arXiv subjects

Matt Landreman

Publications and source records attributed to Matt Landreman.

At least 55 records · Page 3Linked to original sources

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

Magnetic fields with precise quasisymmetry for plasma confinement

Quasisymmetry is an unusual symmetry that can be present in toroidal magnetic fields, enabling confinement of charged particles and plasma. Here it is shown that both quasi-axisymmetry and quasi-helical symmetry can be achieved to a much higher precision than previously thought over a significant volume, resulting in exceptional confinement. For a 1 Tesla mean field far from axisymmetry (vacuum rotational transform $>$ 0.4), symmetry-breaking mode amplitudes throughout a volume of aspect ratio 6 can be made as small as the typical $\sim 50$ $μ$T geomagnetic field.

physics.plasm-ph↗

arXiv2021

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

A method is demonstrated to optimize a stellarator's geometry to eliminate magnetic islands and achieve other desired physics properties at the same time. For many physics quantities that have been used in stellarator optimization, including quasisymmetry, neoclassical transport, and magnetohydrodynamic stability, it is convenient to use a magnetic equilibrium representation that assures the existence of magnetic surfaces. However, this representation hides the possible presence of magnetic islands, which are typically undesirable. To include both surface-based objectives and island widths in a single optimization, two fixed-boundary equilibrium calculations are run at each iteration of the optimization: one that enforces the existence of magnetic surfaces (VMEC [S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983)]), and one that does not (SPEC [S. R. Hudson, et al, Phys. Plasmas 19, 112502 (2012)]). By penalizing the island residues in the objective function, the two magnetic field representations are brought into agreement during the optimization. An example is presented in which, particularly on the surface where quasisymmetry was targeted, quasisymmetry is achieved more accurately than in previously published examples.

physics.plasm-ph↗

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 neoclassically optimized compact stellarator with four planar coils

A neoclassically optimized compact stellarator with simple coils has been designed. The magnetic field of the new stellarator is generated by only four planar coils including two interlocking coils of elliptical shape and two circular poloidal field coils. The interlocking coil topology is the same as that of the Columbia Non-neutral Torus (CNT). The new configuration was obtained by minimizing the effective helical ripple directly via the shape of the two interlocking coils. The optimized compact stellarator has very low effective ripple in the plasma core implying excellent neoclassical confinement. This is confirmed by the results of the drift-kinetic code SFINCS showing that the particle diffusion coefficient of the new configuration is one order of magnitude lower than CNT's.

physics.plasm-ph↗

arXiv2021

Figures of merit for stellarators near the magnetic axis

A new paradigm for rapid stellarator configuration design has been recently demonstrated, in which the shapes of quasisymmetric or omnigenous flux surfaces are computed directly using an expansion in small distance from the magnetic axis. To further develop this approach, here we derive several other quantities of interest that can be rapidly computed from this near-axis expansion. First, the $\nabla\vec{B}$ and $\nabla\nabla\vec{B}$ tensors are computed, which can be used for direct derivative-based optimization of electromagnetic coil shapes to achieve the desired magnetic configuration. Moreover, if the norm of these tensors is large compared to the field strength for a given magnetic field, the field must have a short length scale, suggesting it may be hard to produce with coils that are suitably far away. Second, we evaluate the minor radius at which the flux surface shapes would become singular, providing a lower bound on the achievable aspect ratio. This bound is also shown to be related to an equilibrium beta limit. Finally, for configurations that are constructed to achieve a desired magnetic field strength to first order in the expansion, we compute the error field that arises due to second order terms.

physics.plasm-ph↗

arXiv2021

An adjoint method for determining the sensitivity of island size to magnetic field variations

An adjoint method to calculate the gradient of island width in stellarators is presented and applied to a set of magnetic field configurations. The underlying method of calculation of the island width is that of Cary & Hanson (1991) (with a minor modification), and requires that the residue of the island centre be small. Therefore, the gradient of the residue is calculated in addition. Both the island width and the gradient calculations are verified using an analytical magnetic field configuration introduced in Reiman & Greenside (1986). The method is also applied to the calculation of the shape gradient of the width of a magnetic island in an NCSX vacuum configuration with respect to positions on a coil. A gradient-based optimization is applied to a magnetic field configuration studied in Hanson & Cary (1984) to minimize stochasticity by adding perturbations to a pair of helical coils. Although only vacuum magnetic fields and an analytical magnetic field model are considered in this work, the adjoint calculation of the island width gradient could also be applied to a magnetohydrodynamic (MHD) equilibrium if the derivative of the magnetic field with respect to the equilibrium parameters was known. Using the island width gradient calculation presented here, more general gradient-based optimization methods can be applied to design stellarators with small magnetic islands. Moreoever, the sensitivity of the island size may itself be optimized to ensure that coil tolerances with respect to island size are kept as high as possible.

physics.plasm-ph↗

arXiv2021

Gradient-based optimization of 3D MHD equilibria

Using recently developed adjoint methods for computing the shape derivatives of functions that depend on MHD equilibria (Antonsen et al. 2019; Paul et al. 2020), we present the first example of analytic gradient-based optimization of fixed-boundary stellarator equilibria. We take advantage of gradient information to optimize figures of merit of relevance for stellarator design, including the rotational transform, magnetic well, and quasisymmetry near the axis. With the application of the adjoint method, we reduce the number of equilibrium evaluations by the dimension of the optimization space ($\sim 50-500$) in comparison with a finite-difference gradient-based method. We discuss regularization objectives of relevance for fixed-boundary optimization, including a novel method that prevents self-intersection of the plasma boundary. We present several optimized equilibria, including a vacuum field with very low magnetic shear throughout the volume.

physics.plasm-ph↗

arXiv2020

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

A problem arising in several engineering areas is to design magnets outside a volume that produce a desired magnetic field inside it. One instance of this problem is stellarator design, where it has recently been shown that permanent magnets can provide the required shaping of the magnetic field. Here we demonstrate a robust and efficient algorithm REGCOIL_PM to calculate the spatial distribution of these permanent magnets. The procedure involves a small number of fixed-point iterations, with a linear least-squares problem solved at each step. The method exploits the Biot-Savart Law's exact linearity in magnetization density and approximate linearity in magnet size, for magnets far from the target region. No constraint is placed on the direction of magnetization, so Halbach solutions are found naturally, and the magnitude of the magnetization can be made uniformly equal to a target value.

physics.plasm-ph↗

arXiv2020

Magnetic well and Mercier stability of stellarators near the magnetic axis

We have recently demonstrated that by expanding in small distance from the magnetic axis compared to the major radius, stellarator shapes with low neoclassical transport can be generated efficiently. To extend the utility of this new design approach, here we evaluate measures of magnetohydrodynamic interchange stability within the same expansion. In particular, we evaluate magnetic well, Mercier's criterion, and resistive interchange stability near a magnetic axis of arbitrary shape. In contrast to previous work on interchange stability near the magnetic axis, which used an expansion of the flux coordinates, here we use the inverse expansion in which the flux coordinates are the independent variables. Reduced expressions are presented for the magnetic well and stability criterion in the case of quasisymmetry. The analytic results are shown to agree with calculations from the VMEC equilibrium code. Finally, we show that near the axis, Glasser, Greene, & Johnson's stability criterion for resistive modes approximately coincides with Mercier's ideal condition.

physics.plasm-ph↗

arXiv2020

Impurity temperature screening in stellarators close to quasisymmetry

Impurity temperature screening is a favorable neoclassical phenomenon involving an outward radial flux of impurity ions from the core of fusion devices. Quasisymmetric magnetic fields are intrinsically ambipolar and give rise to temperature screening for low enough $η^{-1}\equiv d\ln n/d\ln T$. In contrast, neoclassical fluxes in generic stellarators will depend on the radial electric field, which is predicted to be inward for ion-root plasmas, potentially leading to impurity accumulation. Here we examine the impurity particle flux in a number of approximately quasisymmetric stellarator configurations and parameter regimes while varying the amount of symmetry-breaking in the magnetic field. The amount of symmetry-breaking (from perfect quasisymmetry) required to lose temperature screening is dependent on the particular configuration and flux surface. Neoclassical fluxes have been obtained using the SFINCS drift-kinetic equation solver with the electrostatic potential $Φ=Φ(ψ)$, where $ψ$ is a flux surface label. Results indicate that achieving temperature screening is possible, but unlikely, at reactor-relevant conditions in the core. Thus, the small departures from symmetry in nominally quasisymmetric stellarators are large enough to significantly alter the neoclassical impurity transport. A further look at these fluxes when compared to a gyro-Bohm turbulence estimate suggests that neoclassical fluxes are small in optimized configurations compared to turbulent fluxes. Therefore, although neoclassical impurity accumulation is expected in most situations, the strength of the turbulence may render this irrelevant.

physics.plasm-ph↗

arXiv2019

Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. Part II: Applications

The shape gradient is a local sensitivity function that provides the change in a figure of merit associated with a perturbation to the shape of the object. The shape gradient can be used for gradient-based optimization, sensitivity analysis, and tolerance calculations. However, it is generally expensive to compute from finite-difference derivatives for shapes which are described by many parameters, as is the case for stellarator geometry. In an accompanying work (Antonsen et al. 2019), generalized self-adjointness relations are obtained for MHD equilibria. These describe the relation between perturbed equilibria due to changes in the rotational transform or toroidal current profiles, displacements of the plasma boundary, modifications of currents in the vacuum region, or the addition of bulk forces. These are applied to efficiently compute the shape gradient of functions of magnetohydrodynamic (MHD) equilibria with an adjoint approach. In this way, the shape derivative with respect to any perturbation applied to the plasma boundary or coil shapes can be computed with only one additional MHD equilibrium solution. We demonstrate that this approach is applicable for several figures of merit of interest for stellarator configuration optimization: the magnetic well, the magnetic ripple on axis, the departure from quasisymmetry, the effective ripple in the low-collisionality $1/ν$ regime ($ε_{\text{eff}}^{3/2})$ (Nemov et al. 1999), and several finite collisionality neoclassical quantities. Numerical verification is demonstrated for the magnetic well figure of merit with the VMEC code (Hirshman & Whitson 1983) and for the magnetic ripple with modification of the ANIMEC code (Cooper et al. 1992). Comparisons with the direct approach demonstrate that to obtain agreement within several percent, the adjoint approach provides a factor of $\mathcal{O}(10^3)$ in computational savings.

physics.plasm-ph↗

arXiv2019

Constructing stellarators with quasisymmetry to high order

A method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren and Boozer's expansion about the magnetic axis (Phys Fluids B 3, 2805 (1991)), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landreman, Sengupta and Plunk (J Plasma Phys 85, 905850103 (2019)), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer's ideal scaling of quasisymmetry-breaking with the cube of inverse aspect ratio. We also demonstrate a strongly nonaxisymmetric configuration (vacuum $ι> 0.4$) in which symmetry-breaking mode amplitudes throughout a finite volume are $< 2\times 10^{-7}$, the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. The approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization.

physics.plasm-ph↗

arXiv2019

An adjoint method for neoclassical stellarator optimization

Stellarators are a promising route to steady-state fusion power. However, to achieve the required confinement, the magnetic geometry must be highly optimized. This optimization requires navigating high-dimensional spaces, often necessitating the use of gradient-based methods. The gradient of the neoclassical fluxes is expensive to compute with classical methods, requiring $O(N)$ flux computations, where $N$ is the number of parameters. To reduce the cost of the gradient computation, we present an adjoint method for computing the derivatives of moments of the neoclassical distribution function for stellarator optimization. The linear adjoint method allows derivatives of quantities which depend on solutions of a linear system, such as moments of the distribution function, to be computed with respect to many parameters from the solution of only two linear systems. This reduces the cost of computing the gradient to the point that the finite-collisionality neoclassical fluxes can be used within an optimization loop. With the neoclassical adjoint method, we compute solutions of the drift kinetic equation and an adjoint drift kinetic equation to obtain derivatives of neoclassical quantities with respect to geometric parameters. When the number of parameters in the derivative is large ($\mathcal{O}(10^2)$), this adjoint method provides up to a factor of 200 reduction in cost. We demonstrate adjoint-based optimization of the field strength to obtain minimal bootstrap current on a surface. With adjoint-based derivatives, we also compute the local sensitivity to magnetic perturbations on a flux surface and identify regions where tight tolerances on error fields are required for control of the bootstrap current or radial transport. Furthermore, the solve for the ambipolar electric field is accelerated using a Newton method with derivatives obtained from the adjoint method.

physics.plasm-ph↗

arXiv2019

Optimized quasisymmetric stellarators are consistent with the Garren-Boozer construction

Most quasisymmetric stellarators to date have been designed by numerically optimizing the plasma boundary shape to minimize symmetry-breaking Fourier modes of the magnetic field strength $B$. At high aspect ratio, a faster approach is to directly construct the plasma shape from the equations of quasisymmetry near the magnetic axis derived by Garren and Boozer [Phys Fluids B 3, 2805 (1991)]. Here we show that the core shape and rotational transform of many optimization-based configurations can be accurately described by this direct-construction approach. This consistency supports use of the near-axis construction as an accurate analytical model for modern stellarator configurations.

physics.plasm-ph↗

arXiv2019

Adjoint approach to calculating shape gradients for 3D magnetic confinement equilibria

The shape gradient quantifies the change in some figure of merit resulting from differential perturbations to a shape. Shape gradients can be applied to gradient-based optimization, sensitivity analysis, and tolerance calculation. An efficient method for computing the shape gradient for toroidal 3D MHD equilibria is presented. The method is based on the self-adjoint property of the equations for driven perturbations of MHD equilibria and is similar to the Onsager symmetry of transport coefficients. Two versions of the shape gradient are considered. One describes the change in a figure of merit due to an arbitrary displacement of the outer flux surface; the other describes the change in the figure of merit due to the displacement of a coil. The method is implemented for several example figures of merit and compared with direct calculation of the shape gradient. In these examples the adjoint method reduces the number of equilibrium computations by factors of $\mathcal{O}(N)$, where $N$ is the number of parameters used to describe the outer flux surface or coil shapes.

physics.plasm-ph↗

arXiv2018

Direct construction of optimized stellarator shapes. II. Numerical quasisymmetric solutions

Quasisymmetric stellarators are appealing intellectually and as fusion reactor candidates since the guiding center particle trajectories and neoclassical transport are isomorphic to those in a tokamak, implying good confinement. Previously, quasisymmetric magnetic fields have been identified by applying black-box optimization algorithms to minimize symmetry-breaking Fourier modes of the field strength $B$. Here instead we directly construct magnetic fields in cylindrical coordinates that are quasisymmetric to leading order in distance from the magnetic axis, without using optimization. The method involves solution of a 1-dimensional nonlinear ordinary differential equation, originally derived by Garren and Boozer [Phys. Fluids B 3, 2805 (1991)]. We demonstrate the usefulness and accuracy of this optimization-free approach by providing the results of this construction as input to the codes VMEC and BOOZ_XFORM, confirming the purity and scaling of the magnetic spectrum. The space of magnetic fields that are quasisymmetric to this order is parameterized by the magnetic axis shape along with three other real numbers, one of which reflects the on-axis toroidal current density, and another one of which is zero for stellarator symmetry. The method here could be used to generate good initial conditions for conventional optimization, and its speed enables exhaustive searches of parameter space.

physics.plasm-ph↗