Computing local sensitivity and tolerances for stellarator physics properties using shape gradients

Landreman, Matt; Paul, Elizabeth

doi:10.1088/1741-4326/aac197

Cited by 21 publications

(34 citation statements)

References 47 publications

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“…The ability to optimize without being confined to a winding surface is a new capability within FOCUS that was not available for the coil sets designed for HSX and W7-X. Previous results showed that areas of low sensitivity can be calculated for stellarator equilibria using shape gradients (Landreman & Paul 2018). Fortuitously, these regions of low sensitivity are also the areas were divertor heat fluxes tend to exit the plasma (Bader et al 2017), allowing for the construction of a non-resonant divertor as described in § 4.4.…”

Section: Coil Constructionmentioning

confidence: 99%

Advancing the physics basis for quasi-helically symmetric stellarators

et al. 2020

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A new optimized quasi-helically symmetric configuration is described that has the desirable properties of improved energetic particle confinement, reduced turbulent transport by three-dimensional shaping and non-resonant divertor capabilities. The configuration presented in this paper is explicitly optimized for quasi-helical symmetry, energetic particle confinement, neoclassical confinement and stability near the axis. Post optimization, the configuration was evaluated for its performance with regard to energetic particle transport, ideal magnetohydrodynamic stability at various values of plasma pressure and ion temperature gradient instability induced turbulent transport. The effects of discrete coils on various confinement figures of merit, including energetic particle confinement, are determined by generating single-filament coils for the configuration. Preliminary divertor analysis shows that coils can be created that do not interfere with expansion of the vessel volume near the regions of outgoing heat flux, thus demonstrating the possibility of operating a non-resonant divertor.

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Section: Coil Constructionmentioning

confidence: 99%

Advancing the physics basis for quasi-helically symmetric stellarators

et al. 2020

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“…The function is defined on a flux surface, ; thus it is sensible to express in the following way, Here describes the local perturbation to the field strength, and quantifies the corresponding change to the moment . The function is analogous to the shape gradient, which quantifies the change in a figure of merit which results from a differential perturbation to a shape (Landreman & Paul 2018). The shape gradient will be discussed further in § 5.2.2.…”

Section: Applications Of the Adjoint Methodsmentioning

confidence: 99%

“…In this way, can inform where perturbations to the magnetic field strength can be tolerated. The sensitivity function could be related directly to a local magnetic tolerance using the method described in section 9 of Landreman & Paul (2018). In contrast with that work, here we are considering perturbations to the field strength on any flux surface rather than at the plasma boundary.…”

Section: Applications Of the Adjoint Methodsmentioning

confidence: 99%

An adjoint method for neoclassical stellarator optimization

et al. 2019

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Received xx; revised xx; accepted xx)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 (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.

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“…For example, Ω = {R c mn , Z s mn } could be assumed, where these are the Fourier coefficients in a cosine and sine representation of the cylindrical coordinates (R, Z) of S P . Upon discretization of the right-hand side on a surface, the above takes the form of a linear system that can be solved for G (Landreman & Paul 2018). However, this approach requires performing at least one additional equilibrium calculation for each parameter with a finite-difference approach.…”

Section: )mentioning

confidence: 99%

Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. Part 2. Applications

et al. 2020

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The shape gradient is a local sensitivity function defined on the surface of an object which provides the change in a characteristic quantity, or 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 that 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 ( 3/2 eff ) (Nemov et al. 1999), and several finite collisionality neoclassical quantities. Numerical verification of this method 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 in order to obtain agreement within several percent, the adjoint approach provides a factor of O(10 3 ) in computational savings.

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Computing local sensitivity and tolerances for stellarator physics properties using shape gradients

Cited by 21 publications

References 47 publications

Advancing the physics basis for quasi-helically symmetric stellarators

Advancing the physics basis for quasi-helically symmetric stellarators

An adjoint method for neoclassical stellarator optimization

Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria. Part 2. Applications

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