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.

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

2019 ◽  
Vol 85 (2) ◽  
Author(s):  
Thomas Antonsen ◽  
Elizabeth J. Paul ◽  
Matt Landreman
Keyword(s):  
Figure Of Merit ◽  
Transport Coefficients ◽  
Three Dimensional ◽  
Direct Calculation ◽  
Flux Surface ◽  
Shape Gradient ◽  
Onsager Symmetry ◽  
Mhd Equilibria ◽  
Gradient Based ◽  
Adjoint Approach

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 three-dimensional magnetohydrodynamic (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$O(N)$, where$N$is the number of parameters used to describe the outer flux surface or coil shapes.

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 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 The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
P. Kim ◽  
R. Jorge ◽  
W. Dorland
Keyword(s):  
Magnetic Field ◽  
Original Work ◽  
Three Dimensional ◽  
Radial Distance ◽  
Analytical Form ◽  
One Dimensional ◽  
Modern Notation ◽  
Magnetic Well ◽  
First Time ◽  
Dimensional Integral

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

Three-Dimensional CFD Model of Pressure Drop in µTAS Devices in a Microchannel

2011 ◽  
Vol 133 (3) ◽  
Author(s):  
Damena D. Agonafer ◽  
J. Yeom ◽  
M. A. Shannon
Keyword(s):  
Pressure Drop ◽  
Figure Of Merit ◽  
Flow Resistance ◽  
Three Dimensional ◽  
Design Rule ◽  
Dynamics Model ◽  
Enhance Heat Transfer ◽  
Micro Total Analysis Systems ◽  
Total Analysis ◽  
Adsorption Desorption

Microposts are utilized to enhance heat transfer, adsorption/desorption, and surface chemical reactions. In a previous study [Yeom et al., J. Micromech. Microeng., 19, p. 065025 (2009)], based in part on an experimental study, an analytical expression was developed to predict the pressure drop across a microchannel filled with arrays of posts with the goal of fabricating more efficient micro-total analysis systems (µTAS) devices for a given pumping power. In particular, a key figure of merit for the design of micropost-filled reactors, based on the flow resistance models was reported thus providing engineers with a design rule to develop efficient µTAS devices. The study did not include the effects of the walls bounding the microposts. In this paper, a three-dimensional computational fluid dynamics model is used to include the effects of three-dimensionality brought about by the walls of the µTAS devices that bound the microposted structures. In addition, posts of smaller size that could not be fabricated for the experiments were also included. It is found that the two- and three-dimensional effects depend on values of the aspect ratio and the blockage ratios. The Reynolds number considered in the experiment that ranged from 1 to 10 was extended to 300 to help determine the range of Re for which the FOM model is applicable.

scholarly journals Joint Discussion 6: Sun and Heliosphere - Challenges for Solar-Terrestrial Physics, Magneto-and Hydrodynamics

1995 ◽  
Vol 10 ◽  
pp. 291-293
Author(s):  
Martin C.E. Huber ◽  
Arne Pedersen ◽  
Claus Fröhlich
Keyword(s):  
Spatial Scales ◽  
Plasma Heating ◽  
European Space Agency ◽  
Three Dimensional ◽  
Plasma Boundary ◽  
Small Scale ◽  
Space Agency ◽  
Terrestrial Magnetosphere ◽  
Surrounding Space

There is one astrophysical system, where the sites of a star’s mass loss can be localised and observed in detail, and where the behaviour of the resulting stellar wind in the star’s environment and around orbiting obstacles can be investigated in situ: it is the Sun, the heliosphere and the surroundings of planets — among the latter most prominently the terrestrial magnetosphere. Indeed, within a year or so a fleet of satellites equipped with sophisticated remote-sensing and in-situ instruments will make this astronomical paradigm, or more precisely, the solar-terrestrial system accessible to intensive, multi-disciplinary study.Four identical CLUSTER spacecraft, orbiting the Earth within the magnetosphere, the surrounding space and the particularly interesting plasma boundary layers will perform a three-dimensional in-situ study of plasma-heating, particle-acceleration and other small-scale plasma processes (Schmidt and Goldstein,1988). A number of other missions — some of them already in orbit, like GEOTAIL and WIND, some to be launched within one or two years, like INTERBALL and POLAR — will provide information about the Earth’s magnetosphere and the solar wind on larger spatial scales. These missions are described in a Brochure issued jointly by the European Space Agency, NASA, the Japanese Institute of Space and Astronomical Science and the Rssian Space Agency, which can be obtained from A. Pedersen at the above address.

Atomic-Scale Three-Dimensional Phononic Crystals With a Large Thermoelectric Figure of Merit

2008 ◽  
Author(s):  
Jean-Numa Gillet ◽  
Sebastian Volz
Keyword(s):  
Thermal Conductivity ◽  
Figure Of Merit ◽  
Phononic Crystal ◽  
Three Dimensional ◽  
Phononic Crystals ◽  
Atomic Scale ◽  
Thermoelectric Figure Of Merit ◽  
Thermoelectric Figure ◽  
Low Thermal Conductivity ◽  
Cmos Compatible

The design of thermoelectric materials led to extensive research on superlattices with a low thermal conductivity. Indeed, the thermoelectric figure of merit ZT varies with the inverse of the thermal conductivity but is directly proportional to the power factor. Unfortunately, as nanowires, superlattices cancel heat conduction in only one main direction. Moreover they often show dislocations owing to lattice mismatches, which reduces their electrical conductivity and avoids a ZT larger than unity. Self-assembly is a major epitaxial technology to design ultradense arrays of germanium quantum dots (QDs) in silicon for many promising electronic and photonic applications as quantum computing. Accurate positioning of the self-assembled QD can now be achieved with few dislocations. We theoretically demonstrate that high-density three-dimensional (3-D) arrays of self-assembled Ge QDs, with a size of only some nanometers, in a Si matrix can also show an ultra-low thermal conductivity in the three spatial directions. This property can be considered to design new CMOS-compatible thermoelectric devices. To obtain a realistic and computationally-manageable model of these nanomaterials, we simulate their thermal behavior with atomic-scale 3-D phononic crystals. A phononic-crystal period (supercell) consists of diamond-like Si cells. At each supercell center, we substitute Si atoms by Ge atoms to form a box-like nanoparticle. Since this phononic crystal is periodic, we compute its phonon dispersion curves by classical lattice dynamics. Non-periodicities can be introduced with statistical distributions. From the flat dispersion curves, we obtain very small group velocities; this reduces the thermal conductivity in our phononic crystal compared to bulk Si. However, owing to the wave-particle duality at very small scales in quantum mechanics, another reduction arises from multiple scattering of the particle-like phonons in nanoparticle clusters. At room temperature, the thermal conductivity in an example phononic crystal can be reduced by a factor of at least 165 compared to bulk Si or below 0.95 W/mK. This value, which is lower than the classical Einstein limit of single crystalline Si, is an upper limit of the thermal conductivity since we use an incoherent-scattering approach for the nanoparticles. Because of its very low thermal conductivity, we hope to obtain a much larger ZT than unity in our atomic-scale 3-D phononic crystal. Indeed, this silicon-based nanomaterial is crystalline with a power factor that can be optimized by doping using CMOS-compatible processes. Future research on the phononic-crystal electrical conductivity has to be performed in order to compute the full ZT with a good accuracy.

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