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At least 37 records · Page 2

DeePMD-kit v2: A software package for deep potential models

DeePMD-kit is a powerful open-source software package that facilitates molecular dynamics simulations using machine learning potentials known as Deep Potential (DP) models. This package, which was released in 2017, has been widely used in the fields of physics, chemistry, biology, and material science for studying atomistic systems. The current version of DeePMD-kit offers numerous advanced features, such as DeepPot-SE, attention-based and hybrid descriptors, the ability to fit tensile properties, type embedding, model deviation, DP-range correction, DP long range, graphics processing unit support for customized operators, model compression, non-von Neumann molecular dynamics, and improved usability, including documentation, compiled binary packages, graphical user interfaces, and application programming interfaces. This article presents an overview of the current major version of the DeePMD-kit package, highlighting its features and technical details. Additionally, this article presents a comprehensive procedure for conducting molecular dynamics as a representative application, benchmarks the accuracy and efficiency of different models, and discusses ongoing developments.

97 MATHEMATICS AND COMPUTING↗

Near conformal perturbation theory in SYK type models

We present a systematic procedure to extract the dynamics of the low energy soft mode in SYK type models with a single energy scale J and emergent reparametrization symmetry in the IR. This is given in the framework of the perturbative scheme of arXiv:1608.07567 based on a specific (off-shell) breaking of conformal invariance in the UV, adjusted to yield the exact large-N saddle point. While this breaking term formally vanishes on-shell, it has a non-trivial effect on correlation functions and the effective action. In particular, it leads to the Schwarzian action with a specific coupling to bi-local matter. The method is applied to the evaluation of O(1) corrections to the correlation function of bi-locals. As a byproduct we confirm precise agreement with the explicit, symmetry breaking procedure. We provide a verification in the large q limit (Liouville theory), where the correlators can be calculated exactly at all length scales. In this case, our scheme illuminates how the enhanced O(J) and the subleading O(1) contributions originate from the Schwarzian dynamics of the soft mode and its interaction with h = 2 (bi-local) matter.

1/N expansion↗

Dynamic Species Reduction for Multi-Cycle CFD Simulations (Final Technical Report)

This project primarily sought to address some of the computational cost concerns of detailed simulations by developing improved methods of handling species transport, and chemical kinetics evaluation in a commercial 3D Computational Fluid Dynamics (CFD) environment. Secondary goals were to apply these techniques to fuels and conditions of interest to better understand the key species and reactions required to adequately capture cycle to cycle coupling. Improved modeling will lead to better understanding of these combustion modes and their dependence on fuel composition, which can then enable more clean and efficient engines, ultimately benefiting the consumer as well as the general public. Two approaches were used to address the computational cost. A “Dynamic Species Reduction” (DSR) procedure was developed to remove chemical species from the simulation during periods when chemical reactions were not expected to be important, particularly during gas exchange when temperatures are low and little fuel remains. This modelling procedure automatically detects relevant species to retain in the simulation domain based on their local concentration, removes species below a specified concentration threshold, and then adjusts the remaining species mass to conserve not only the number of H, C, and O atoms in each cell, but also the relative proportions between species and the heating value of the mixture in every cell. The second procedure was a “Product Directed Remapping” (PDR) updates the algorithm used to group individual computational cells for chemical kinetics evaluation to account for non-uniform temperature distributions and the CO to CO 2 ratio in the cells. The model techniques developed in this work successfully demonstrated computational performance improvements for a range of conditions relevant for Highly Dilute SI and HCCI engine operation. Runtime reductions of 10% were observed for small mechanisms, with further reductions of up to 36% observed for larger mechanisms. Improvements were primarily related to reducing the number of chemical species tracked during the gas exchange process using the Dynamic Species Reduction method. With smaller benefits observed from changes to the kinetics binning and evaluation strategy in the post combustion region using the PDR method. The results of this work show the potential for improved computational runtime for complicated simulations. They can and should be extended to additional conditions and new bio-derived and renewable fuels as they are developed and new kinetic mechanisms become available.

10 SYNTHETIC FUELS↗

A review of fluids under nanoconfinement: Reactivity, geomechanics, phase transitions, and flow

Due to its many industrial applications, the physics associated with the nanoconfinement of fluids has garnered great attention from the scientific community in roughly the last fifteen years. One pioneer of that research has been the Multi-Scale Fluid-Solid Interactions in Architected and Natural Materials (MUSE) Energy Frontier Research Center, which produced more than 50 journal articles from 2018 to 2023. We structure the findings of MUSE into four categories and contextualize the work with the broader literature. Here, the four categories are (1) the dependence of chemical reactions on the degree of confinement, which given the lower freedom of motion of molecules yet improved availability of reaction sites generally lead to enhancement from bulk reactions; (2) the mechanical properties of the porous matrix, which are dependent on the heterogeneity of the system in terms of both composition and structure; (3) the phase transitions under nanoconfinement, which are influenced by the pore size and morphology, surface chemistry, and substrate composition; and (4) the physical controls of slip flow, which demonstrate the importance of several factors, such as wettability. These aspects of nanoconfinement are becoming clearer, but there are other elements such as mathematical theory that are still being developed.

58 GEOSCIENCES↗

Simulation-Based Recovery Action Analysis Using the EMRALD Dynamic Risk Assessment Tool

Recovery human action is defined as the action that prevents deviant conditions from producing unwanted effects. Analyzing recovery actions has been a critical part in human reliability analysis (HRA). However, there are a couple of limitations to treating recovery actions using only the current HRA methods available. Representatively, the existing recovery analysis does not specifically consider recovery actions as they have occurred in actual nuclear power plants (NPPs). To handle the challenges in the existing recovery analyses, this study suggests a way to analyze recovery actions under a dynamic HRA method, the Procedure-based Risk Investigation MEthod-Human Reliability Analysis (PRIME-HRA) method. The PRIME-HRA method suggests a way on how to develop dynamic simulation models using dynamic risk assessment tools such as the Event Modeling Risk Assessment Using Linked Diagram (EMRALD) [1] and the Human Unimodel for Nuclear Technology to Enhance Reliability (HUNTER) [2]. EMRALD and HUNTER are the dynamic probabilistic risk assessment and HRA tools developed at Idaho National Laboratory. In this paper, differences on analyzing recovery actions in the Technique for Human Error-Rate Prediction (THERP), the Cause-Based Decision Tree (CBDT) and the Korean Standard HRA (K-HRA) and challenges of these approaches are introduced. How we have developed the PRIME-HRA is also introduced in this paper. Then, the proposed approach to analyzing recovery human actions in dynamic context is partially discussed with an example.

99 GENERAL AND MISCELLANEOUS↗

Probing Quantum Telecloning on Superconducting Quantum Processors

Quantum information cannot be perfectly cloned, but approximate copies of quantum information can be generated. Quantum telecloning combines approximate quantum cloning, more typically referred to as quantum cloning, and quantum teleportation. Quantum telecloning allows approximate copies of quantum information to be constructed by separate parties, using the classical results of a Bell measurement made on a prepared quantum telecloning state. Quantum telecloning can be implemented as a circuit on quantum computers using a classical coprocessor to compute classical feedforward instructions using if statements based on the results of a midcircuit Bell measurement in real time. We present universal symmetric optimal 1 → M telecloning circuits and experimentally demonstrate these quantum telecloning circuits for M = 2 up to M = 10, natively executed with real-time classical control systems on IBM Quantum superconducting processors, known as dynamic circuits. We perform the cloning procedure on many different message states across the Bloch sphere, on seven IBM Quantum processors, optionally using the error suppression technique X–X sequence digital dynamical decoupling. Two circuit optimizations are utilized: one that removes ancilla qubits for M = 2,3 , and one that reduces the total number of gates in the circuit but still uses ancilla qubits. Parallel single-qubit tomography with maximum likelihood estimation density matrix reconstruction is used in order to compute the mixed-state density matrices of the clone qubits, and clone quality is measured using quantum fidelity. These results present one of the largest and most comprehensive noisy intermediate-scale quantum computer experimental analyses on (single qubit) quantum telecloning to date. The clone fidelity sharply decreases to 0.5 for M > 5 , but for M = 2 , we are able to achieve a mean clone fidelity of up to 0.79 using dynamical decoupling.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Foundations of magnetohydrodynamics

In this tutorial, a derivation of magnetohydrodynamics (MHD) valid beyond the usual ideal gas approximation is presented. Non-equilibrium thermodynamics is used to obtain conservation equations and linear constitutive relations. When coupled with Maxwell's equations, this provides closed fluid equations in terms of material properties of the plasma, described by the equation of state and transport coefficients. These properties are connected to microscopic dynamics using the Irving–Kirkwood procedure and Green–Kubo relations. Symmetry arguments and the Onsager–Casimir relations allow one to vastly simplify the number of independent coefficients. Importantly, expressions for current density, heat flux, and stress (conventionally Ohm's law, Fourier's law, and Newton's law) take different forms in systems with a non-ideal equation of state. The traditional form of the MHD equations, which is usually obtained from a Chapman–Enskog solution of the Boltzmann equation, corresponds to the ideal gas limit of the general equations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Small-Signal Analysis of Current-Limiting Grid-Forming Inverters

Grid-forming (GFM) inverters are equipped with a current limiter to protect the device during grid disturbances. Because the intervention of the current limiter can compromise the transient stability of the GFM inverter, various types of current limiter designs and additional frequency-stabilization control methods have been proposed in recent literature to aid in transient GFM inverter stability; however, the small-signal stability implications of adding these additional control blocks during, in particular, off-nominal conditions are not fully understood. To address this challenge, this paper presents a generic small-signal model of a GFM inverter that incorporates various types of current limiters and frequency stability-enhancing controls. With the proposed model, we analyze the root causes of inverter instability, which reveals two critical design trade-offs. First, due to current-limiter engagement caused by a disturbance, the GFM inverter dynamics are altered, which can lead to voltage oscillations. Second, additional frequency-stabilization controls can improve the small-signal stability of the GFM inverter, but at the cost of reduced power and voltage support to the grid. Comprehensive hardware experiments validate the theoretical concept and analysis. The findings in this work underscore the importance of incorporating both transient-response and small-signal dynamics requirements into the design procedure of GFM current limiters and frequency-stabilization controls.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Extreme-scale stochastic optimization and simulation via learning-enhanced decomposition and parallelization (Final Technical Report)

Stochastic optimization and simulation models ubiquitously arise in designing and operating complex service/engineering systems. They can be extreme in scale due to high-dimensional data and decisions, and can also involve decisions made sequentially in response to newly revealed data, both causing significant computational challenge. The objective of this research is to explore a unified framework that integrates machine learning with discrete optimization and risk-averse modeling, to improve the efficiency of decomposition paradigms for stochastic optimization and simulations at extreme scale. The models we consider represent a broad class of complex decision-making problems, where 0-1 or continuous decisions are made before and/or after knowing multiple sources of uncertainties that could be correlated. We will employ machine learning methods to dynamically decide and prioritize computational procedures, including cut generation, branching, and bounding of the optimal objective. Furthermore, the research will shed new lights on the traditional decomposition algorithms for extreme-scale computing. Deliverables of the research include new modeling and computational methods for advancing the state-of-the-art research in optimization and simulation, bringing many relevant risk-averse, data-driven optimization problems in practice within the range of tractability. Examples include distributed computing server scheduling and sensor deployment for monitoring critical infrastructures. Success in this effort will enable progress in solving multiple extreme-scale problems in the complex system design and operations arising from DoE missions in energy, environment, and national security.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Automatically parallelizing batch inference on deep neural networks using Fiats and Fortran 2023 `do concurrent`

This paper introduces novel programming strategies that leverage features of the Fortran 2023 standard of the International Standards Organization (ISO) to automatically parallelize computations on deep neural networks. The paper focuses on the interplay of object-oriented, parallel, and functional programming paradigms in the Fiats deep learning library. We demonstrate how several infrequently used language features play a role in enabling efficient, parallel execution. Specifically, the ability to explicitly declare that a procedure is pure facilitates inference in the context of the language’s loop-parallelism construct `do concurrent`. Also, explicitly prohibiting the overriding of a parent type’s type-bound procedures eliminates the need for dynamic dispatch in performance-critical code. Finally, this paper uses batch inference calculations on a neural network surrogate for atmospheric aerosol dynamics to demonstrate that LLVM Flang compiler’s automatic parallelization of `do concurrent` achieves roughly the same performance and scalability as achieved by OpenMP compiler directives. We also demonstrate that double-precision inference costs 37–72% longer runtime than default-real precision with most values in the range 57-60%.

Rouson, Damian↗

Efficient Sampling Allocation Procedures for Optimal Quantile Selection

We propose a dynamic sampling allocation and selection paradigm for finding the alternative with the optimal quantile in a Bayesian framework. Myopic allocation policies (MAPs), analogous to existing methods in classic ranking and selection for selecting the alternative with the optimal mean, and computationally efficient selection policies are derived for selecting the alternative with the optimal quantile. Under certain conditions, we prove that the proposed MAPs and selection procedures are consistent, which means that the best quantile would be eventually correctly selected as the sample size goes to infinity. Numerical experiments demonstrate that the proposed schemes can significantly improve the performance.

Computer Science↗

An Approach to Dynamic Human Reliability Analysis and Its Data Collection Framework

Human reliability analysis (HRA) is a method for evaluating human errors in a variety of complex systems such as nuclear power plants, military systems, aircraft, and chemical plants. Most HRA methods currently used by regulatory institutes or utilities are called static HRA and are carried out by simple worksheets or simple calculators. To date, there are many unsolved or intrinsic challenges in static HRA. For example, existing static HRA does not realistically model and evaluate human actions as they would be performed at actual systems. There is no method with HRA to objectively estimate the time required for human actions despite being essential to HRA processes. In addition, many HRA methods still rely on a dataset generated prior to the 1980s, from unrelated industry experience or simply from expert judgment. Accordingly, this study attempted to research how to overcome the challenges of existing HRA via dynamic risk assessment (a.k.a., simulation-based or computation-based risk assessment) techniques. First, this study developed a dynamic HRA method, named as PRocedure-based Investigation Method of EMRALD Risk Assessment – HRA (PRIMERA-HRA). The PRIMERA-HRA mainly concentrates on providing HRA analysts with specific guidelines on how to reasonably model human actions, assign human reliability data and evaluate output of simulation within a dynamic probabilistic risk assessment tool, called as Event Modeling Risk Assessment using Linked Diagrams (EMRALD). Second, this study also developed a module for performance shaping factors (i.e., the key concept in HRA quantification) applicable to dynamic HRA, then implemented it based on PRIMERA-HRA within the EMRALD tool. Third, this study developed an HRA data collection framework to support dynamic HRA, called as Simplified Human Error Experimental Program (SHEEP). Originally, the SHEEP study aimed to support static HRA and its data collection, but recently extended the scope to the new technologies such as dynamic HRA or HRA for advanced reactors. SHEEP focuses on the use of data collected from simplified simulators to complement—but not replace—data collection studies using full-scope simulators and actual operators. To date, many experiments were conducted under the SHEEP framework. Multiple analyses, such as human performance analysis, human error analysis, task complexity analysis, learning effect analysis and time distribution analysis, were also carried out using the collected data. Then, based on the major insights, an approach to inferring full-scope data based on simplified simulator data was proposed. The PRIMERA-HRA and SHEEP research are expected to evaluate human actions more realistically than existing static HRA, provide an opportunity to collect more HRA data with reasonable cost and labor, then contribute to enhance the quality of HRA.

99 - GENERAL AND MISCELLANEOUS↗

An Approach to Dynamic Human Reliability Analysis and Its Data Collection Framework

Human reliability analysis (HRA) is a method for evaluating human errors in a variety of complex systems such as nuclear power plants, military systems, aircraft, and chemical plants. Most HRA methods currently used by regulatory institutes or utilities are called static HRA and are carried out by simple worksheets or simple calculators. To date, there are many unsolved or intrinsic challenges in static HRA. For example, existing static HRA does not realistically model and evaluate human actions as they would be performed at actual systems. There is no method with HRA to objectively estimate the time required for human actions despite being essential to HRA processes. In addition, many HRA methods still rely on a dataset generated prior to the 1980s, from unrelated industry experience or simply from expert judgment. Accordingly, this study attempted to research how to overcome the challenges of existing HRA via dynamic risk assessment (a.k.a., simulation-based or computation-based risk assessment) techniques. First, this study developed a dynamic HRA method, named as PRocedure-based Investigation Method of EMRALD Risk Assessment – HRA (PRIMERA-HRA). The PRIMERA-HRA mainly concentrates on providing HRA analysts with specific guidelines on how to reasonably model human actions, assign human reliability data and evaluate output of simulation within a dynamic probabilistic risk assessment tool, called as Event Modeling Risk Assessment using Linked Diagrams (EMRALD). Second, this study also developed a module for performance shaping factors (i.e., the key concept in HRA quantification) applicable to dynamic HRA, then implemented it based on PRIMERA-HRA within the EMRALD tool. Third, this study developed an HRA data collection framework to support dynamic HRA, called as Simplified Human Error Experimental Program (SHEEP). Originally, the SHEEP study aimed to support static HRA and its data collection, but recently extended the scope to the new technologies such as dynamic HRA or HRA for advanced reactors. SHEEP focuses on the use of data collected from simplified simulators to complement—but not replace—data collection studies using full-scope simulators and actual operators. To date, many experiments were conducted under the SHEEP framework. Multiple analyses, such as human performance analysis, human error analysis, task complexity analysis, learning effect analysis and time distribution analysis, were also carried out using the collected data. Then, based on the major insights, an approach to inferring full-scope data based on simplified simulator data was proposed. The PRIMERA-HRA and SHEEP research are expected to evaluate human actions more realistically than existing static HRA, provide an opportunity to collect more HRA data with reasonable cost and labor, then contribute to enhance the quality of HRA.

99 - GENERAL AND MISCELLANEOUS↗

Data-Centric Approach to Capture Non-Polynomial Nonlinear Dynamics

We propose an analytical construction of observable functions in the extended dynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method for approximating the spectral properties of the Koopman operator. The choice of observable functions is fundamental for applying EDMD to nonlinear problems arising in systems and control. Existing methods either start from a set of dictionary functions and look for the subset that best fits the underlying nonlinear dynamics or rely on machine learning algorithms to “learn” observable functions. Conversely, in this paper, we start from the dynamical system model and lift it through the Lie derivatives, rendering it into a polynomial form. This proposed transformation into a polynomial form is exact and provides an adequate set of observable functions. The strength of the proposed approach is its applicability to a broader class of nonlinear dynamical systems, particularly those with nonpolynomial functions and compositions thereof. Moreover, it retains the physical interpretability of the underlying dynamical system and can be readily integrated into existing numerical libraries. We demonstrate the proposed approach with an application to electric power systems. The modeled system consists of a single generator connected to an infinite bus, where nonlinear terms include sine and cosine functions. The results demonstrate the effectiveness of the proposed procedure in off-attractor nonlinear dynamics for estimation and prediction; the observable functions obtained from the proposed construction outperform methods that use dictionary functions comprising monomials or radial basis functions.

extended dynamic mode decomposition↗

Non-Boussinesq subgrid-scale model with dynamic tensorial coefficients

A major drawback of Boussinesq-type subgrid-scale stress models used in large-eddy simulations is the inherent assumption of alignment between large-scale strain rates and filtered subgrid-stresses. A priori analyses using direct numerical simulation (DNS) data have shown that this assumption is invalid locally as subgrid-scale stresses are poorly correlated with the large-scale strain rates [J. Bardina, J. Ferziger, and W. Reynolds, Improved subgrid-scale models for large-eddy simulation, in Proceedings of the 13th Fluid and Plasmadynamics Conference, AIAA (1980); C. Meneveau and K. Katz, Scale-invariance and turbulence models for large-eddy simulation, Ann. Rev. Fluid Mech. 32, 1 (2000)]. In the present work, a new, non-Boussinesq subgrid-scale model is presented where the model coefficients are computed dynamically. Some previous non-Boussinesq models have observed issues in providing adequate dissipation of turbulent kinetic energy [e.g., Bardina et al., Proceedings of the 13th Fluid and Plasmadynamics Conference (1980); R. A. Clark, J. Ferziger, and W.C. Reynolds. Evaluation of subgrid-scale models using an accurately simulated turbulent flow, J. Fluid Mech. 91, 1 (1979); S. Stolz and N. A. Adams, An approximate deconvolution procedure for large-eddy simulation, Phys. Fluids 11, 1699 (1999)]; however, the present model is shown to provide sufficient dissipation using dynamic coefficients. Modeled subgrid-scale Reynolds stresses satisfy the consistency requirements of the governing equations for large-eddy simulation (LES), vanish in laminar flow and at solid boundaries, and have the correct asymptotic behavior in the near-wall region of a turbulent boundary layer. The new model, referred to as the dynamic tensor-coefficient Smagorinsky model (DTCSM), has been tested in simulations of canonical flows: decaying and forced homogeneous isotropic turbulence, and wall-modeled turbulent channel flow at high Reynolds numbers. The results show favorable agreement with DNS data. It has been shown that DTCSM offers similar predictive capabilities as the dynamic Smagorinsky model for canonical flows. In order to assess the performance of DTCSM in more complex flows, wall-modeled simulations of high Reynolds number flow over a Gaussian bump (Boeing speed bump) exhibiting smooth-body flow separation are performed. Predictions of surface pressure and skin friction, compared against DNS and experimental data, show improved accuracy from DTCSM in comparison to existing static coefficient (Vreman) and dynamic Smagorinsky model. The computational cost of performing LES with this model is up to 15% higher than the dynamic Smagorinsky model.

42 ENGINEERING↗

Digitalization, Management and Synthesis of Thermal Hydraulic Legacy Data Using Dynamical System Scaling

A unique attribute of the nuclear industry is the extent by which designers, developers, operators, and regulators pay attention to demonstrating the safety case. The safety case for nuclear systems is a written, traceable expression of safety, demonstrating the system’s protective measures against uncontrolled radiological releases. The nuclear installation resiliency or safety takes overriding priority over function and performance. The preparation of the safety case heavily relies on extensive experimental campaigns that challenge the target design under a spectrum of postulated accident scenarios. For safety reasons, these experiments are typically conducted in sub-scale test apparatuses intended to reproduce the postulated accident scenarios and event sequences realistically but subject to proven similarity criteria, often referred to as scaling analyses. Since the dawn of the nuclear industry, significant resources have been committed to the construction and operation of such test facilities and a massive amount of data has been generated. Such test data have been and are still used to validate the fundamental assumptions of nuclear power plant phenomena. As we move into the digital world, capturing this multi-decades worth of information in a transparent, easily accessible, and usable manner for the public and industry stakeholders is of paramount importance. Over the last few years FPoliSolutions has been actively developing an enterprise digital data ontology platform (FPoliDON) intended to do just that. The concept was introduced in a paper at the previous NURETH edition and has been significantly enhanced since. With the vision of helping designers use the data to assess their safety case evaluation models, the platform was designed not only to ontologically organize the experimental data, but also to guide the user in the complex analytics associated with the interpretation and use of such test results. A new feature which has been recently added is the ability for the user to access a library of sophisticated mathematical procedures for performing scaling analyses. For this purpose, the Dynamical System Scaling (DSS) procedure invented by Dr. Jose’ Reyes was formulated as one of the services in the digital platform. These methods have been demonstrated to provide an excelled mathematical apparatus to identify similarity criteria or quantify distortions between measured data and the target prototypical conditions. Moreover, DSS can provide a level of synthesis across SETs and IETs which has the promise to yield efficiency in the evaluation code assessment activities. This is often a daunting and expensive task to set up evaluation models that support the safety case of current and future nuclear power plants.

Reyes, Jose↗

Variational Self-Consistent Theory for Beam-Loaded Cavities

A new variational theory is presented for beam loading in microwave cavities. The beam{field interaction is formulated as a dynamical interaction whose stationarity according to Hamilton’s principle will naturally lead to steady-state solutions that indicate how a cavity’s resonant frequency, Q and optimal coupling coefficient will detune as a result of the beam loading. A driven cavity Lagrangian is derived from first principles, including the effects of cavity wall losses, input power and beam interaction. The general formulation is applied to a typical klystron input cavity to predict the appropriate detuning parameters required to maximize the gain (or modulation depth) in the average Lorentz factor boost, (Δ γ). Numerical examples are presented, showing agreement with the general detuning trends previously observed in the literature. The developed formulation carries several advantages for beam-loaded cavity structures. It provides a self-consistent model for the dynamical (nonlinear) beam{field interaction, a procedure for maximizing gain under beam-loading conditions, and a useful set of parameters to guide cavity-shape optimization during the design of beam-loaded systems. Enhanced clarity of the physical picture underlying the problem seems to be gained using this approach, allowing straightforward inclusion or exclusion of different field configurations in the calculation and expressing the final results in terms of measurable quantities. Two field configurations are discussed for the klystron input cavity, using finite magnetic confinement or no confinement at all. Formulating the problem in a language that is directly accessible to the powerful techniques found in Hamiltonian dynamics and canonical transformations may potentially carry an additional advantage in terms of analytical computational gains, under suitable conditions

43 PARTICLE ACCELERATORS↗