Direct Optimization of Fast-Ion Confinement in Stellarators

TL;DR

The paper addresses direct optimization of fast-ion confinement in stellarators by embedding trajectory-based alpha-particle losses into the design loop. It formulates a boundary-shape optimization over Fourier coefficients $\mathbf{w}$ to minimize $\mathcal{J}(\mathbf{w}) = \mathbb{E}[\mathcal{J}_{\text{energy}}]$ with $\mathcal{J}_{\text{energy}} = 3.5\,e^{-2\mathcal{T}(\mathbf{x},v_{\parallel},\mathbf{w})/t_{\max}}$ and traces collisionless guiding-center trajectories in Boozer coordinates using SIMSOPT/VMEC; the objective is evaluated via MC/SAA/QMC/Simpson quadrature. The study reports two vacuum configurations, A and B, that achieve low alpha losses and favorable $\epsilon_{\text{eff}}$ values without relying on quasi-symmetry, illustrating that direct trajectory optimization can produce good confinement. It also discusses the computational challenges and outlines variance-reduction, symplectic tracing, and multi-fidelity strategies to accelerate the process, highlighting practical pathways to incorporate direct fast-ion metrics in early-stage stellarator design. The results imply that direct, trajectory-based objectives can complement or surpass proxy metrics in achieving robust fast-ion confinement in future reactors.

Abstract

Confining energetic ions such as alpha particles is a prime concern in the design of stellarators. However, directly measuring alpha confinement through numerical simulation of guiding-center trajectories has been considered to be too computationally expensive and noisy to include in the design loop, and instead has been most often used only as a tool to assess stellarator designs post hoc. In its place, proxy metrics, simplified measures of confinement, have often been used to design configurations because they are computationally more tractable and have been shown to be effective. Despite the success of proxies, it is unclear what is being sacrificed by using them to design the device rather than relying on direct trajectory calculations. In this study, we optimize stellarator designs for improved alpha particle confinement without the use of proxy metrics. In particular, we numerically optimize an objective function that measures alpha particle losses by simulating alpha particle trajectories. While this method is computationally expensive, we find that it can be used successfully to generate configurations with low losses.

Figures (9)

• Figure 1: (left) Radial probability density $f_s(s)$ derived from the fusion reaction rate. (right) Density over Boozer coordinates $\theta$ and $\zeta$, $f_{\theta,\zeta}$ for configuration A which will be discussed in \ref{['sec:numerical']}.
• Figure 2: Wall-clock time required to trace a single particle until a terminal time $t_{\max}$ using a single Intel(R) Xeon(R) CPU E5-2620 v3 @ 2.40GHz processor on a computing cluster. The processor was allotted 2GB of memory. Timing results were averaged over $2000$ particles randomly generated throughout a four field period configuration, all of which were confined to their terminal time $t_{\max}$. The total time of an objective evaluation also includes the fixed time of evaluating VMEC, computing the Boozer transformation, and building interpolants of the $\mathbf{B}$-field, which took $19.07$ seconds for this configuration.
• Figure 3: Energy of alpha particles at the time they are lost. Data points were generated by tracing particles with collisions from 10 configurations: NCSX, ARIES-CS, a QA from NYU, CFQS, a QH from IPP, a QA from IPP, HSX, Wistell-A, LHD, and W7-X. $20,000$ particles were traced for each configuration. The solid black line indicates the regressed mean of the data, and the dashed red line is the energy decay model $3.5\exp(-2t\nu_{s}^{\alpha/e})$ where the slowing-down time $1/\nu_s^{\alpha/e} \approx 0.057$sec was computed analytically using the volume averaged density and temperature NRLPlasma2019. The energy model very closely matches the mean particle energy.
• Figure 4: (Left) Approximations of the objective function $\mathcal{J}_{1/4}$ using Simpson's rule, QMC, SAA, and MC across a one dimensional slice of space, around a point $\mathbf{w}_0$. The curves were computed by tracing $4096$ particles per point, with particles on a $16^3$ mesh for Simpson's rule. The shaded region is the $95\%$ confidence interval for the objective value computation with MC. The black line (Actual) represents the actual value of the objective, and is computed using MC with $32,000$ samples. (Right) Relative error of MC, QMC, and Simpson's rule in computing the objective at a single point. The relative error was computed against the QMC estimate with $2^{15}$ particles. The MC curve represents the expected relative error of the MC estimator given the sample size, and was computed by the statistical method of bootstrapping james2013introduction.
• Figure 5: Three dimensional views of configuration A (left) and configuration B (middle). Cross sections of configuration A (solid) and B (dashed) at four cylindrical angles $\phi$ across a field period (right).