Topology optimization for inverse magnetostatics as sparse regression: application to electromagnetic coils for stellarators

Topology optimization, a technique to determine where material should be placed within a predefined volume in order to minimize a physical objective, is used across a wide range of scientific fields and applications. A general application for topology optimization is inverse magnetostatics; a desired magnetic field is prescribed, and a distribution of steady currents is computed to produce that target field. In the present work, electromagnetic coils are designed by magnetostatic topology optimization, using volume elements (voxels) of electric current, constrained so the current is divergence-free. Compared to standard electromagnet shape optimization, our method has the advantage that the nonlinearity in the Biot-Savart law with respect to position is avoided, enabling convex cost functions and a useful reformulation of topology optimization as sparse regression. To demonstrate, we consider the application of designing electromagnetic coils for a class of plasma experiments known as stellarators. We produce topologically-exotic coils for several new stellarator designs and show that these solutions can be interpolated into a filamentary representation and then further optimized

Greedy permanent magnet optimization

A number of scientific fields rely on placing permanent magnets in order to produce a desired magnetic field. We have shown in recent work that the placement process can be formulated as sparse regression. However, binary, grid-aligned solutions are desired for realistic engineering designs. We now show that the binary permanent magnet problem can be formulated as a quadratic program with quadratic equality constraints (QPQC), the binary, grid-aligned problem is equivalent to the quadratic knapsack problem with multiple knapsack constraints (MdQKP), and the single-orientation-only problem is equivalent to the unconstrained quadratic binary problem (BQP). We then provide a set of simple greedy algorithms for solving variants of permanent magnet optimization, and demonstrate their capabilities by designing magnets for stellarator plasmas. The algorithms can a-priori produce sparse, grid-aligned, binary solutions. Despite its simple design and greedy nature, we provide an algorithm that outperforms the state-of-the-art algorithms while being substantially faster, more flexible, and easier-to-use

Permanent magnet optimization for stellarators as sparse regression

A common scientific inverse problem is the placement of magnets that produce a desired magnetic field inside a prescribed volume. This is a key component of stellarator design, and recently permanent magnets have been proposed as a potentially useful tool for magnetic field shaping. Here, we take a closer look at possible objective functions for permanent magnet optimization, reformulate the problem as sparse regression, and propose an algorithm that can efficiently solve many convex and nonconvex variants. The algorithm generates sparse solutions that are independent of the initial guess, explicitly enforces maximum strengths for the permanent magnets, and accurately produces the desired magnetic field. The algorithm is flexible, and our implementation is open-source and computationally fast. We conclude with two new permanent magnet configurations for the NCSX and MUSE stellarators. Our methodology can be additionally applied for effectively solving permanent magnet optimizations in other scientific fields, as well as for solving quite general high-dimensional, constrained, sparse regression problems, even if a binary solution is required

The Evaluation of the Role of Beta-Hydroxy Fatty Acids on Chronic Inflammation and Insulin Resistance

β-hydroxy fatty acids are a major component of lipid A moiety of lipopolysaccaride. We aimed to investigate the role of free β-hydroxy fatty acids on inflammation, as well as to evaluate their effects on cytokine release from human blood cells, and whether they exist in plasma of patients with chronic inflammatory diseases with/without insulin resistance. Peripheral venous blood was incubated with β-hydroxy lauric and β-hydroxy myristic acids (each 100 ng, 1 μg, 10 μg/mL) up to 24 hours. Cytokines were measured from culture media and plasma. Free fatty acids and biochemical parameters were also measured from patients' plasma. Only β-hydroxy lauric acid significantly stimulated interleukin-6 production at 10 μg/mL compared to control (533.9 ± 218.1 versus 438.3 ± 219.6 pg/mL, P < .05). However, free β-hydroxy lauric and myristic acids were not found in patients' plasma. Therefore, free β-hydroxy lauric and myristic acids do not seem to have a role on sterile inflammation in chronic inflammatory diseases associated with insulin resistance

Nonlinear parametric models of viscoelastic fluid flows

Reduced-order models have been widely adopted in fluid mechanics, particularly in the context of Newtonian fluid flows. These models offer the ability to predict complex dynamics, such as instabilities and oscillations, at a considerably reduced computational cost. In contrast, the reduced-order modeling of non-Newtonian viscoelastic fluid flows remains relatively unexplored. This work leverages the sparse identification of nonlinear dynamics algorithm to develop interpretable reduced-order models for viscoelastic flows. In particular, we explore a benchmark oscillatory viscoelastic flow on the four-roll mill geometry using the classical Oldroyd-B fluid. This flow exemplifies many canonical challenges associated with non-Newtonian flows, including transitions, asymmetries, instabilities, and bifurcations arising from the interplay of viscous and elastic forces, all of which require expensive computations in order to resolve the fast timescales and long transients characteristic of such flows. First, we demonstrate the effectiveness of our data-driven surrogate model to predict the transient evolution and accurately reconstruct the spatial flow field for fixed flow parameters. We then develop a fully parametric, nonlinear model capable of capturing the dynamic variations as a function of the Weissenberg number. While the training data is predominantly concentrated on a limit cycle regime for moderate Wi, we show that the parameterized model can be used to extrapolate, accurately predicting the dominant dynamics in the case of high Weissenberg numbers. The proposed methodology represents an initial step in the field of reduced-order modeling for viscoelastic flows with the potential to be further refined and enhanced for the design, optimization, and control of a wide range of non-Newtonian fluid flows using machine learning and reduced-order modeling techniques

Grad-Shafranov equilibria via data-free physics informed neural networks

A large number of magnetohydrodynamic (MHD) equilibrium calculations are often required for uncertainty quantification, optimization, and real-time diagnostic information, making MHD equilibrium codes vital to the field of plasma physics. In this paper, we explore a method for solving the Grad-Shafranov equation by using Physics-Informed Neural Networks (PINNs). For PINNs, we optimize neural networks by directly minimizing the residual of the PDE as a loss function. We show that PINNs can accurately and effectively solve the Grad-Shafranov equation with several different boundary conditions. We also explore the parameter space by varying the size of the model, the learning rate, and boundary conditions to map various trade-offs such as between reconstruction error and computational speed. Additionally, we introduce a parameterized PINN framework, expanding the input space to include variables such as pressure, aspect ratio, elongation, and triangularity in order to handle a broader range of plasma scenarios within a single network. Parametrized PINNs could be used in future work to solve inverse problems such as shape optimization