Direct stellarator coil design using global optimization: application to a comprehensive exploration of quasi-axisymmetric devices

Many stellarator coil design problems are plagued by multiple minima, where the locally optimal coil sets can sometimes vary substantially in performance. As a result, solving a coil design problem a single time with a local optimization algorithm is usually insufficient and better optima likely do exist. To address this problem, we propose a global optimization algorithm for the design of stellarator coils and outline how to apply box constraints to the physical positions of the coils. The algorithm has a global exploration phase that searches for interesting regions of design space and is followed by three local optimization algorithms that search in these interesting regions (a "global-to-local" approach). The first local algorithm (phase I), following the globalization phase, is based on near-axis expansions and finds stellarator coils that optimize for quasisymmetry in the neighborhood of a magnetic axis. The second local algorithm (phase II) takes these coil sets and optimizes them for nested flux surfaces and quasisymmetry on a toroidal volume. The final local algorithm (phase III) polishes these configurations for an accurate approximation of quasisymmetry. Using our global algorithm, we study the trade-off between coil length, aspect ratio, rotational transform, and quality of quasi-axisymmetry. The database of stellarators, which comprises almost 140,000 coil sets, is available online and is called QUASR, for "QUAsi-symmetric Stellarator Repository"

A parallel, adaptive discontinuous Galerkin method for hyperbolic problems on unstructured meshes

This thesis is concerned with the parallel, adaptive solution of hyperbolic conservation laws on unstructured meshes. First, we present novel algorithms for cell-based adaptive mesh refinement (AMR) on unstructured meshes of triangles on graphics processing units (GPUs). Our implementation makes use of improved memory management techniques and a coloring algorithm for avoiding race conditions. The algorithm is entirely implemented on the GPU, with negligible communication between device and host. We show that the overhead of the AMR subroutines is small compared to the high-order solver and that the proportion of total run time spent adaptively refining the mesh decreases with the order of approximation. We apply our code to a number of benchmarks as well as more recently proposed problems for the Euler equations that require extremely high resolution. We present the solution to a shock reflection problem that addresses the von Neumann triple point paradox. We also study the problem of shock disappearance and self-similar diffraction of weak shocks around thin films. Next, we analyze the stability and accuracy of second-order limiters for the discontinuous Galerkin method on unstructured triangular grids. We derive conditions for a limiter such that the numerical solution preserves second order accuracy and satisfies the local maximum principle. This leads to a new measure of cell size that is approximately twice as large as the radius of the inscribed circle. It is shown with numerical experiments that the resulting bound on the time step is tight. We also consider various combinations of limiting points and limiting neighborhoods and present numerical experiments comparing the accuracy, stability, and efficiency of the corresponding limiters. We show that the theory for strong stability preserving (SSP) time stepping methods employed with the method of lines-type discretizations of hyperbolic conservation laws may result in overly stringent time step restrictions. We analyze a fully discrete finite volume method with slope reconstruction and a second order SSP Runge-Kutta time integrator to show that the maximum stable time step can be increased over the SSP limit. Numerical examples show that this result extends to two-dimensional problems on triangular meshes. Finally, we propose a moment limiter for the discontinuous Galerkin method applied to hyperbolic conservation laws in two and three dimensions. The limiter works by finding directions in which the solution coefficients can be separated and limits them independently of one another by comparing to forward and backward reconstructed differences. The limiter has a precomputed stencil of constant size, which provides computational advantages in terms of implementation and runtime. We provide examples that demonstrate stability and second order accuracy of solutions

Single-stage gradient-based stellarator coil design: stochastic optimization

We extend the single-stage stellarator coil design approach for quasi-symmetry on axis from [Giuliani et al, 2020] to additionally take into account coil manufacturing errors. By modeling coil errors independently from the coil discretization, we have the flexibility to consider realistic forms of coil errors. The corresponding stochastic optimization problems are formulated using a risk-neutral approach and risk-averse approaches. We present an efficient, gradient-based descent algorithm which relies on analytical derivatives to solve these problems. In a comprehensive numerical study, we compare the coil designs resulting from deterministic and risk-neutral stochastic optimization and find that the risk-neutral formulation results in more robust configurations and reduces the number of local minima of the optimization problem. We also compare deterministic and risk-neutral approaches in terms of quasi-symmetry on and away from the magnetic axis, and in terms of the confinement of particles released close to the axis. Finally, we show that for the optimization problems we consider, a risk-averse objective using the Conditional Value-at-Risk leads to results which are similar to the risk-neutral objective

Direct stellarator coil optimization for nested magnetic surfaces with precise quasisymmetry

We present a robust optimization algorithm for the design of electromagnetic coils that generate vacuum magnetic fields with nested flux surfaces and precise quasisymmetry. The method is based on a bilevel optimization problem, where the outer coil optimization is constrained by a set of inner least-squares optimization problems whose solutions describe magnetic surfaces. The outer optimization objective targets coils that generate a field with nested magnetic surfaces and good quasisymmetry. The inner optimization problems identify magnetic surfaces when they exist, and approximate surfaces in the presence of magnetic islands or chaos. We show that this formulation can be used to heal islands and chaos, thus producing coils that result in magnetic fields with precise quasisymmetry. We show that the method can be initialized with coils from the traditional two stage coil design process, as well as coils from a near axis expansion optimization. We present a numerical example where island chains are healed and quasisymmetry is optimized up to surfaces with aspect ratio 6. Another numerical example illustrates that the aspect ratio of nested flux surfaces with optimized quasisymmetry can be decreased from 6 to approximately 4. The last example shows that our approach is robust and a cold-start using coils from a near-axis expansion optimization

On the optimal CFL number of SSP methods for hyperbolic problems

The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.apnum.2018.08.015 © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/We show that the theory for strong stability preserving (SSP) time stepping methods employed with the method of lines-type discretizations of hyperbolic conservation laws may result in overly stringent time step restrictions. We analyze a fully discrete finite volume method with slope reconstruction and a second order SSP Runge–Kutta time integrator to show that the maximum stable time step can be increased over the SSP limit. Numerical examples indicate that this result extends to two-dimensional problems on triangular meshes.Natural Sciences and Engineering Research Council of Canada ["341373-07"]Alexander Graham Bell PGS-DNVIDIA Corporatio

Relationships Between Neuromuscular Function and Functional Balance Performance in Firefighters

The purpose of the present study was to examine the relationships between neuromuscular function and functional balance performance in firefighters. Fifty career firefighters (35.1±7.5yr) performed isometric leg extension and flexion muscle actions to examine peak torque (PT), and absolute (aTQ) and normalized (nTQ; %PT) rapid torque variables at 50, 100, 150, and 200ms. A performance index (PI) was determined from the functional balance assessment completion time. Partial correlations were used to examine the relationship between the PI and the maximal and rapid TQ variables for each muscle and the composite value, while controlling for demographic data related to the PI. Multiple regression analyses examined the relative contributions of the maximal and rapid aTQ variables, and demographic data on the PI. After controlling for age and %BF, the majority of the later aTQ and nTQ variables (100– 200ms) and PT were associated with the PI (r=−0.501–−0.315). Age, %BF, and aTQ100 explained 42– 50% of the variance in the PI. Lower rapid strength, increased age, and poorer body composition were related to worse performance during the functional balance assessment. Strategies to improve rapid strength and %BF, especially in aging firefighters may impact dynamic balance abilities in firefighters