnetic confinement fusion, namely the optimization of stellarator coil designs to maximize the estimated confinement of energetic particles. The meta estimators outperform all of their constituent variance reduction techniques alone, ultimately yielding two orders of magnitude speedup compared to standard Monte Carlo estimation at the same computational budget.

physics.comp-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 $ε_{\mathrm{eff}}^{3/2}$ less than approximately $5\times 10^{-9}.$

physics.plasm-ph↗

arXiv2022

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↗

arXiv2021

Accelerating the estimation of energetic particle confinement statistics in stellarators using multifidelity Monte Carlo

In the design of stellarators, energetic particle confinement is a critical point of concern which remains challenging to study from a numerical point of view. Standard Monte Carlo analyses are highly expensive because a large number of particle trajectories need to be integrated over long time scales, and small time steps must be taken to accurately capture the features of the wide variety of trajectories. Even when they are based on guiding center trajectories, as opposed to full-orbit trajectories, these standard Monte Carlo studies are too expensive to be included in most stellarator optimization codes. We present the first multifidelity Monte Carlo scheme for accelerating the estimation of energetic particle confinement in stellarators. Our approach relies on a two-level hierarchy, in which a guiding center model serves as the high-fidelity model, and a data-driven linear interpolant is leveraged as the low-fidelity surrogate model. We apply multifidelity Monte Carlo to the study of energetic particle confinement in a 4-period quasi-helically symmetric stellarator, assessing various metrics of confinement. Stemming from the very high computational efficiency of our surrogate model as well as its sufficient correlation to the high-fidelity model, we obtain speedups of up to 10 with multifidelity Monte Carlo compared to standard Monte Carlo.

physics.plasm-ph↗

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↗

arXiv2019

Elimination of MHD current sheets by modifications to the plasma wall in a fixed boundary model

Models of magnetohydrodynamic (MHD) equilibia that for computational convenience assume the existence of a system of nested magnetic flux surfaces tend to exhibit singular current sheets. These sheets are located on resonant flux surfaces that are associated with rational values of the rotational transform. We study the possibility of eliminating these singularities by suitable modifications of the plasma boundary, which we prescribe in a fixed boundary setting. We find that relatively straightforward iterative procedures can be used to eliminate weak current sheets that are generated at resonant flux surfaces by the nonlinear interactions of resonating wall harmonics. These types of procedures may prove useful in the design of fusion devices with configurations that enjoy improved stability and transport properties.

physics.plasm-ph↗

arXiv2019

Taylor States in Stellarators: A Fast High-order Boundary Integral Solver

We present a boundary integral equation solver for computing Taylor relaxed states in non-axisymmetric solid and shell-like toroidal geometries. The computation of Taylor states in these geometries is a key element for the calculation of stepped pressure stellarator equilibria. The integral representation of the magnetic field in this work is based on the generalized Debye source formulation, and results in a well-conditioned second-kind boundary integral equation. The integral equation solver is based on a spectral discretization of the geometry and unknowns, and the computation of the associated weakly-singular integrals is performed with high-order quadrature based on a partition of unity. The resulting scheme for applying the integral operator is then coupled with an iterative solver and suitable preconditioners. Several numerical examples are provided to demonstrate the accuracy and efficiency of our method, and a direct comparison with the leading code in the field is reported.

math.NA↗

arXiv2015

Tokamak elongation: how much is too much? II Numerical results

The analytic theory presented in Paper I is converted into a form convenient for numerical analysis. A fast and accurate code has been written using this numerical formulation. The results are presented by first defining a reference set of physical parameters based on experimental data from high performance discharges. Numerically obtained scaling relations of maximum achievable elongation versus inverse aspect ratio are obtained for various values of poloidal beta, wall radius and feedback capability parameter in ranges near the reference values. It is also shown that each value of maximum elongation occurs at a corresponding value of optimized triangularity, whose scaling is also determined as a function of inverse aspect ratio. The results show that the theoretical predictions of maximum elongation are slightly higher than experimental observations for high performance discharges as measured by high average pressure. The theoretical optimized triangularity values are noticeably lower. We suggest that the explanation is associated with the observation that high performance involves not only MHD considerations, but also transport as characterized by confinement time. Operation away from the MHD optimum may still lead to higher performance if there are more than compensatory gains in the confinement time. Unfortunately, while the empirical scaling of the confinement time with the elongation has been determined, the dependence on the triangularity has still not been quantified. This information is needed in order to perform more accurate overall optimizations in future experimental designs.

physics.plasm-ph↗

arXiv2015

A Spectral Transform Method for Singular Sturm-Liouville Problems with Applications to Energy Diffusion in Plasma Physics

We develop a spectrally accurate numerical method to compute solutions of a model partial differential equation used in plasma physics to describe diffusion in velocity space due to Fokker-Planck collisions. The solution is represented as a discrete and continuous superposition of normalizable and non-normalizable eigenfunctions via the spectral transform associated with a singular Sturm-Liouville operator. We present a new algorithm for computing the spectral density function of the operator that uses Chebyshev polynomials to extrapolate the value of the Titchmarsh-Weyl $m$-function from the complex upper half-plane to the real axis. The eigenfunctions and density function are rescaled and a new formula for the limiting value of the $m$-function is derived to avoid amplification of roundoff errors when the solution is reconstructed. The complexity of the algorithm is also analyzed, showing that the cost of computing the spectral density function at a point grows less rapidly than any fractional inverse power of the desired accuracy. A WKB analysis is used to prove that the spectral density function is real analytic. Using this new algorithm, we highlight key properties of the partial differential equation and its solution that have strong implications on the optimal choice of discretization method in large-scale plasma physics computations.

math.CA↗

arXiv2014

ECOM: a fast and accurate solver for toroidal axisymmetric MHD equilibria

We present ECOM (Equilibrium solver via COnformal Mapping), a fast and accurate fixed boundary solver for toroidally axisymmetric magnetohydrodynamic equilibria with or without a toroidal flow. ECOM combines conformal mapping and Fourier and integral equation methods on the unit disk to achieve exponential convergence for the poloidal flux function as well as its first and second partial derivatives. As a consequence of its high order accuracy, for dense grids and tokamak-like elongations ECOM computes key quantities such as the safety factor and the magnetic shear with higher accuracy than the finite element based code CHEASE [H. Lütjens \textit{et al.}, Computer physics communications 97, 219 (1996)] at equal run time. ECOM has been developed to provide equilibrium quantities and details of the flux contour geometry as inputs to stability, wave propagation and transport codes.

physics.plasm-ph↗

arXiv2014

Accurate Spectral Numerical Schemes for Kinetic Equations with Energy Diffusion

We examine the merits of using a family of polynomials that are orthogonal with respect to a non-classical weight function to discretize the speed variable in continuum kinetic calculations. We consider a model one-dimensional partial differential equation describing energy diffusion in velocity space due to Fokker-Planck collisions. This relatively simple case allows us to compare the results of the projected dynamics with an expensive but highly accurate spectral transform approach. It also allows us to integrate in time exactly, and to focus entirely on the effectiveness of the discretization of the speed variable. We show that for a fixed number of modes or grid points, the non-classical polynomials can be many orders of magnitude more accurate than classical Hermite polynomials or finite-difference solvers for kinetic equations in plasma physics. We provide a detailed analysis of the difference in behavior and accuracy of the two families of polynomials. For the non-classical polynomials, if the initial condition is not smooth at the origin when interpreted as a three-dimensional radial function, the exact solution leaves the polynomial subspace for a time, but returns (up to roundoff accuracy) to the same point evolved to by the projected dynamics in that time. By contrast, using classical polynomials, the exact solution differs significantly from the projected dynamics solution when it returns to the subspace. We also explore the connection between eigenfunctions of the projected evolution operator and (non-normalizable) eigenfunctions of the full evolution operator, as well as the effect of truncating the computational domain.

math.NA↗