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NASA NTRS1983

Viscous, resistive MHD stability computed by spectral techniques

Expansions in Chebyshev polynomials are used to study the linear stability of one dimensional magnetohydrodynamic (MHD) quasi-equilibria, in the presence of finite resistivity and viscosity. The method is modeled on the one used by Orszag in accurate computation of solutions of the Orr-Sommerfeld equation. Two Reynolds like numbers involving Alfven speeds, length scales, kinematic viscosity, and magnetic diffusivity govern the stability boundaries, which are determined by the geometric mean of the two Reynolds like numbers. Marginal stability curves, growth rates versus Reynolds like numbers, and growth rates versus parallel wave numbers are exhibited. A numerical result which appears general is that instability was found to be associated with inflection points in the current profile, though no general analytical proof has emerged. It is possible that nonlinear subcritical three dimensional instabilities may exist, similar to those in Poiseuille and Couette flow.

Dahlburg, R. B.↗

DOE OSTI2024

Material Model Parameters Optimization in Liquid Mercury Target Dynamics Simulation With Machine Learning Surrogates

A pulsed spallation target is subjected to very short (∼0.7μs) but intense loads (23.3 kJ) from repeated proton pulses, which knock away neutrons from the mercury atoms’ nuclei for a wide range application in physics, engineering, medicine, petroleum exploration, biology, chemistry, etc. The effect of this pulsed loading on the stainless-steel target module which contains the flowing mercury target material is difficult to predict not only due to its short but intense explosive-like physical reaction, but also the nonlinear material behavior of the liquid mercury in the structure. Injecting small helium bubbles in the mercury has been an efficient method of mitigating the pressure wave at high power level stage. However, prediction of the resultant loading on the target is more difficult when helium gas is intentionally injected into the mercury. A 2-phase material model that incorporates the Rayleigh-Plesset (R-P) model is expected to address this complex multi-physics dynamics problem by including the bubble dynamics in the liquid mercury. A parameter sensitivity study was firstly employed to understand their impact on the simulation strains. The investigated parameters included E, μ, γ, σ, n, VFgas, and gas cumulative volume curve control parameters a and b. Verification and validation results from sparse polynomial expansions (SPE) method and directional Gaussian smoothing (DGS) optimization show that the surrogate model had training error of ∼7% and validation error of ∼15%, indicating that machine learning methods and surrogate models can help optimize the uncertain parameters in the complex 2-phase material model. This approach is expected to fill the knowledge gap between unknown liquid-gas mixture material model and measured vessel strain responses.

Lin, Lianshan↗

DOE OSTI2025

Study of Quadrupole Fringe Fields in the Interaction Region of the Hadron Storage Ring of the Electron Ion Collider

Fringe fields in quadrupole magnets are usually neglected in studies of beam dynamics at accelerators. However, the extreme optical parameters present in the final focus of a collider such as the Electron–Ion Collider (EIC) may give rise to effects that should not be overlooked. The calculation of quadrupole fringe fields presented in this study follows the procedure outlined in Ref. [1], specialized to the case of a straight reference orbit (i.e., with no dipole field component). A right-handed Cartesian coordinate system is employed, with the $z$-axis aligned with the quadrupole axis and $x$ and $y$ denoting the horizontal and vertical transverse coordinates, respectively. The magnetic quadrupole field gradient, $G = \partial B_y / \partial x$, transitions from its peak value inside the quadrupole—where it is nearly independent of the longitudinal coordinate $z$—to zero at some distance beyond the magnet edge. Consequently, $G$ is treated as a function of $z$. The region over which this variation occurs is defined as the quadrupole fringe field region. The study begins with the development of a description of the magnetic field in the fringe region using a power-series expansion in the transverse coordinates $x$and $y$, consistent with the longitudinally varying gradient. A model for the $z$-dependence of the gradient is then proposed and adjusted to reproduce magnetic field data obtained from three-dimensional field calculations. To evaluate the impact of the fringe fields on beam dynamics, the corresponding vector potential is derived and incorporated into a Hamiltonian formulation of particle motion. The significance of the fringe fields is quantified by calculating the amplitude-dependent tune shift from the Hamiltonian. Using linear beam optics parameters of the Hadron Storage Ring (HSR) of the EIC, the tune shift due to the fringe fields of all quadrupole magnets in the IR-6 interaction region is evaluated. Finally, the resulting tune shifts are compared with those arising from other nonlinear field components present in the HSR.

43 PARTICLE ACCELERATORS↗

DOE OSTI2020

Bi-fidelity Gradient-Based Approach for Nonlinear Well Logging Inverse Problems

Solving a non-linear inverse problem is challenging in computational science and engineering. Sampling based methods require a large number of model valuations; gradientbased methods require fewer model evaluations but only find the local minima. Multifidelity optimization combines the low fidelity model and the high-fidelity model to achieve both high accuracy and high efficiency. In this paper, we present a bi-fidelity approach to solve non-linear inverse problems. In the bi-fidelity inversion method, the low-fidelity model is used to acquire a good initial guess, and the high-fidelity model is used to locate the global minimum. Combined with a multi-start optimization scheme, the proposed approach significantly increases the possibility of finding the global minimum for nonlinear inverse problems with many local minima. The method is tested with two toy problems and then applied to an electromagnetic well logging inverse problem, which is difficult to solve using traditional gradient-based methods. The bi-fidelity method provides promising inversion results and can be easily applied to traditional gradient-based methods.

42 ENGINEERING↗

NASA NTRS1983

Viscous, resistive magnetohydrodynamic stability computed by spectral techniques

Expansions in Chebyshev polynomials are used to study the linear stability of one-dimensional magnetohydrodynamic quasi-equilibria, in the presence of finite resistivity and viscosity. The method is modeled on the one used by Orszag in accurate computation of solutions of the Orr-Sommerfeld equation. Two Reynolds-like numbers involving Alfven speeds, length scales, kinematic viscosity, and magnetic diffusivity govern the stability boundaries, which are determined by the geometric mean of the two Reynolds-like numbers. Marginal stability curves, growth rates versus Reynolds-like numbers, and growth rates versus parallel wave numbers are exhibited. A numerical result that appears general is that instability has been found to be associated with inflection points in the current profile, though no general analytical proof has emerged. It is possible that nonlinear subcritical three-dimensional instabilities may exist, similar to those in Poiseuille and Couette flow.

Dahlburg, R. B.↗

DOE OSTI2020

Uncertainty analysis of correlated parameters in automated reaction mechanism generation

Abstract Uncertainty analysis is a useful tool for inspecting and improving detailed kinetic mechanisms because it can identify the greatest sources of model output error. Owing to the very nonlinear relationship between kinetic and thermodynamic parameters and computed concentrations, model predictions can be extremely sensitive to uncertainties in some parameters while uncertainties in other parameters can be irrelevant. Error propagation becomes even more convoluted in automatically generated kinetic models, where input uncertainties are correlated through kinetic rate rules and thermodynamic group values. Local and global uncertainty analyses were implemented and used to analyze error propagation in Reaction Mechanism Generator (RMG), an open‐source software for generating kinetic models. A framework for automatically assigning parameter uncertainties to estimated thermodynamics and kinetics was created, enabling tracking of correlated uncertainties. Local first‐order uncertainty propagation was implemented using sensitivities computed natively within RMG. Global uncertainty analysis was implemented using adaptive Smolyak pseudospectral approximations as implemented in the MIT Uncertainty Quantification Library to efficiently compute and construct polynomial chaos expansions to approximate the dependence of outputs on a subset of uncertain inputs. Cantera was used as a backend for simulating the reactor system in the global analysis. Analyses were performed for a phenyldodecane pyrolysis model. Local and global methods demonstrated similar trends; however, many uncertainties were significantly overestimated by the local analysis. Both local and global analyses show that correlated uncertainties based on kinetic rate rules and thermochemical groups drastically reduce a model's degrees of freedom and have a large impact on the determination of the most influential input parameters. These results highlight the necessity of incorporating uncertainty analysis in the mechanism generation workflow.

Gao, Connie W.↗

DOE OSTI2026

Taylor approximation variance reduction for approximation errors in PDE-constrained Bayesian inverse problems

In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e. the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e. before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.

Bayesian approximation error↗

DOE OSTI2021

Emission of electromagnetic waves as a stopping mechanism for nonlinear collisionless ionization waves in a high-β regime

A high energy density plasma embedded in a neutral gas is able to launch an outward-propagating nonlinear electrostatic ionization wave that traps energetic electrons. The trapping maintains a strong sheath electric field, enabling rapid and long-lasting wave propagation aided by field ionization. Here, using 1D3V kinetic simulations, we examine the propagation of the ionization wave in the presence of a transverse MG-level magnetic field with the objective to identify qualitative changes in a regime where the initial thermal pressure of the plasma exceeds the pressure of the magnetic field (Beta > 1). Our key finding is that the magnetic field stops the propagation by causing the energetic electrons sustaining the wave to lose their energy by emitting an electromagnetic wave. The emission is accompanied by the magnetic field expulsion from the plasma and an increased electron loss from the trapping wave structure. The described effect provides a mechanism mitigating rapid plasma expansion for those applications that involve an embedded plasma, such as high-flux neutron production from laser-irradiated deuterium gas jets.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1993

Spatiotemporal behavior and nonlinear dynamics in a phase conjugate resonator

The work described can be divided into two parts. The first part is an investigation of the transient behavior and stability property of a phase conjugate resonator (PCR) below threshold. The second part is an experimental and theoretical study of the PCR's spatiotemporal dynamics above threshold. The time-dependent coupled wave equations for four-wave mixing (FWM) in a photorefractive crystal, with two distinct interaction regions caused by feedback from an ordinary mirror, was used to model the transient dynamics of a PCR below threshold. The conditions for self-oscillation were determined and the solutions were used to define the PCR's transfer function and analyze its stability. Experimental results for the buildup and decay times confirmed qualitatively the predicted behavior. Experiments were carried out above threshold to study the spatiotemporal dynamics of the PCR as a function of Pragg detuning and the resonator's Fresnel number. The existence of optical vortices in the wavefront were identified by optical interferometry. It was possible to describe the transverse dynamics and the spatiotemporal instabilities by modeling the three-dimensional-coupled wave equations in photorefractive FWM using a truncated modal expansion approach.

Liu, Siuying Raymond↗

NASA NTRS1993

Influence of transverse-shear and large-deformation effects on the low-speed impact response of laminated composite plates

An analytical procedure is presented for determining the transient response of simply supported, rectangular laminated composite plates subjected to impact loads from airgun-propelled or dropped-weight impactors. A first-order shear-deformation theory is included in the analysis to represent properly any local short-wave-length transient bending response. The impact force is modeled as a locally distributed load with a cosine-cosine distribution. A double Fourier series expansion and the Timoshenko small-increment method are used to determine the contact force, out-of-plane deflections, and in-plane strains and stresses at any plate location due to an impact force at any plate location. The results of experimental and analytical studies are compared for quasi-isotropic laminates. The results indicate that using the appropriate local force distribution for the locally loaded area and including transverse-shear-deformation effects in the laminated plate response analysis are important. The applicability of the present analytical procedure based on small deformation theory is investigated by comparing analytical and experimental results for combinations of quasi-isotropic laminate thicknesses and impact energy levels. The results of this study indicate that large-deformation effects influence the response of both 24- and 32-ply laminated plates, and that a geometrically nonlinear analysis is required for predicting the response accurately.

Ambur, Damodar R.↗

DOE OSTI2025

Autonomous Nanoparticle Synthesis Guided by In Situ Multiscale Structural Characterization

Autonomous synthesis platforms promise rapid exploration of vast parameter spaces; yet, integrating in situ structural characterization in closed-loop synthesis optimization remains challenging. We demonstrate a realization of such a closed-loop platform coupled with a droplet-flow microreactor, in situ X-ray scattering methods (SAXS/WAXS), and Gaussian process optimization to synthesize citrate-reduced Au nanoparticles with targeted characteristics. The system efficiently explored ∼19,000 synthesis recipes through 365 experiments, achieving precise control over size (4–60 nm) and polydispersity (σ < 0.11) across large citrate/gold ratios, exceeding traditional synthesis boundaries (1–10). Beyond confirming classical Turkevich–Frens trends, partial-dependence analysis revealed strong nonlinear coupling among precursor, citrate, and pH effects. Combining quantitative SAXS/WAXS analysis with electron microscopy characterization, we uncovered that crystallite size (d c ) and particle size (d) follow d c = 0.18d + β, where synthesis chemistry controls the intercept β while maintaining a universal slope. This parallel-band structure enables independent tuning of crystallite domain size at fixed particle diameter through a combination of chloride, gold precursor, citrate, and pH contributions (cross-validated Spearman ρ = 0.7 ± 0.1). High-resolution electron microscopy shows multiple lattice-fringe orientations within single particles, directly confirming polycrystalline domains and the ability to tune d c at the fixed d. The platform’s validation includes indistinguishable static versus flowing measurements, stable droplet transport at 100 °C, and <5% run-to-run variation, establishing a robust framework for mapping and controlling multiscale nanoparticle structure across expansive chemical spaces. In conclusion, the developed closed-loop platform can be applied to a borad range of nanosyntheis processes.

77 NANOSCIENCE AND NANOTECHNOLOGY↗

DOE OSTI2026

Gradient-informed Hamiltonian Monte Carlo for multicomponent CALPHAD model optimization and uncertainty quantification

CALPHAD model parameter optimization is inherently challenging due to non-smooth objective functions, high-dimensional parameter spaces, and the need for uncertainty quantification (UQ). Traditional weighted nonlinear least squares approaches are computationally efficient but local, whereas black-box global optimizers and ensemble Markov Chain Monte Carlo (MCMC) methods provide broader exploration at substantial computational cost. The objective of this work is to combine the global exploration capability of gradient-informed Hamiltonian Monte Carlo – specifically the No-U-Turn Sampler (NUTS) – with local deterministic refinement using BFGS to efficiently optimize multicomponent CALPHAD models with minimal manual intervention. Analytic gradients are computed via the Jansson derivative framework. The methodology is demonstrated on the Cr—Fe binary system and extended to the Cr—Fe—Ni ternary system with 32 degrees of freedom. For Cr—Fe, NUTS achieves comparable or superior optimality relative to ensemble MCMC while requiring over an order-of-magnitude fewer likelihood evaluations. Parameter uncertainties are quantified through NUTS sampling and propagated to thermodynamic observables using local expansion, demonstrating a novel modular approach that combines binary and ternary parameter subsets without requiring global relaxation. These results establish gradient-informed exploration as a scalable strategy for multicomponent CALPHAD optimization and provide a practical route towards efficient higher-order database development with quantified uncertainty.

36 MATERIALS SCIENCE↗

NASA NTRS1982

Wave-driven winds from cool stars. I - Some effects of magnetic field geometry

The wave-driven wind theory of Hartmann and MacGregor (1980) is extended to include effects due to non-radial divergence of the flow. Specifically, isothermal expansion within a flow tube whose cross-sectional area increases outward faster than the square of the radius near the stellar surface is considered. It is found that the qualitative conclusions of Hartmann and MacGregor concerning the physical properties of Alfven wave-driven winds are largely unaffected. In particular, mass fluxes of similar magnitude are obtained, and wave dissipation is still necessary to produce acceptably small terminal velocities. Increasingly divergent flow geometries generally lead to higher initial wind speeds and slightly lower terminal velocities. For some cases of extremely rapid flow tube divergence, steady supersonic wind solutions which extend to infinity with vanishing gas pressure cannot be obtained. In addition, departures from spherical symmetry can cause the relative Alfven wave amplitude delta-B/B to become approximately greater than 1 within several stellar radii of the base of the wind, suggesting that nonlinear processes may contribute to the wave dissipation required by the theory.

Hartmann, L.↗

DOE OSTI2020

Improved single-particle phase-space distributions for viscous fluid dynamic models of relativistic heavy ion collisions

Comparison of hydrodynamic calculations with experimental data inevitably requires a model for converting the fluid to particles. For shear viscous fluids, several different conversion models have been proposed, such as the quadratic Grad corrections, the Strickland-Romatschke (SR) ansatz, self-consistent shear corrections from linearized kinetic theory, or corrections from the relaxation time approach. We show, via comparison to nonlinear 2 → 2 kinetic theory evolution, that even for low specific shear viscosities 0.03 ≲ $\textit{η/s}$ ≲ 0.2 , these four models have varying accuracy and still significant errors in reproducing particle phase space densities from hydrodynamic fields. Moreover, we demonstrate that the overall reconstruction error of additive shear viscous $f = f_{eq} + δf$ models dramatically reduces by up to one order of magnitude, if one ensures through exponentiation that $\textit{f}$ is always positive. Further, we also illustrate how even more accurate viscous $\textit{δf}$ models can be constructed by incorporating the first time derivatives of hydrodynamic fields to account for the past evolution of the system. Such time derivatives are readily available in hydrodynamic simulations, though usually not included in the output. Though our comparisons are limited to a one-component massless system undergoing a (0 + 1)-dimensional (0 + 1D) longitudinal boost-invariant expansion, we expect that the improved models will be useful in 2 + 1D and 3 + 1D hydrodynamic studies as well.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2024

Jets from shocked metal surfaces with grooves: Missing experiments

Many studies have investigated the mass outflows generated when a planar shock transits an imperfect (“defected”) metal surface, where the defects are symmetric triangular or sinusoidal grooves. Yet a fundamental question remains unanswered: how does the quantity of outflow mass and its maximum velocity vary as a function of the groove cross-sectional aspect ratio? We identify two sets of missing experiments that must be addressed to answer the question. The aspect ratio (groove depth over width) is equivalently represented by θ, the cross-sectional half angle, or by η 0 k, the amplitude multiplied by an effective wavenumber. Low θ (high η 0 k) grooves comprise the first set of missing experiments, which are necessary to determine the validity of theoretical predictions of the nonlinear regime (η 0 k ≥ 1, θ < 57.5°). The second set of missing experiments are those in which the volume of the groove (or equivalently, the axial cross-sectional area) has been held constant as θ or η 0 k are varied. Such experiments are necessary to independently measure the effects of variations in groove volume and groove aspect ratio on the resulting jets.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS2017

Fully Coupled Aero-Thermochemical-Elastic Simulations of an Eroding Graphite Nozzle

A multiphysics simulation capability has been developed that incorporates mutual interactions between aerodynamics, structural response from aero/thermal loading, ablation/pyrolysis, heating, and surface-to-surface radiation to perform high-fidelity, fully coupled aerothermoelastic ablation simulations, which to date had been unattainable. The multiphysics framework couples CHAR (a 3-D implicit charring ablator solver), Loci/CHEM (a computational fluid dynamics solver for high-speed chemically reacting flows), and Abaqus (a nonlinear structural dynamics solver) to create a fully coupled aerothermoelastic charring ablative solver. The solvers are tightly coupled in a fully integrated fashion to resolve the effects of the ablation pyrolysis and charring process and chemistry products upon the flow field, the changes in surface geometry due to recession upon the flow field, and thermal-structural analysis of the body from the induced aerodynamic heating from the flow field. The multiphysics framework was successfully demonstrated on a solid rocket motor graphite nozzle erosion application. Comparisons were made with available experimental data that measured the throat erosion during the motor firing. The erosion data is well characterized, as the test rig was equipped with a windowed nozzle section for real-time X-ray radiography diagnostics of the instantaneous throat variations for deducing the instantaneous erosion rates. The nozzle initially undergoes a nozzle contraction due to thermal expansion before ablation effects are able to widen the throat. A series of parameters studies were conducted using the coupled simulation capability to determine the sensitivity of the nozzle erosion to different parameters. The parameter studies included the shape of the nozzle throat (flat versus rounded), the material properties, the effect of the choice of turbulence model, and the inclusion or exclusion of the mechanical thermal expansion. Overall, the predicted results match the experiment very well, and the predictions were able to bound the data within acceptable limits.

Blades, E. L.↗

DOE OSTI2023

Intricate structure of the plasma Rayleigh–Taylor instability in shock tubes

Spikes and bubbles grow on unstable interfaces that are accelerated in high-energy-density conditions. If a shock propagates ahead of the interface, the plasma can be heated to extreme conditions where conduction and radiation fluxes influence the hydrodynamics. For example, a National Ignition Facility experiment found reduced single-mode nonlinear mixed-width growth in conditions scaled from a supernova explosion. We present high-resolution two-dimensional radiation hydrodynamic simulations with the Flash code that quantitatively reproduce the experiment. Radiative fluxes are primarily responsible for ablating the spike and removing the mushroom caps. The ablated plasma increases the mixed mass and forms a low-density halo with spikes forming in both directions. This is considerably more complex than classical instability. The halo is sensitive to ablative physics, so radiographing it may aid in the verification of energy transport modeling. Although ablation changes the spike shape, it has little effect on the overall mixed width for these parameters. This is because ablation enhances the bubble velocity but it has the opposite effect on the spike. The radiation transport instead suppresses the growth via increasing the shocked foam density, thus decreasing the Atwood number. Furthermore, a terminal velocity model including the rarefaction expansion agrees with the experimental mixed-width growth.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS2002

Observer-Based Magnetic Bearing Controller Developed for Aerospace Flywheels

A prototype of a versatile, observer-based magnetic bearing controller for aerospace flywheels was successfully developed and demonstrated on a magnetic bearing test rig (see the photograph) and an actual flywheel module. The objective of this development included a fast, yet low risk, control development process, and a robust, high-performance controller for a large variety of flywheels. This required a good system model, an efficient development procedure, and a model-based controller that addressed the key problems associated with flywheel and bearing imbalance, sensor error, and vibration. The model used in this control development and tuning procedure included the flexible rotor dynamics and motor-induced vibrations. Such a model was essential for low-risk scheduling of speed-dependent control parameters and for reliable evaluation of novel control strategies. The successfully tested control prototype utilized an extended Kalman filter to estimate the true rotor principal-axis motion from the raw sensor position feedback. For control refinement, the extended Kalman filter also estimated and eliminated the combined effects of mass-imbalance and sensor runouts from the input data. A key advantage of the design based on the extended Kalman filter is its ability to accurately estimate both the rotor's principal-axis position and gyroscopic rates with the least amount of phase lag. This is important for control parameter scheduling to dampen the gyroscopic motions. Because of large uncertainties in the magnetic bearing and imbalance characteristics, this state-estimation scheme alone is insufficient for containing the rotor motion within the desired 1-mil excursion radius. A nonlinear gain adjustment based on an estimation of the principal-axis orbit size was needed to provide a coarse (nonoptimal), but robust, control of the orbit growth. Control current minimization was achieved with a (steepest gradient) search of synchronous errors in the principal-axis position input data. Actual flywheel tests of this observer-based controller (developed entirely in-house) at the NASA Glenn Research Center showed that the model correctly predicted the rotor orbit growth as a function of rotational speed, and it demonstrated the capability of gain adjustments to arrest this growth. Data from these tests on an actual flywheel module spun to 26,000 rpm proved that the controller was able to contain the shaft motion to within much less than 0.5 mils of radial excursion with axis currents less than 300 mA in root-mean-square estimate. The test speed range was limited because of thermal expansion concerns for this particular flywheel unit, not because of any deficiency in the controller. Simulations for this unit indicated that the controller should be robust up to its top operating speed of 60,000 rpm. Aside from these important achievements, and most significantly, it took less than 1 week to adapt this controller from the simple test rig to the actual flywheel and to demonstrate full five-axis levitation and control. This demonstration showed that both the controller and the model-based development and tuning framework are easily adaptable to a wide range of rotors and bearing configurations and, hence, are capable of reducing design risks and costs for many future flywheel technology developments.

Le, Dzu K.↗