scholarly journals Computing the shape gradient of stellarator coil complexity with respect to the plasma boundary

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Arthur Carlton-Jones ◽  
Elizabeth J. Paul ◽  
William Dorland
Keyword(s):  
Null Space ◽  
Radial Distance ◽  
Plasma Boundary ◽  
Optimization Approach ◽  
Surface Parameters ◽  
Fixed Boundary ◽  
Plasma Surface ◽  
Shape Gradient ◽  
Derivative Free ◽  
Boundary Optimization

Coil complexity is a critical consideration in stellarator design. The traditional two-step optimization approach, in which the plasma boundary is optimized for physics properties and the coils are subsequently optimized to be consistent with this boundary, can result in plasma shapes which cannot be produced with sufficiently simple coils. To address this challenge, we propose a method to incorporate considerations of coil complexity in the optimization of the plasma boundary. Coil complexity metrics are computed from the current potential solution obtained with the REGCOIL code (Landreman, Nucl. Fusion, vol. 57, 2017, 046003). While such metrics have previously been included in derivative-free fixed-boundary optimization (Drevlak et al., Nucl. Fusion, vol. 59, 2018, 016010), we compute the local sensitivity of these metrics with respect to perturbations of the plasma boundary using the shape gradient (Landreman & Paul, Nucl. Fusion, vol. 58, 2018, 076023). We extend REGCOIL to compute derivatives of these metrics with respect to parameters describing the plasma boundary. In keeping with previous research on winding surface optimization (Paul et al., Nucl. Fusion, vol. 58, 2018, 076015), the shape derivatives are computed with a discrete adjoint method. In contrast with the previous work, derivatives are computed with respect to the plasma surface parameters rather than the winding surface parameters. To further reduce the resolution required to compute the shape gradient, we present a more efficient representation of the plasma surface which uses a single Fourier series to describe the radial distance from a coordinate axis and a spectrally condensed poloidal angle. This representation is advantageous over the standard cylindrical representation used in the VMEC code (Hirshman & Whitson, Phys. Fluids, vol. 26, 1983, pp. 3553–3568), as it provides a uniquely defined poloidal angle, eliminating a null space in the optimization of the plasma surface. In comparison with previous spectral condensation methods (Hirshman & Breslau, Phys. Plasmas, vol. 5, 1998, p. 2664), the modified poloidal angle is obtained algebraically rather than through the solution of a nonlinear optimization problem. The resulting shape gradient highlights features of the plasma boundary that are consistent with simple coils and can be used to couple coil and fixed-boundary optimization.

scholarly journals Gradient-based optimization of 3D MHD equilibria

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Elizabeth J. Paul ◽  
Matt Landreman ◽  
Thomas Antonsen
Keyword(s):  
Plasma Boundary ◽  
Plasma Phys ◽  
Vacuum Field ◽  
Fixed Boundary ◽  
Mhd Equilibria ◽  
Gradient Based ◽  
Boundary Optimization ◽  
Novel Method ◽  
Rotational Transform ◽  
Magnetic Well

Using recently developed adjoint methods for computing the shape derivatives of functions that depend on magnetohydrodynamic (MHD) equilibria (Antonsen et al., J. Plasma Phys., vol. 85, issue 2, 2019; Paul et al., J. Plasma Phys., vol. 86, issue 1, 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 quasi-symmetry near the axis. With the application of the adjoint method, we reduce the number of equilibrium evaluations by the dimension of the optimization space ( ${\sim }50\text {--}500$ ) 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.

scholarly journals RF cavity design exploiting a new derivative-free trust region optimization approach

2015 ◽  
Vol 6 (6) ◽  
pp. 915-924 ◽  
Author(s):  
Abdel-Karim S.O. Hassan ◽  
Hany L. Abdel-Malek ◽  
Ahmed S.A. Mohamed ◽  
Tamer M. Abuelfadl ◽  
Ahmed E. Elqenawy
Keyword(s):  
Trust Region ◽  
Optimization Approach ◽  
Rf Cavity ◽  
Derivative Free ◽  
Cavity Design

scholarly journals Combined plasma–coil optimization algorithms

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
S. A. Henneberg ◽  
S. R. Hudson ◽  
D. Pfefferlé ◽  
P. Helander
Keyword(s):  
Free Boundary ◽  
Optimization Algorithms ◽  
Energy Functional ◽  
Equilibrium Calculation ◽  
Weak Formulation ◽  
Fixed Boundary ◽  
Minimization Principle ◽  
Boundary Equilibrium ◽  
Boundary Optimization ◽  
Coil Optimization

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.

scholarly journals Modal optimization approach for composite aeroelastic wing model based on a derivative-free BFGS method

2019 ◽  
Vol 1300 ◽  
pp. 012085
Author(s):  
Binbin Lv ◽  
Pengxuan Lei ◽  
Hongtao Guo ◽  
Xiping Kou ◽  
Li Yu
Keyword(s):  
Optimization Approach ◽  
Bfgs Method ◽  
Model Based ◽  
Wing Model ◽  
Derivative Free

scholarly journals End-effector Cartesian stiffness shaping - sequential least squares programming approach

2021 ◽  
Vol 18 (1) ◽  
pp. 1-14
Author(s):  
Nikola Knezevic ◽  
Branko Lukic ◽  
Kosta Jovanovic ◽  
Leon Zlajpah ◽  
Tadej Petric
Keyword(s):  
Least Squares ◽  
Stiffness Matrix ◽  
Null Space ◽  
Joint Stiffness ◽  
Mechanical Impedance ◽  
Programming Algorithm ◽  
Programming Approach ◽  
Optimization Approach ◽  
End Effector ◽  
Cartesian Stiffness

Control of robot end-effector (EE) Cartesian stiffness matrix (or the whole mechanical impedance) is still a challenging open issue in physical humanrobot interaction (pHRI). This paper presents an optimization approach for shaping the robot EE Cartesian stiffness. This research targets collaborative robots with intrinsic compliance - serial elastic actuators (SEAs). Although robots with SEAs have constant joint stiffness, task redundancy (null-space) for a specific task could be used for robot reconfiguration and shaping the stiffness matrix while still keeping the EE position unchanged. The method proposed in this paper to investigate null-space reconfiguration's influence on Cartesian robot stiffness is based on the Sequential Least Squares Programming (SLSQP) algorithm, which presents an expansion of the quadratic programming algorithm for nonlinear functions with constraints. The method is tested in simulations for 4 DOF planar robot. Results are presented for control of the EE Cartesian stiffness initially along one axis, and then control of stiffness along both planar axis - shaping the main diagonal of the EE stiffness matrix.

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

2020 ◽  
Vol 86 (1) ◽  
Author(s):  
Elizabeth J. Paul ◽  
Thomas Antonsen ◽  
Matt Landreman ◽  
W. Anthony Cooper
Keyword(s):  
Figure Of Merit ◽  
Three Dimensional ◽  
Plasma Boundary ◽  
Plasma Phys ◽  
Numerical Verification ◽  
Shape Gradient ◽  
Characteristic Quantity ◽  
Mhd Equilibria ◽  
Adjoint Approach ◽  
Magnetic Well

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 typical stellarator geometry. In an accompanying work (Antonsen, Paul & Landreman J. Plasma Phys., vol. 85 (2), 2019), generalized self-adjointness relations are obtained for magnetohydrodynamic (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 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 $1/\unicode[STIX]{x1D708}$ regime $(\unicode[STIX]{x1D716}_{\text{eff}}^{3/2})$ (Nemov et al. Phys. Plasmas, vol. 6 (12), 1999, pp. 4622–4632) 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 Phys. Fluids, vol. 26 (12), 1983, p. 3553) and for the magnetic ripple with modification of the ANIMEC code (Cooper et al. Comput. Phys. Commun., vol. 72 (1), 1992, pp. 1–13). Comparisons with the direct approach demonstrate that, in order to obtain agreement within several per cent, the adjoint approach provides a factor of $O(10^{3})$ in computational savings.

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