An adjoint method for gradient-based optimization of stellarator coil shapes

Paul, Elizabeth; Landreman, Matt; Bader, A.; Dorland, W.

doi:10.1088/1741-4326/aac1c7

Cited by 38 publications

(59 citation statements)

References 44 publications

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“…It is possible to identify regions where coil position is most sensitive (Landreman & Paul 2018; Paul et al. 2018), and it may be possible to target equilibria to relax specific regions such as those with strong concavities (see § 7 of Paul et al. 2018).…”

Section: Resultsmentioning

confidence: 99%

Stellarator equilibria with reactor relevant energetic particle losses

et al. 2019

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Stellarator configurations with reactor relevant energetic particle losses are constructed by simultaneously optimizing for quasisymmetry and an analytically derived metric ( $\unicode[STIX]{x1D6E4}_{c}$ ), which attempts to align contours of the second adiabatic invariant, $J_{\Vert }$ with magnetic surfaces. Results show that with this optimization scheme it is possible to generate quasihelically symmetric equilibria on the scale of ARIES-CS which completely eliminate all collisionless alpha particle losses within normalized radius $r/a=0.3$ . We show that the best performance is obtained by reducing losses at the trapped–passing boundary. Energetic particle transport can be improved even when neoclassical transport, as calculated using the metric $\unicode[STIX]{x1D716}_{\text{eff}}$ , is degraded. Several quasihelically symmetric equilibria with different aspect ratios are presented, all with excellent energetic particle confinement.

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Section: Resultsmentioning

confidence: 99%

Stellarator equilibria with reactor relevant energetic particle losses

et al. 2019

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“…To begin, it is assumed that the continuous adjoint approach outlined in section 3.1 is used. The approach taken here is analogous to that used in appendix A of Paul et al (2018), in which an additional adjoint equation is used to compute derivatives at a fixed constraint function for optimization of stellarator coil shapes.…”

Section: Appendix E Derivatives At Ambipolaritymentioning

confidence: 99%

“…The approach taken here is analogous to that used in appendix A of Paul et al. (2018), in which an additional adjoint equation is used to compute derivatives at a fixed constraint function for optimization of stellarator coil shapes.Consider the set of unknowns computed with SFINCS, , which depends on parameters and . The total differential of satisfies which follows from (2.11).…”

Section: Dkes Trajectoriesmentioning

confidence: 99%

“…They have only recently been implemented for stellarator design, namely for the design of coil shapes (Paul et al. 2018) and efficiently computing shape gradients for MHD equilibria (Antonsen, Paul & Landreman 2019). The numerical method is quite general and has the potential to greatly impact many inverse design problems in magnetic confinement fusion.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Coil perturbations under eigenvectors might not be clear and it would require much efforts to investigate all the eigenvectors. Recently, Paul et al [23] presented a technique for visualization of sensitivities. This could be a good complement of our Hessian method.…”

Section: Summary and Future Plansmentioning

confidence: 99%

Hessian matrix approach for determining error field sensitivity to coil deviations

Zhu

Hudson

Lazerson

et al. 2018

Plasma Phys. Control. Fusion

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Abstract. The presence of error fields has been shown to degrade plasma confinement and drive instabilities. Error fields can arise from many sources, but are predominantly attributed to deviations in the coil geometry. In this paper, we introduce a Hessian matrix approach for determining error field sensitivity to coil deviations. A primary cost function used for designing stellarator coils, the surface integral of normalized normal field errors, was adopted to evaluate the deviation of the generated magnetic field from the desired magnetic field. The FOCUS code [Zhu et al., Nucl. Fusion 58(1):016008 (2018)] is utilized to provide fast and accurate calculations of the Hessian. The sensitivities of error fields to coil displacements are then determined by the eigenvalues of the Hessian matrix. A proof-of-principle example is given on a CNT-like configuration. We anticipate that this new method could provide information to avoid dominant coil misalignments and simplify coil designs for stellarators.

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An adjoint method for gradient-based optimization of stellarator coil shapes

Cited by 38 publications

References 44 publications

Stellarator equilibria with reactor relevant energetic particle losses

Stellarator equilibria with reactor relevant energetic particle losses

An adjoint method for neoclassical stellarator optimization

Hessian matrix approach for determining error field sensitivity to coil deviations

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