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DOE OSTI2024

Prediction of pellet mass thresholds for ELM triggering in low-collisionality, ITER-like discharges

Abstract In ITER, pellets are calculated to require more than 8 times the mass than currently planned to reliably trigger edge-localized modes (ELMs). Unmitigated heat flux impulses from ELMs are intolerable in ITER at full power and current. Therefore, ITER operation relies on multiple approaches to control ELM heat fluxes. One method is pellet ELM pacing to instigate small rapid ELMs with low heat flux. Predicting the performance of pellet pacing is critical for ITER, which is expected to operate in a regime with a low-collisionality, peeling-limited pedestal. However, to trigger ELMs the local pressure increase in the expanding pellet cloud pushes the equilibrium over the ballooning stability limit. In this work, linear and nonlinear M3D-C1 simulations are used to predict pellet mass thresholds in DIII-D discharges and ITER scenarios with peeling-limited pedestals. It is found that the distance of the equilibrium’s operational point from the ballooning branch of the pedestal stability boundary strongly changes thresholds. Linear M3D-C1 simulations find a strong dependence of the pellet mass threshold on the poloidal injection location for ITER’s 15 MA, Q = 10 scenario. The required pellet mass at the planned injection locations is 8 to 17 times larger than currently considered. However, such linear simulations do not include pellet ablation physics or time evolution of density and temperature. A new scheme of 2D nonlinear simulations, coupled with linear stability analysis at various steps throughout the nonlinear time evolution, was developed to include such physics and improve on the linear results. These new nonlinear-to-linear simulations confirm previous findings. This result suggests that pellet ELM triggering in ITER could require pellets much larger than those currently planned, which makes ELM-pacing operationally challenging. On the other hand, fueling pellets injected from the high-field side will likely not unintentionally trigger ELMs in an otherwise ELM-stable plasma.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Jacobian-free Newton–Krylov method for the simulation of non-thermal plasma discharges with high-order time integration and physics-based preconditioning

A preconditioning framework for the numerical simulation of non-thermal streamer discharges is developed using the Jacobian-free Newton-Krylov (JFNK) method. A reduced plasma fluid model is considered, consisting of electrons, one positive ion, one negative ion, and the electrostatic potential. Here, the plasma kinetics model includes ionization, electron-ion recombination, electron attachment, electron detachment, and ion-ion recombination. The governing equations are made dimensionless, discretized in space with finite differences, and integrated in time with a fully implicit method based on high-order backward differentiation formulas. The preconditioning framework is based on a linearized form of the governing equations and physics-based operator splitting. The efficiency of the preconditioning strategy is assessed through two test cases: streamer propagation between parallel plates and an axisymmetric pin-to-pin discharge. The fully implicit approach overcomes traditional restrictions in the time step size due to processes such as electron drift, electron diffusion, and dielectric relaxation. Excellent performance is observed through relevant statistics of the JFNK solver, although the number of linear iterations increases for the pin-to-pin discharge when nonlinear numerical boundary conditions are imposed at the electrodes. Performance studies show scalability with O(100-1000) processors for O(10M) unknowns with ample room for optimization.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

The role of ion and electron-scale turbulence in setting heat and particle transport in the DIII-D ITER baseline scenario

In this work, the heat and particle transport in a DIII-D ITER Baseline Scenario (IBS) discharge has been investigated using both linear and nonlinear gyrokinetic simulations performed with the CGYRO code [J. Candy et al: Journal Comp. Phys. (2016)].These simulations were used to investigate the role that ion-scale ($k_θp_s$ < 1:0) and electron-scale ($k_θp_s$ > 1:0) turbulence play in determining heat and particle transport in the core of conditions believed to directly extrapolate to ITER operation. This investigation spans over nearly half of the plasma minor radius, from ρ = 0:45 - 0:85. To probe the nature of the transport and turbulence in these conditions and to validate the model against experimental fluxes, scans of $a/L_{T_i}$, $a/L_{T_e}$, and $a/L_n$ were performed at all radial locations. Long wavelength turbulence is found to be dominated by ITG modes with strong non-adiabatic electron effects and appears unable to reproduce ion and electron heat fluxes and electron particle fluxes simultaneously at most radial positions when single parameter scans are performed. To investigate the potential role of short wavelength turbulence, a series of electron-scale simulations are performed that indicate that experimentally relevant levels of electron heat flux could arise at sub-ion scales. Quantitative comparison of the simulated fluxes obtained from ion and electron-scale simulations with experimental levels along with the sensitivity of simulations results to changes in inputs is presented. Through extensive sensitivity scans and comparison with multiple transport channels, this work demonstrates a clear need for self-consistent modification of multiple inputs and suggests multi-scale interactions play a significant role at many of the radial locations studied. The results of this analysis help shape our understanding of the model fidelity needed to predict turbulence and transport reactor-relevant conditions and have important implications for the prediction of future fusion devices.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

An Incremental Gradient Method for Optimization Problems With Variational Inequality Constraints

We consider minimizing a sum of agent-specific nondifferentiable merely convex functions over the solution set of a variational inequality (VI) problem in that each agent is associated with a local monotone mapping. This problem finds an application in computation of the best equilibrium in nonlinear complementarity problems arising in transportation networks. We develop an iteratively regularized incremental gradient method where at each iteration, agents communicate over a directed cycle graph to update their solution iterates using their local information about the objective and the mapping. The proposed method is single-timescale in the sense that it does not involve any excessive hard-to-project computation per iteration. We derive nonasymptotic agent-wise convergence rates for the suboptimality of the global objective function and infeasibility of the VI constraints measured by a suitably defined dual gap function. Finally, the proposed method appears to be the first fully iterative scheme equipped with iteration complexity that can address distributed optimization problems with VI constraints over cycle graphs.

convergence↗

DOE OSTI2022

JET EROSION OF PARTICLE BEDS: PROJECTING CRITICAL SUSPENSION VELOCITIES FROM EFFECTIVE CLEARING / CLEANING RADII

Here we explore the relationship between effective cleaning/clearing radii, ECR, and critical suspension velocities, Ucs. Although one set of physics controls both the radial extent of erosion (i.e., the ECR) and the velocity needed to erode from the center of jet impingement to any given location (i.e., Ucs, typically defined from nozzle center to the vessel center), the relationship between these two remains unexplored quantitatively. Here we advance the model of Kuhn, et al., (PNNL-22816, 2013) as described by Pease, et al. (FEDSM2017-69444, 2017) to evaluate the relationship between the effective clearing/cleaning radius and the nozzle velocity on flat surfaces including flat bottomed vessels. Two governing dimensionless groups are identified. We present both a closed form analytical but transcendental solution and a non-iterative approximation modeled after the Serghides approximation of Colebrook’s nonlinear equation for both ECR versus nozzle velocity and Ucs. Comparison of the model to data from flume testing on a flat surface finds reasonable agreement.

jet, erosion, critical suspension velocity, effect↗

DOE OSTI2023

Control-oriented core-SOL-divertor model to address integrated burn and divertor control challenges in ITER

The real-time regulation of a burning plasma’s temperature and density, or burn control, will be necessary to produce high fusion power in future tokamaks like ITER. This is made more challenging due to the plasma’s nonlinear characteristics and the interdependence between the core-plasma and edge-plasma regions. For example, a raising plasma temperature leads to increasing reactivity and therefore to more alpha-particle heating, which further increases temperature. Furthermore, a raise of the fusion power increases the heat flow through the scrape-off-layer (SOL), which can compromise the integrity of the divertor without proper safeguards. For control design, a model-based approach is attractive because it can directly incorporate the nonlinear, coupled, burning-plasma dynamics into the design. To facilitate this design approach, a control-oriented core-SOL-divertor (CSD) model is presented in this work. In this CSD model, a core-plasma model captures the nonlinear dynamics of the core’s density and temperature, and a SOL-divertor model defines the plasma conditions at the separatrix and divertor including the heat load on the target plates. The core-plasma and SOL-divertor models are coupled through the exchange of various variables. In particular, the SOL-divertor model yields the separatrix temperature and the influx of recycled particles into the core-plasma. Further, these variables influence the power and particle balances captured by the core-plasma model. In return, the core-plasma model determines the intensity of the heat and particles fluxes across the separatrix, and this outflow strongly impacts the SOL-divertor model. Therefore, the power and density of the core-plasma, which can be readily modulated through external heating systems and pellet injection, can be viewed as control knobs for the SOL-divertor region in addition to the gas puffing. In simulations of the CSD model, it is demonstrated how external actuation can be utilized to meet burn control and divertor control objectives simultaneously.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

A Fast Temporal Decomposition Procedure for Long-Horizon Nonlinear Dynamic Programming

We propose a fast temporal decomposition procedure for solving long-horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit step size is accepted locally for the approximate search direction and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme (Na et al. 2022). However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, whereas we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Robust Iterative Method for Symmetric Quantum Signal Processing in All Parameter Regimes

Here, this paper addresses the problem of solving nonlinear systems in the context of symmetric quantum signal processing (QSP), a powerful technique for implementing matrix functions on quantum computers. Symmetric QSP focuses on representing target polynomials as products of matrices in SU(2) that possess symmetry properties. We present a novel Newton’s method tailored for efficiently solving the nonlinear system involved in determining the phase factors within the symmetric QSP framework. Our method demonstrates rapid and robust convergence in all parameter regimes, including the challenging scenario with ill-conditioned Jacobian matrices, using standard double precision arithmetic operations. For instance, solving symmetric QSP for a highly oscillatory target function α cos(1000x) (polynomial degree ≈ 1433) takes 6 iterations to converge to machine precision when α = 0.9, and the number of iterations only increases to 18 iterations when α = 1 – 10 -9 with a highly ill-conditioned Jacobian matrix. Leveraging the matrix product state structure of symmetric QSP, the computation of the Jacobian matrix incurs a computational cost comparable to a single function evaluation. Moreover, we introduce a reformulation of symmetric QSP using real-number arithmetics, further enhancing the method’s efficiency. Extensive numerical tests validate the effectiveness and robustness of our approach, which has been implemented in the QSPPACK software package.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

An adaptive discontinuous Petrov-Galerkin method for the Grad-Shafranov equation

In this work, we propose and develop an arbitrary-order adaptive discontinuous Petrov--Galerkin (DPG) method for the nonlinear Grad--Shafranov equation. An ultraweak formulation of the DPG scheme for the equation is given based on a minimal residual method. The DPG scheme has the advantage of providing more accurate gradients compared to conventional finite element methods, which is desired for numerical solutions to the Grad--Shafranov equation. The numerical scheme is augmented with an adaptive mesh refinement approach, and a criterion based on the residual norm in the minimal residual method is developed to achieve dynamic refinement. Nonlinear solvers for the resulting system are explored and a Picard iteration with Anderson acceleration is found to be efficient to solve the system. Finally, the proposed algorithm is implemented in parallel on MFEM using a domain-decomposition approach, and our implementation is general, supporting arbitrary order of accuracy and general meshes. Furthermore, numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithm.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

An Adaptive Newton-Based Free-Boundary Grad–Shafranov Solver

Equilibria in magnetic confinement devices result from force balancing between the Lorentz force and the plasma pressure gradient. In an axisymmetric configuration like a tokamak, such an equilibrium is described by an elliptic equation for the poloidal magnetic flux, commonly known as the Grad–Shafranov equation. It is challenging to develop a scalable and accurate free-boundary Grad–Shafranov solver, since it is a fully nonlinear optimization problem that simultaneously solves for the magnetic field coil current outside the plasma to control the plasma shape. In this work, we develop a Newton-based free-boundary Grad–Shafranov solver using adaptive finite elements and preconditioning strategies. The free-boundary interaction leads to the evaluation of a domain-dependent nonlinear form of which its contribution to the Jacobian matrix is achieved through shape calculus. The optimization problem aims to minimize the distance between the plasma boundary and specified control points while satisfying two nontrivial constraints, which correspond to the nonlinear finite element discretization of the Grad–Shafranov equation and a constraint on the total plasma current involving a nonlocal coupling term. The linear system is solved by a block factorization, and AMG is called for subblock elliptic operators. The unique contributions of this work include the treatment of a global constraint, preconditioning strategies, nonlocal reformulation, and the implementation of adaptive finite elements. Furthermore, it is found that the resulting Newton solver is robust, successfully reducing the nonlinear residual to 1e-6 and lower in a small handful of iterations while addressing the challenging case to find a Taylor state equilibrium where conventional Picard-based solvers fail to converge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

thornado+FLASH-X: A Hybrid Discontinuous Galerkin–Implicit-explicit and Finite-volume Framework for Neutrino-radiation Hydrodynamics in Core-collapse Supernovae

We present neutrino-transport algorithms implemented in the toolkit for high-order neutrino-radiation hydrodynamics (thornado) and their coupling to self-gravitating hydrodynamics within the adaptive mesh refinement–based multiphysics simulation framework FLASH-X. thornado, developed primarily for simulations of core-collapse supernovae (CCSNe), employs a spectral, six-species two-moment formulation with algebraic closure and special-relativistic observer corrections accurate to $\mathcal{O}(v/c)$, and uses discontinuous Galerkin (DG) methods for phase-space discretization combined with implicit-explicit time stepping. A key development is a nonlinear neutrino–matter coupling algorithm based on nested fixed-point iteration with Anderson acceleration, enabling fully implicit treatment of collisional processes, including energy-coupling interactions such as neutrino–electron scattering and pair production. Coupling to finite-volume (FV) hydrodynamics is achieved through a hybrid DG-FV representation of the fluid variables and operator-split evolution within FLASH-X. The implementation is verified using basic transport tests with idealized opacities and relaxation and deleptonization problems with tabulated microphysics. Spherically symmetric CCSN simulations demonstrate accuracy and robustness of the coupled scheme, including close agreement with the CCSN simulation code Chimera. An axisymmetric CCSN simulation further demonstrates the viability of DG-based neutrino transport for multidimensional supernova modeling within FLASH-X. thornado’s neutrino-transport solver is GPU-enabled using OpenMP offloading or OpenACC, and all CCSN applications included in this work use the GPU implementation. Together, these results establish a foundation for future enhancements in physics fidelity, numerical algorithms, and computational performance, for increasingly realistic large-scale CCSN simulations.

Endeve, Eirik [Oak Ridge National Laboratory (ORNL↗

DOE OSTI2023

Constraint energy minimizing generalized multiscale finite element method for multi-continuum Richards equations

In fluid flow simulation, the multi-continuum model is a useful strategy. When the heterogeneity and contrast of coefficients are high, the system becomes multiscale, and some kinds of reduced order methods are demanded. Combining these techniques with nonlinearity, we will consider in this paper a dual-continuum model which is generalized as a multi-continuum model for a coupled system of nonlinear Richards equations as unsaturated flows, in complex heterogeneous fractured porous media; and we will solve it by a novel multiscale approach utilizing the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). In particular, such a nonlinear system will be discretized in time and then linearized by Picard iteration (whose global convergence is proved theoretically). Subsequently, we tackle the resulting linearized equations by the CEM-GMsFEM and obtain proper offline multiscale basis functions to span the multiscale space (which contains the pressure solution). More specifically, we first introduce two new sources of samples, and the GMsFEM is used over each coarse block to build local auxiliary multiscale basis functions via solving local spectral problems, that are crucial for detecting high-contrast channels. Second, per oversampled coarse region, local multiscale basis functions are created through the CEM as constrainedly minimizing an energy functional. Various numerical tests for our approach reveal that the error converges with the coarse-grid size and that only few oversampling layers as well as basis functions are needed.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

The role of edge resonant magnetic perturbations in edge-localized-mode suppression and density pump-out in low-collisionality DIII-D plasmas

Two-fluid nonlinear MHD simulations using the TM1 code demonstrate that the formation of magnetic islands at the top and bottom of the H-mode pedestal, together with the strong screening of resonant fields in the gradient region of the pedestal, can account for ELM suppression and density pump-out by n = 2 resonant magnetic perturbations (RMPs) in low-collisionality DIII-D ITER similar shape (ISS) plasmas. Using experimentally relevant transport coefficients, neoclassical resistivity, electron collisionality, and RMP amplitudes, nonlinear MHD simulations reproduce the observed level of density reduction (density pump-out) in DIII-D due to the formation of narrow magnetic islands in the resistive foot of pedestal. For large amplitude RMPs ( B r / B t > 1 × 10 -4 ) simulations predict resonant field penetration and resulting in pressure reduction at the top of the pedestal, consistent with experimental observations at the onset of ELM suppression. The predicted reduction in the height and width of the pedestal by magnetic island enhanced collisional transport provides a quantitative mechanism for the stabilization of the peeling-ballooning mode (PBM). Importantly, these simulations predict strong screening of resonant fields in the steep gradient region of the pedestal due to strong E × B rotation and diamagnetic flows. Yet, if the plasma resistivity is made artificially larger (~ 10X) than neoclassical, the simulations predict magnetic stochasticity throughout the plasma edge and the collapse of the pedestal due to the reduction in the penetration threshold with increasing resistivity. A scaling relation for resonant field penetration at the pedestal top, using several hundred nonlinear simulations, reproduces the density and E × B rotation dependence of the ELM suppression threshold observed in DIII-D. Simulations using ITER model equilibria indicate that the penetration threshold at the top of the ITER pedestal will be lower than in DIII-D due to the anticipated lower perpendicular flow velocities in ITER, resulting in the weaker screening of resonant fields.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Dakota A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis: Version 6.12 Theory Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis: Version 6.12 User's Manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a user's manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis (V.6.14) (Theory Manual)

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Dakota A Multilevel Parallel Object-Oriented Framework for Design Optimization Parameter Estimation Uncertainty Quantification and Sensitivity Analysis (V.6.14) (User's Manual)

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a users manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.13 Theory Manual

97 MATHEMATICS AND COMPUTING↗