Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria

Antonsen, Thomas M.; Paul, Elizabeth; Landreman, Matt

doi:10.1017/s0022377819000254

Cited by 16 publications

(42 citation statements)

References 35 publications

(63 reference statements)

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“…For perturbed MHD equilibria, the displacement vector ξ describes the motion of the boundary (ξ · n| S P = δr · n| S P ). Thus the shape derivative with respect to the plasma boundary can be expressed with the replacement δr → ξ (Appendix C of Antonsen et al (2019)). Therefore we see that the boundary term in (3.13) is already in the form of a shape gradient (2.4).…”

Section: Adjoint Relations For Mhd Equilibriamentioning

confidence: 99%

“…In an accompanying work (Antonsen, Paul & Landreman 2019), two adjoint relations are derived: one involving perturbations to the plasma boundary, referred to as the fixed-boundary adjoint relation, and the other involving perturbations to currents in the vacuum region, known as the free-boundary adjoint relation. These can be considered generalizations of the self-adjointness of the force operator that arises in linearized MHD (Bernstein et al 1958).…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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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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Section: Adjoint Relations For Mhd Equilibriamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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

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“…The shape gradient can be used for fixed-boundary optimization of equilibria or for analysis of sensitivity to perturbations of magnetic surfaces. It can be computed using a second adjoint method, where a perturbed MHD force balance equation is solved with the addition of a bulk force which depends on derivatives computed from the neoclassical adjoint method (Antonsen et al 2019). While the continuous neoclassical adjoint method described in this work arises from the self-adjointness of the linearized Fokker-Planck operator, the adjoint method for MHD equilibria arises from the self-adjointness of the MHD force operator.…”

Section: Optimization Of Mhd Equilibriamentioning

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%

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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Combined plasma–coil optimization algorithms

et al. 2021

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Combined plasma–coil optimization approaches for designing stellarators are discussed and a new method for calculating free-boundary equilibria for multiregion relaxed magnetohydrodynmics (MRxMHD) is proposed. Four distinct categories of stellarator optimization, two of which are novel approaches, are the fixed-boundary optimization, the generalized fixed-boundary optimization, the quasi-free-boundary optimization, and the free-boundary (coil) optimization. These are described using the MRxMHD energy functional, the Biot–Savart integral, the coil-penalty functional and the virtual casing integral and their derivatives. The proposed free-boundary equilibrium calculation differs from existing methods in how the boundary-value problem is posed, and for the new approach it seems that there is not an associated energy minimization principle because a non-symmetric functional arises. We propose to solve the weak formulation of this problem using a spectral-Galerkin method, and this will reduce the free-boundary equilibrium calculation to something comparable to a fixed-boundary calculation. In our discussion of combined plasma–coil optimization algorithms, we emphasize the importance of the stability matrix.

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Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria

Cited by 16 publications

References 35 publications

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

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

An adjoint method for neoclassical stellarator optimization

Combined plasma–coil optimization algorithms

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