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Results for “Fokker–Planck solver”

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Simulating energetic ions and enhanced fusion rates from ion-cyclotron resonance heating with a full-wave/Fokker–Planck model

Reproducing fast-ion enhanced fusion rates from ion-cyclotron resonance heating (ICRH) in tokamaks requires the self-consistent coupling of a full-wave solver and a Fokker–Planck solver, which evolves multiple simultaneously resonant ion species. We introduce a new self-consistent model that iterates the TORIC full-wave solver with the CQL3D Fokker–Planck solver using the integrated plasma simulator (IPS). This model evolves the bounce-averaged ion distribution functions in both parallel and perpendicular velocity-space with a quasilinear radio frequency (RF) diffusion operator valid in the ion finite Larmor radius (FLR) limit and the RF electric fields with the resultant non-Maxwellian FLR dielectric tensor. This produces non-Maxwellian ICRH simulations that are fully self-consistent, fast, and interoperable with integrated modeling frameworks, such as TRANSP/GACODE/IPS-FASTRAN. We demonstrate our model's capabilities by validating it against experimental data in Alcator C-Mod. We then perform the first RF heating simulations of SPARC using self-consistent non-Maxwellian ion distributions to investigate the potential to enhance fusion rates using ion cyclotron resonance heating generated fast ions.

Physics↗
DOE OSTI2026

Construction of generalized quasilinear diffusion coefficient using neural networks with physical restrictions

The quasilinear diffusion coefficient (D QL ) derived from our machine learning framework shows comparable trends with the ground truth D QL obtained from GENRAY-CQL3D simulations. Additionally, for the strong absorption cases, the radial current drive profiles generated using the D QL from our model exhibit consistent behavior with those obtained from the original simulation. These findings indicate the potential of our surrogate modeling approach with physical restrictions to replicate key wave–plasma interaction characteristics while reducing computational costs. Traditionally, calculating D QL for wave–particle interactions relies on computationally intensive wave simulations coupled with Fokker–Planck solvers. To address this challenge, we developed a machine learning-based surrogate model with physical restrictions derived from cold plasma theory and bounce-averaged damping effects. First, we establish the propagation domain of Lower Hybrid Waves in the (N∥, ρ) space by identifying the accessibility limit and determining the upper and lower bounds of N∥ using the Potential Power Deposition (PPD) method. Subsequently, leveraging a database constructed using Latin hypercube sampling alongside the underlying physical restrictions (e.g. PPD), machine learning methods including U-Net and Recurrent Neural Networks are employed to design a physics-restricted machine learning framework capable of reconstructing D QL .

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

A physics-constrained deep learning treatment of runaway electron dynamics

An adjoint formulation leveraging a physics-informed neural network (PINN) is employed to advance the density moment of a runaway electron (RE) distribution forward in time. A distinguishing feature of this approach is that once the adjoint problem is solved, its solution can be used to project the RE density forward in time for an arbitrary initial momentum space distribution of REs. Furthermore, by employing a PINN, a parametric solution to the adjoint problem can be learned. Thus, once trained, this adjoint-deep learning framework is able to efficiently project the RE density forward in time across various plasma conditions while still including a fully kinetic description of RE dynamics. As an example application, the temporal evolution of the density of primary electrons is studied, with particular emphasis on evaluating the decay of a RE population when below threshold. Predictions from the adjoint-deep learning framework are found to be in good agreement with a traditional relativistic electron Fokker–Planck solver, for several distinct initial conditions, and across an array of physics parameters. Once trained, the PINN thus provides a means of generating RE density time histories with exceptionally low online execution time.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Simulation Center for Runaway Electron Avoidance and Mitigation (SCREAM SciDAC) (Technical Final Report)

Runaway electrons can severely damage the plasma facing components on ITER during a major disruption and pose a major risk for tokamak fusion. It has been recognized that an adequate disruption mitigation system (DMS) is essential for the safe operation of ITER. The United States is responsible for the design and implementation of the disruption mitigation system on ITER, and in July 2016 the Simulation Center for Runaway Electron Avoidance and Mitigation (SCREAM) was launched by DOE, in a joint Fusion Energy Sciences (FES) and Advanced Scientific Computing Research (ASCR) collaboration. SCREAM was a comprehensive theory and simulation SciDAC center that provided physics guidance in the avoidance and mitigation of runaway electrons, and in tandem with domestic and international experiments, helped establish the qualitative and quantitative bases for safe operational scenarios and viable mitigation techniques. The SCREAM center assembled a national team of experts in runaway electron physics, tokamak disruptions, magnetohydrodynamic (MHD) simulation, and advanced algorithms and computing. The team combined advanced simulation and analysis capability facilitated by direct participation of ASCR SciDAC institutes with theoretical models and code development by FES scientists to focus on the runaway risk for ITER and tokamaks in general. The research scope was focussed on integrated simulations of kinetic runaway electrons, including MHD and fluid models of impurity transport, within a research plan guided by theory. The specific research tasks were (1) establish the fundamental physics of runaway generation, saturation, and dynamical evolution in a tokamak; (2) examine the critical path toward runaway avoidance; and (3) investigate the viability and effectiveness of the leading candidate schemes for runaway mitigation. In all three areas, members of the team carried out scoping studies that established the readiness for rapid and critical advances, especially in the deployment and further development of large-to extreme-scale simulation tools. Our multi-pronged computational approach included (1) relativistic Fokker-Planck solvers with discretization in phase space, (2) self-consistent particle-in-cell techniques, (3) particle-based Monte-Carlo, and (4) MHD-particle hybrid simulations. Cross-check between these different methods provided an additional means for verification and further bolstered the fidelity of our physics prediction. Validation against experimental results brings confidence to the predictive capability for ITER and frequently leads to new ideas for understanding and mitigating the thermal quench driven runaway electron phenomenon.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

X-ray free electron laser linear accelerator without a laser heater

Linear accelerator based X-ray free electron laser (FEL) light sources provide an important tool for scientific discoveries. Most of these light source facilities employ a laser heater to increase the electron beam’s uncorrelated energy spread to suppress microbunching instability through the linear accelerators. In this paper, we first studied the microbunching instability in an X-ray FEL linear accelerator with lower initial peak current (~ 10 A) and moderate final peak currents (1-2 kA). In this regime, the microbunching instability can be substantially mitigated with modest initial uncorrelated energy spread. Here, we then suggested a less expensive method to mitigate the microbunching instability using a section of low beta FODO lattice instead of the laser heater. With the use of the high brightness electron beam from a photoinjector, the intrabeam scattering effect inside the beam through the FODO lattice can generate sufficient uncorrelated energy spread to mitigate the microbunching instability. At last, we demonstrated the feasibility of this method with self-consistent solution of the Fokker–Planck equation through the X-ray FEL linear accelerator.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2025

Excitation of whistler and slow-X waves by runaway electrons in a collisional plasma

Runaway electrons are known to provide robust ideal or collisionless kinetic drive for plasma wave instabilities in both the whistler and slow-X branches, via the anomalous Doppler-shifted cyclotron resonances. In a cold and dense post-thermal-quench plasma, collisional damping of the plasma waves can compete with the collisionless drive. Previous studies have found that, due to their higher wavelength and frequency, slow-X waves suffer stronger collisional damping than the whistlers, while the ideal growth rate of slow-X modes is higher. Here, we study runaway avalanche distributions that maintain the same eigen distribution and increase only in magnitude over time. The distributions are computed from the relativistic Fokker–Planck–Boltzmann solver, upon which a linear dispersion analysis is performed to search for the most unstable or least damped slow-X and whistler modes. Taking into account the effect of plasma density, plasma temperature, and effective charge number, we find that the slow-X modes tend to be excited before the whistlers in a runaway current ramp-up. Furthermore, even when the runaway current density is sufficiently high that both branches are excited, the most unstable slow-X mode has a much higher growth rate than the most unstable whistler mode. The qualitative and quantitative trends uncovered in the current study indicate that even though past experiments and modeling efforts have concentrated on whistler modes, there is a compelling case that slow-X modes should also be a key area of focus in the runaway self-mediation through wave instabilities.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Scalable Implicit Solvers with Dynamic Mesh Adaptation for a Relativistic Drift-Kinetic Fokker–Planck–Boltzmann Model

In this work we consider a relativistic drift-kinetic model for runaway electrons along with a Fokker–Planck operator for small-angle Coulomb collisions, a radiation damping operator, and a secondary knock-on (Boltzmann) collision source. Here, we develop a new scalable fully implicit solver utilizing finite volume and conservative finite difference schemes and dynamic mesh adaptivity. A new data management framework in the PETSc library based on the p4est library is developed to enable simulations with dynamic adaptive mesh refinement (AMR), distributed memory parallelization, and dynamic load balancing of computational work. This framework and the runaway electron solver building on the framework are able to dynamically capture both bulk Maxwellian at the low-energy region and a runaway tail at the high-energy region. To effectively capture features via the AMR algorithm, a new AMR indicator prediction strategy is proposed that is performed alongside the implicit time evolution of the solution. This strategy is complemented by the introduction of computationally cheap feature-based AMR indicators that are analyzed theoretically. Numerical results quantify the advantages of the prediction strategy in better capturing features compared with nonpredictive strategies; and we demonstrate trade-offs regarding computational costs. The robustness with respect to model parameters, algorithmic scalability, and parallel scalability are demonstrated through several benchmark problems including manufactured solutions and solutions of different physics models. We focus on demonstrating the advantages of using implicit time stepping and AMR for runaway electron simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Feasibility of a multigroup Boltzmann–Fokker–Planck solution for electron beam dose calculations

Legacy nuclear-reactor Boltzmann solvers start clinical deployment as an alternative to Monte Carlo (MC) codes and Fermi–Eyges semiemprical models in radiation oncology treatment planning. Today’s certified clinical solvers are limited to photon beams. In this paper, ELECTR, a state-of-the-art multigroup electron cross sections generation module in NJOY is presented and validated against Lockwood’s calorimetric measurements, EGS-nrc and GEANT-4 for 1–20 MeV unidirectional electron beams. The nuclear-reactor DRAGON-5 solver is upgraded to access the library and solve the Boltzmann–Fokker–Planck (BFP) equation. A variety of heterogeneous radiotherapy and radiosurgery phantom configurations were used for validation purpose. Case studies include a thorax benchmark, that of a typical breast Intra-Operative Radiotherapy and a high-heterogeneity patient-like benchmark. For all beams, 100% of the water voxels satisfied the American Association of Physicists in Medicine accuracy criterion for a BFP-MC dose error below 2%. At least, 97.0% of adipose, muscle, bone, lung, tumor and breast voxels satisfied the 2% criterion. The average BFP-MC relative error was about 0.56% for all voxels, beams and materials combined. By irradiating homogeneous slabs from Z = 1 (hydrogen) to Z = 99 (einsteinium), we reported performance and defects of the CEPXS mode [US. Sandia National Lab., SAND-89-1685] in ELECTR for the entire periodic table. For all Lockwood’s benchmarks, NJOY-DRAGON dose predictions are within the experimental data precision for 98% of voxels.

42 ENGINEERING↗
DOE OSTI2024

A physics-informed deep learning description of Knudsen layer reactivity reduction

A physics-informed neural network (PINN) is used to evaluate the fast ion distribution in the hot spot of an inertial confinement fusion target. The use of tailored input and output layers to the neural network is shown to enable a PINN to learn the parametric solution to the Vlasov–Fokker–Planck equation in the absence of any synthetic or experimental data. As an explicit demonstration of the approach, the specific problem of Knudsen layer fusion yield reduction is treated. Here, the predictions from the Vlasov–Fokker–Planck PINN are used to provide a non-perturbative solution of the fast ion tail in the vicinity of the hot spot, thus allowing the spatial profile of the fusion reactivity to be evaluated for a range of collisionalities and hot spot conditions. Excellent agreement is found between the predictions of the Vlasov–Fokker–Planck PINN and the results from traditional numerical solvers with respect to both the energy and spatial distribution of fast ions and the fusion reactivity profile, demonstrating that the Vlasov–Fokker–Planck PINN provides an accurate and efficient means of determining the impact of Knudsen layer yield reduction across a broad range of plasma conditions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

A Physics-Informed Deep Learning Description of Knudsen Layer Reactivity Reduction

A physics-informed neural network (PINN) is used to evaluate the fast ion distribution in the hot spot of an inertial confinement fusion target. The use of tailored input and output layers to the neural network is shown to enable a PINN to learn the parametric solution to the Vlasov–Fokker–Planck equation in the absence of any synthetic or experimental data. As an explicit demonstration of the approach, the specific problem of Knudsen layer fusion yield reduction is treated. Here, the predictions from the Vlasov–Fokker–Planck PINN are used to provide a non-perturbative solution of the fast ion tail in the vicinity of the hot spot, thus allowing the spatial profile of the fusion reactivity to be evaluated for a range of collisionalities and hot spot conditions. Excellent agreement is found between the predictions of the Vlasov–Fokker–Planck PINN and the results from traditional numerical solvers with respect to both the energy and spatial distribution of fast ions and the fusion reactivity profile, demonstrating that the Vlasov–Fokker–Planck PINN provides an accurate and efficient means of determining the impact of Knudsen layer yield reduction across a broad range of plasma conditions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Final Technical Report - Center for Simulation of Fusion Relevant RF Actuators

We have developed a suite of 3D electromagnetic field solvers, both FEM and FDTD based, that account for the RF antenna and vacuum vessel geometries with unprecedented accuracy. Workflows were developed that make it possible to translate CAD models for the antenna and vacuum vessel to physics meshes for RF wave simulation. Nonlinear RF sheath formation has been incorporated self-consistently as a boundary condition in these solvers. We have also carried out extensive studies of the impact of RF sheaths on the ion energy angle distribution at plasma-material interfaces, using high fidelity particle-in-cell codes. Comprehensive simulation models were developed to assess the impact of blob-like edge turbulence on RF wave propagation and the impact of the RF ponderomotive force on the plasma scrape-off layer (SOL). A fluid transport solver for the far-SOL was also developed which accounts for the high parallel to perpendicular heat anisotropy on an unstructured mesh, thus making it possible to precisely represent an antenna structure in the presence of edge transport. Finally we have developed a hierarchy of core wave propagation and absorption models that self-consistently combine continuum Fokker Planck and Monte Carlo treatments of fast ion evolution with ICRF full-wave field solvers and continuum Fokker Planck treatments of fast electron evolution with both full-wave and ray tracing models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Fusion by beam ions in a low collisionality, high mirror ratio magnetic mirror

Abstract The classical problem of neutral beam ions slowing down in a magnetic mirror geometry is revisited to provide predictive capability for the new Wisconsin HTS Axisymmetric Mirror under development at the University of Wisconsin. A Fokker–Planck model named fast beam ion solver (FBIS) is developed to include the spatial non-uniformity of a physical mirror geometry. The mathematical framework allows for efficient orbit averaging of the pitch-angle scattering operator, and permits a determination of the axial profile of the ambipolar potential confining the electrons. The numerical results from FBIS are consistent with earlier work, but further show how mirror-ratio and a near square magnetic well optimizes the fusion gain. The numerical results are also applied to inform the conceptual design of WHAM++, a low capital-cost breakeven-class magnetic mirror device.

Egedal, J. (ORCID:0000000177531529)↗
DOE OSTI2025

Enhanced Collisional Losses from a Magnetic Mirror Using the Lenard-Bernstein Collision Operator

Collisions are crucial in governing particle and energy transport in plasmas confined in a magnetic mirror trap. Modern gyrokinetic codes model transport in magnetic mirrors, but some utilize approximate model collision operators. This study focuses on a Pastukhov-style method of images calculation of particle and energy confinement times using a Lenard-Bernstein model collision operator. Prior work on parallel particle and energy balances used a different Fokker-Planck plasma collision operator. The method must be extended in non-trivial ways to study the Lenard-Bernstein operator. To assess the effectiveness of our approach, we compare our results with a modern finite element solver. Our findings reveal that the particle confinement time scales like a exp( a 2 ) using the Lenard-Bernstein operator, in contrast to the more accurate scaling that the Coulomb collision operator would yield a 2 exp( a 2 ), where a 2 is approximately proportional to the ambipolar potential. We propose that codes solving for collisional losses in magnetic mirrors utilizing the Lenard-Bernstein or Dougherty collision operator scale their collision frequency of any electrostatically confined species. This study illuminates the collision operator’s intricate role in the Pastukhov-style method of images calculation of collisional confinement.

fusion plasma↗
DOE OSTI2026

Non-local transport in radiation-hydrodynamics codes for ICF by efficient coupling to an external Vlasov–Fokker–Planck code

Accurately incorporating non-local transport into radiation-hydrodynamics codes, and indeed any fluid system, has long been elusive. To date, a simplified and accurate theory that can be easily integrated has not been available. This limitation affects modeling in inertial confinement fusion (ICF) and magnetic confinement fusion systems, among others, where non-local transport is well-known to be present. Here, we present a coupling methodology between a full Vlasov–Fokker–Planck (VFP) electron kinetic code and radiation-hydrodynamics (rad-hydro) codes. The VFP code is used to adjust native electron transport in the rad-hydro code, thus enabling improved transport without the need to integrate a full electron VFP solver into the rad-hydro code. This approach necessitates only occasional invocation of the VFP code, reducing computational intensity compared to following the dynamic evolution entirely with the VFP code on fluid time scales. We illustrate that the methodology is more accurate than other simplified methods in thermal decay systems relevant to ICF and can replicate standard theoretical results with high accuracy.

Electronic transport↗
DOE OSTI2022

A self-consistent hybrid model of kinetic striations in low-current argon discharges

A self-consistent hybrid model of standing and moving striations was developed for low-current DC discharges in noble gases. We introduced the concept of surface diffusion in phase space ( r , u ) (where u denotes the electron kinetic energy) described by a tensor diffusion in the nonlocal Fokker–Planck kinetic equation for electrons in the collisional plasma. Electrons diffuse along surfaces of constant total energy ε = u - eφ ( r ) between energy jumps in inelastic collisions with atoms. Numerical solutions of the 1d1u kinetic equation for electrons were obtained by two methods and coupled to ion transport and Poisson solver. We studied the dynamics of striation formation in Townsend and glow discharges in argon gas at low discharge currents using a two-level excitation-ionization model and a ‘full-chemistry’ model, which includes stepwise and Penning ionization. Standing striations appeared in Townsend and glow discharges at low currents, and moving striations were obtained for the discharge currents exceeding a critical value. These waves originate at the anode and propagate towards the cathode. We have seen two types of moving striations with the two-level and full-chemistry models, which resemble the s and p striations previously observed in the experiments. Simulations indicate that processes in the anode region could control moving striations in the positive column plasma. The developed model helps clarify the nature of standing and moving striations in DC discharges of noble gases at low discharge currents and low gas pressures.

Physics↗
DOE OSTI2025

Convex relaxation for Fokker–Planck equation

We propose an approach to directly estimate the moments or marginals for a high-dimensional equilibrium distribution in statistical mechanics by solving the high-dimensional Fokker–Planck equation in terms of low-order cluster moments or marginals. With this approach, we bypass the exponential complexity of estimating the full high-dimensional distribution and directly solve the simplified partial differential equations for low-order moments/marginals. Moreover, the proposed moment/marginal relaxation is fully convex and can be solved via off-the-shelf solvers. We further propose a time-dependent version of the convex programs to study non-equilibrium dynamics. In a specific setting, we show the proposed method can recover a mean-field-type equilibrium density. Numerical results are provided to demonstrate the performance of the proposed algorithm for high-dimensional systems.

Chen, Yian↗
DOE OSTI2024

Linearised Fokker–Planck collision model for gyrokinetic simulations

We introduce a gyrokinetic, linearised Fokker–Planck collision model that satisfies conservation laws and is accurate at arbitrary collisionalities. The differential test-particle component of the operator is exact; the integral field-particle component is approximated using a spherical harmonic and a modified Laguerre polynomial expansion developed by Hirshman and Sigmar (1976 Phys. Fluids 19 1532). The numerical methods of the implementation in the δf-gyrokinetic code stella (Barnes et al 2019 J. Comput. Phys. 391 365–80) are discussed, and conservation properties of the operator are demonstrated. The collision model is then benchmarked against the collision model of the gyrokinetic solver GS2 in the limiting cases of a reduced test-particle collision operator and energy- and momentum-conserving operator. The accuracy of the full collision model is investigated by solving the parallel Spitzer-Härm problem for the transport coefficients. It is shown that retaining collisional energy flux and higher-order terms in the field-particle operator reduces errors in the transport coefficients from 10%–25% for a simple momentum- and energy-conserving model to under 1%.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker–Planck Equations

The Fokker-Planck (FP) equation is a foundational partial differential equation (PDE) in stochastic processes involving Brownian motions. However, the curse of dimensionality (CoD) poses a formidable challenge when dealing with high-dimensional FP equations. Although Monte Carlo simulation and (vanilla) Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit significant numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimensionality increases, surpassing the precision of numerical simulations and resulting in substantial errors. In addition, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a notoriously difficult Hamilton-Jacobi-Bellman (HJB) equation, which is impractical for PINN learning, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in stochastic differential equations (SDEs). The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling, offering an effective means to overcome the CoD. Three fitting methods, Score Matching (SM), Sliced Score Matching (SSM), and Score-PINN, are introduced, each contributing unique advantages in computational complexity, accuracy, and generality. The proposed score-based SDE solver operates in two stages: first, employing score matching or Score-PINN to acquire the score function; and second, solving the LL via an ordinary differential equation (ODE) using the obtained score function. Comparative evaluations across these methods showcase varying trade-offs. The proposed methodology is evaluated across diverse SDEs, including anisotropic Ornstein-Uhlenbeck processes, geometric Brownian motion, and Brownian motion with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy distributions. The numerical results demonstrate the score-based SDE solver’s stability, speed, and performance across different experimental settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

97 MATHEMATICS AND COMPUTING↗